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![Chapter 0
Introduction
Exotic semirings such as (R ∪ {−∞}, max, +) and (R ∪ {+∞}, min, +) have been studied at
length since the 1950s, beginning primarily in the area of operational research. Nowadays the
term ‘tropical mathematics’ is often used to describe their study, though this term originally
referred to one particular discrete version of the max-plus algebra introduced by I. Simon
in 1988 [15]. Their applications span a wide range of fields including optimisation & control,
mathematical physics, algebraic geometry, dynamic programming and mathematical biology [10,
15]. In particular, the study of such algebras in relation to discrete event system theory (both
deterministic and stochastic), graph theory, Markov decision processes, asymptotic analysis and
language theory has lead to some significant progress in these areas over the last 30 years [8].
Many of the concepts developed in conventional linear algebra have been ‘translated’ into the
world of max-plus, including solutions to linear and non-linear systems (both analytical and
numerical), linear dependence and independence, determinants, eigenvalues and eigenvectors
[9]. In 1979 Cuninghame-Green authored the first comprehensive unified account of these results
entitled “Minimax Algebra” [7], building on many papers published over the preceding 20 years
from various disciplines within mathematics, economics and computer science. As recently
as 2006, Heidergott, Olsder and Woude published what they consider the first ‘textbook’ in
the area of max-plus algebra [13], and many of the ideas explored below can be found in this
publication.
In the first chapter of this thesis, we aim to give an overview of max-plus linear algebra and
to build the necessary groundwork required for the applications discussed in the chapters that
follow. In particular, we present two celebrated theorems in the area of max-plus theory. The
first, which can be found in [7], concerns spectral theory and says that under mild conditions,
a matrix over the max-plus algebra has a unique eigenvalue with a simple graph-theoretic
interpretation. The second, originally proved by M. Viot in 1983 [2, 6], relates to the asymptotic
behaviour of sequential powers of max-plus matrices, which turns out to be essentially periodic
and has great implications for the material explored in Chapters 2 & 3.
In chapter 2 we introduce the concept of timed Petri nets & event graphs. For a thorough
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![discussion on the scope of their application readers are referred to [18]; in this thesis we fo-
cus solely on their use in the modelling of the time behaviour of a class of dynamic systems
known as ‘discrete event dynamic systems’. In simple terms, these are systems in which a finite
number of resources (e.g. processors or machines) are shared by several users (e.g. packets or
manufactured objects) which all contribute to the achievement of some common goal (e.g. a
parallel computation or the assembly of a product) [2]. We will see that under certain conditions
these systems, while highly non-linear in the conventional sense, can be ‘linearised’ by using
the max-plus algebra. This observation, first made in [5], is of vital importance and constitutes
one of the main reasons for the continued study of max-plus algebra today. The main content
of Chapter 2 concerns the ‘basic autonomous equation’ which governs the time evolution of
discrete event systems, and the steps towards its solution. We are then able to apply some ideas
from Chapter 1 to explore the long-term behaviour of such systems.
Chapter 3 concerns stochastic event graphs, which can be thought of as a natural extension
to the concepts introduced in Chapter 2. As the name suggests, we now assume a degree of
randomness in the event timings of the systems we are trying to model. Amongst other things,
stochastic event graphs can be used to model many types of queuing systems [3], the most
simple of which being the G/G/1 queue. We introduce several key ‘first order’ theorems which
establish the nature of stationary regimes in terms of the inverse throughput, and explore the
conditions under which such regimes are reached. We end by presenting a ‘second order’ theorem
concerning the stability of inter-event timings (for example, waiting times) in the context of
queuing systems.
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![(a) εR × a = a × εR = εR
Note that the final axiom is not required in the definition of a standard ring since it follows
from the others, but it is needed here.
As the title of this section suggests, the max-plus algebra is a semiring with additive identity
ε := −∞ and multiplicative identity e := 0. It is straightforward to verify that all the axioms
of Definition 1.1 hold in the case of (Rmax, ⊕, ⊗). For example, the first distributive law holds
since
a ⊗ (b ⊕ c) = a + max{b, c}
= max{a + b, a + c}
= (a ⊗ b) ⊕ (a ⊗ c)
and the others follow similarly. For the sake of simplicity we will write Rmax for (Rmax, ⊕, ⊗)
when the context is clear.
Below we list three additional algebraic properties of Rmax which do not form part of the
definition of a semiring:
(i) Commutativity of ⊗:
∀a, b ∈ Rmax : a ⊗ b = b ⊗ a
(ii) Existence of multiplicative inverses:
∀a ∈ Rmax{ε} ∃ b ∈ Rmax such that a ⊗ b = e
(iii) Idempotency of ⊕:
∀a ∈ Rmax : a ⊕ a = a
The first two properties follow directly from the fact that (R, +) forms an abelian group, and
the third property is easily proved: a ⊕ a = max{a, a} = a. Properties (i) and (ii) mean that
we could refer to (Rmax, ⊕, ⊗) as a semifield (i.e. a field without additive inverses), though
this term can be ambiguous and is seldom used in mathematical literature. Note also that in
general, any semiring in which addition is idempotent we call an idempotent semiring. The term
dioid (originating from the phrase double monoid) was introduced by Baccelli et al. in 1992 to
mean idempotent semiring [2], but we do not use this word here.
The crucial difference between a semiring and a ring in general is that an element of the former
need not have an additive inverse. Note that this does not say that additive inverses can never
exist - there may be a non-empty subset of R containing elements which do have additive
inverses (which could be thought of as the additive analogue to the set of units in a standard
ring). However, the following lemma immediately tells us that no elements of Rmax (apart from
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![way:
a⊗ n
m :=
n
m
× a
which is well-defined, assuming m = ε.
Next, we can equip the max-plus algebra with a natural order relation as follows:
Definition 1.3. For a, b ∈ Rmax, we say a ≤ b if a ⊕ b = b.
It is easily verified that the max-plus operations ⊕ and ⊗ preserve this order, i.e. ∀a, b, c ∈ Rmax,
a ≤ b ⇒ a ⊕ c ≤ b ⊕ c and a ⊗ c ≤ b ⊗ c.
Finally, infinite sums in max-plus are defined by i∈I xi := sup{xi : i ∈ I} for any possibly
infinite (even uncountable) family {xi}i∈I of elements of Rmax, when the supremum exists. In
general, we say that an idempotent semiring is complete if any such family has a supremum,
and if the product distributes over infinite sums. The max-plus semiring Rmax is not complete
(a complete idempotent semiring must have a maximal element), but it can be embedded in
the complete semiring (Rmax, ⊕, ⊗), where Rmax := Rmax ∪ {+∞}.
1.2 Vectors and Matrices over Rmax
1.2.1 Definitions and Structure
Let n, m ∈ N. We denote the set of n × m matrices over Rmax by Rn×m
max . For i ∈ {1, . . . , n},
j ∈ {1, . . . , m}, the element of a matrix A ∈ Rn×m
max in row i and column j is denoted by [A]ij,
or simply aij for notational convenience. Thus A ∈ Rn×m
max can be written as
a11 a12 · · · a1m
a21 a22 · · · a2m
...
...
...
...
an1 an2 · · · anm
where a11, . . . , anm ∈ Rmax. In a similar vein, the elements of Rn
max := Rn×1
max are called max-plus
vectors, and we write the i-th element of a vector x ∈ Rn
max as [x]i, or simply xi.
Typical concepts and operations from conventional algebra are defined for max-plus matrices
in the usual way (replacing + and × with ⊕ and ⊗ respectively), as outlined in the following
definitions.
Definition 1.4. The n × n max-plus identity matrix, denoted En, is defined by
[En]ij =
0 i = j
ε i = j
We will write E := En whenever the context is clear.
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![Definitions 1.5. (i) For A, B ∈ Rn×m
max , their sum A ⊕ B is defined by
[A ⊕ B]ij = aij ⊕ bij = max aij, bij
(ii) For A ∈ Rn×k
max and B ∈ Rk×m
max , their product A ⊗ B is defined by
[A ⊗ B]il =
k
j=1
(aij ⊗ bjl) = max
j=1,...,k
(aij + bjl)
(iii) The transpose of a matrix A ∈ Rn×m
max is denoted by A and is defined as usual by
[A ]ij = [A]ji
(iv) For A ∈ Rn×n
max and k ∈ N, the k-th power of A, denoted A⊗k, is defined by
A⊗k
= A ⊗ A ⊗ · · · ⊗ A
k times
For k = 0, A⊗0 := En.
(v) For A ∈ Rn×m
max and α ∈ Rmax, α ⊗ A is defined by
[α ⊗ A]ij = α ⊗ [A]ij
We now look at a crucial result concerning the algebraic structure of square matrices over Rmax.
Proposition 1.6. (Rn×n
max , ⊕, ⊗) is an idempotent semiring with multiplicative identity En.
Proof. The axioms of Definition 1.1 all follow from the semiring structure of Rmax, and are
readily verified. For example, for A, B, C ∈ Rn×n
max we have that
[A ⊗ (B ⊕ C)]il =
n
j=1
(aij ⊗ (bjl ⊕ cjl))
=
n
j=1
(aij ⊗ bjl) ⊕ (aij ⊗ cjl)
=
n
j=1
(aij ⊗ bjl) ⊕
n
j=1
(aij ⊗ cjl)
= [(A ⊗ B) ⊕ (A ⊗ C)]il
and so A ⊗ (B ⊕ C) = (A ⊗ B) ⊕ (A ⊗ C). The other axioms follow similarly.
Note that since addition in (Rn×n
max , ⊕, ⊗) is idempotent, we can apply Lemma 1.2 once again to
see that no element of Rn×n
max has an additive inverse. However, unlike in Rmax, multiplication
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![of matrices over Rmax is not commutative. For example
1 e
ε −2
2 −1
3 ε
=
3 e
1 ε
=
3 2
4 3
=
2 −1
3 ε
1 e
ε −2
Also unlike Rmax, matrices over Rmax do not necessarily have multiplicative inverses (i.e. they
are not necessarily invertible). We explore this in the next section.
1.2.2 Matrix Inversion
Definition 1.7. Let A, B ∈ Rn×n
max . B is a right inverse of A if A ⊗ B = E, and B is a left
inverse of A if B ⊗ A = E.
Definition 1.8. A max-plus permutation matrix is a matrix A ∈ Rn×n
max with each row and
each column containing exactly one entry equal to e, with all other entries equal to ε. If σ :
{1, . . . , n} → {1, . . . , n} is a permutation, the max plus permutation matrix Pσ is defined by
[Pσ]ij :=
e i = σ(j)
ε i = σ(j)
As the name suggests, left multiplication by Pσ permutes the rows of a matrix: the i-th row of
a matrix A ∈ Rn×n
max will appear as the σ(i)-th row of Pσ ⊗ A. For example, if n = 2 and σ is
defined by σ(1) = 2, σ(2) = 1:
ε e
e ε
1 2
3 4
=
3 4
1 2
Similarly, it is straightforward to see that right multiplication by Pσ permutes the columns of
a matrix.
Definition 1.9. A matrix A ∈ Rn×n
max is diagonal if [A]ij = ε for all i = j. If a1, . . . , an ∈
Rmax{ε}, the diagonal matrix D(a1, . . . , an) is defined by
[D(a1, . . . , an)]ij :=
ai i = j
ε i = j
Combining these two definitions, if σ is a permutation and a1, . . . , an ∈ Rmax {ε}, Pσ ⊗
D(a1, . . . , an) gives a matrix in which each row and each column contains exactly one finite
entry. This class of matrices (sometimes referred to as generalised permutation matrices) in
max-plus turns out to be of some significance, as the theorem below shows.
Theorem 1.10. A matrix A ∈ Rn×n
max has a right inverse if and only if A = Pσ ⊗ D(a1, . . . , an)
for some permutation σ and a1, . . . , an ∈ Rmax{ε}.
Proof. Suppose A = Pσ ⊗ D(a1, . . . , an) for some permutation σ and a1, . . . , an ∈ Rmax{ε}.
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![Recalling from Section 1.1.1 that multiplicative inverses exist in Rmax, define B ∈ Rn×n
max by
[B]ij =
[A]⊗−1
ji if [A]ji = ε
ε otherwise
Then for i, j = 1, . . . , n we have that
[A ⊗ B]ij = max
k=1,...,n
aik ⊗ bkj
=
e j = i
ε j = i
Since if j = i, at least one of aik, bkj is equal to ε for each k = 1, . . . , n (since A only has one
finite element per column and row). Thus A ⊗ B = E, and B is a right inverse of A.
Conversely, suppose A has inverse B ∈ Rn×n
max . For i, j = 1, . . . , n we have
n
k=1
[A]ik ⊗ [B]kj = [E]ij
and therefore for each i = 1, . . . , n there is a (least) index c(i) (1 ≤ c(i) ≤ n) such that [A]ic(i)
and [B]c(i)i are both finite, since [E]ii = e. Moreover we cannot have [A]hc(i) finite with h = i,
since then
[A ⊗ B]hi ≥ [A]hc(i) ⊗ [B]c(i)i > ε = [E]hi
which contradicts our assumption that B is a right inverse of A. It follows that the mapping i →
c(i) is a bijection, i.e. each column of A is labelled c(i) for some i and contains exactly one finite
element, and each row of A contains exactly one finite element. That is, A = Pσ ⊗D(a1, . . . , an)
for some permutation σ and a1, . . . , an ∈ Rmax{ε}.
Theorem 1.11. For A, B ∈ Rn×n
max , A ⊗ B = E if and only if B ⊗ A = E (i.e. right and left
inverses are equivalent), and A uniquely determines B.
Proof. Suppose that A has right inverse BR ∈ Rn×n
max . Then by Theorem 1.10, we know that
A = Pσ ⊗ D(a1, . . . , an) for some permutation σ and a1, . . . , an ∈ Rmax{ε}. Now, as before,
define BL ∈ Rn×n
max by
[BL]ij =
[A]⊗−1
ji if [A]ji = ε
ε otherwise
and using the same reasoning as before we observe that BL is a left inverse of A. Finally, note
that
BR = E ⊗ BR = (BL ∗ A) ⊗ BR = BL ⊗ (A ⊗ BR) = BL ⊗ E = BL
showing that BR is uniquely determined, and is also a left inverse.
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![Note that, crudely put, the permanent is the max-plus analogue of the determinant with the
minuses simply removed. We can understand the formula to give the maximal sum of the
diagonal values for all permutations of the columns of A. The permanent has been studied at
length both in the case of conventional algebra (see [17]) and in max-plus & related semirings
(see [19]).
Note that if A ∈ Rn×n
max is invertible then by Theorem 1.10, A = Pσ ⊗ D(a1, . . . , an) and so
perm(A) = n
i=1 ai = ε. However, unlike in the case of determinants in conventional matrix
algebra, the converse is not necessarily true.
The second concept in max-plus related to the determinant, known as the dominant, can be
thought of as a refinement of the permanent. It is defined below.
Definition 1.14. Let A ∈ Rn×n
max and let the matrix zA be defined by [zA]ij = zaij . The dominant
of A, denoted dom(A), is defined as
dom(A) =
highest exponent in det(zA) if det(zA) = 0
ε otherwise
The dominant can be used to prove max-plus analogues of major results such as Cramér’s
Theorem and the Cayley-Hamilton Theorem. We do not have the space to include these here;
for a comprehensive discussion readers are again referred to [19].
1.3 Graph-theoretic Interpretations in Max-Plus
As in conventional linear algebra, when working with vectors and matrices it is often natural
to interpret definitions and theorems graphically. It turns out that in the case of max-plus
algebra, it is not only natural to do so but also rather insightful. We will only really be able to
appreciate this when we come to look at the eigenvalue problem in the next section, but firstly
we must define all of the graph-theoretic concepts that we will require.
Definitions 1.15. (i) A directed graph G is a pair (V, E) where V is the set of vertices (or
nodes) and E ⊆ V × V is the set of edges (or arcs).
(ii) A path from vertex i to vertex j is a sequence of edges p = (i1, . . . , is+1) with i1 = i and
is+1 = j, such that (ik, ik+1) ∈ E for all k ∈ {1, . . . , s}.
(iii) The length of a path p = (i1, . . . , is+1), denoted |p|l, is equal to s. The set of paths from
vertex i to vertex j of length k is denoted Pk(i, j).
(iv) The weight of a path p from vertex i to vertex j of length d is given by
|p|w =
d
k=1
aik+1,ik
where i1 = i and id+1 = j.
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![(v) The average weight of a path p is given by |p|w
|p|l
.
(vi) A circuit of length s is a path of length s which starts and finishes at the same vertex, i.e.
a path c = (i1, . . . , is+1) such that i1 = is+1.
(vii) A circuit c = (i1, . . . , is+1) is elementary if i1, . . . , is are distinct, and s ≥ 1. We denote
the set of elementary circuits in G(A) by C(A).
(viii) For A ∈ Rn×n
max , the communication graph (or the precedence graph) of A, denoted G(A),
is the graph with vertex set V(A) = {1, . . . , n} and edge set E(A) = {(i, j) : aji = ε}. The
weight of the edge (i, j) ∈ E(A) is given by the entry aji.
Note that the (i, j)-th entry of the matrix A specifies the weight of the edge in G(A) from vertex
j to vertex i. This is common practice in the area of max-plus and graph theory but may not
appear intuitive to those new to the subject.
We now move on to looking at two particular matrices that play a vital role in relating graph
theory to max-plus linear algebra. For A ∈ Rn×n
max , let
A+
:=
∞
k=1
A⊗k
The element [A+]ji gives the maximal weight of any path from i to j in G(A). This statement
is non-trivial, but follows directly from the theorem below.
Theorem 1.16. Let A ∈ Rn×n
max . Then ∀k ∈ N:
[A⊗k
]ji =
max |p|w : p ∈ Pk(i, j) if Pk(i, j) = ∅
ε if Pk(i, j) = ∅
Proof. We use induction on k. Let i, j ∈ {1, . . . , n}. When k = 1, P1(i, j) either contains a
single path of length 1, namely the edge (i, j), or is empty if no such edge exists. In the first case,
the weight of the path is by definition [A]ji, and in the second case max |p|w : p ∈ Pk(i, j) = ε,
which is again equal to the value [A]ji (since there is no edge from i to j).
Now suppose the result holds for some k. Firstly, assume that Pk+1(i, j) = ∅. A path p ∈
Pk+1(i, j) can be split up into a subpath of length k running from i to some vertex l, and a
path consisting of a single edge from l to j. More formally:
p = ˆp ◦ (l, j) with ˆp ∈ Pk(i, l)
The maximal weight of any path in Pk+1(i, j) can thus be obtained from
max
l=1,...,n
[A]jl + max{|ˆp|w : ˆp ∈ Pk(i, l)}
= max
l=1,...,n
[A]jl + [A⊗k
]li (Inductive hypothesis)
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![=
n
l=1
[A]jl ⊗ [A⊗k
]li
= [A ⊗ A⊗k
]ji
= [A⊗(k+1)
]ji
which is what we wanted to prove. Finally, consider the case when Pk+1(i, j) = ∅; i.e. when
there exists no path of length k + 1 from i to j. This implies that for any vertex l, either there
is no path of length k from i to l or there is no edge from l to j (or possibly both). Hence
for any l, at least one of the values [A]jl, [A⊗k]li equals ε. Therefore [A⊗(k+1)]ji = ε, and this
completes the proof.
Note that Theorem 1.16 immediately tells us that A+ is not necessarily well-defined. For
example, if there exists a circuit c = (i1, . . . , is+1) in G(A) in which every edge has positive
weight, then [A⊗k]ji diverges (i.e. tends to +∞) as k → ∞ for any i, j ∈ {i1, . . . , is+1} (since
we can loop around the circuit c as many times a we like, creating a path of higher and higher
weight). The next lemma provides us with a sufficient condition for A+ to be well-defined, and
also reduces the complexity of the infinite sum.
Lemma 1.17. Let A ∈ Rn×n
max be such that any circuit in G(A) has non-positive average weight
(i.e. less than or equal to e). Then we have
A+
= A⊗1
⊕ A⊗2
⊕ A⊗3
⊕ · · · ⊕ A⊗n
∈ Rn×n
max
Proof. Since A is of dimension n, any path p in G(A) from i to j of length greater than n
necessarily contains at least one circuit. We have assumed that all of the circuits in G(A) have
non-positive weights, so removing the circuits in p yields a path from i to j of length at most
n, and of greater average weight. It follows that
[A+
]ji ≤ max [A⊗k
]ji : k ∈ {0, . . . , n}
and the reverse inequality is immediate from the definition of A+. This concludes the proof.
Before moving on, we prove one simple property of A+ that will come in handy later on.
Proposition 1.18. For A ∈ Rn×n
max , we have that A+ ⊗ A+ = A+.
Proof. Consider two vertices i, l ∈ {1, . . . , n}. A path of maximal weight from i to l can be split
up as a path of maximal weight from i to j plus a path of maximal weight from j to l, for any
j ∈ {1, . . . , n} for which the sum of the two path weights is maximal. Indeed this relationship
holds if and only if j is in the path of maximal weight from i to l, but for our purposes we can
simply take the maximum over all vertices.
By Theorem 1.16, the weight of such a path is given by [A+]li. Thus in max-plus notation
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![(recalling that ⊗ is commutative for scalars α ∈ Rmax), we can write
[A+
]li =
n
j=1
[A+
]ji ⊗ [A+
]lj
=
n
j=1
[A+
]lj ⊗ [A+
]ji = [A+
⊗ A+
]li
and therefore A+ = A+ ⊗ A+ as required.
We now introduce one more definition which is closely related to the object A+ defined above.
This will prove to be an integral concept throughout the rest of this chapter and beyond, and
as such, this is one of the most important definitions in this thesis.
Definition 1.19. For A ∈ Rn×n
max , let
A∗
:=
∞
k=0
A⊗k
= E ⊕ A+
Clearly, A∗ and A+ only differ on the leading diagonal. By Theorem 1.16, the (j, i)-th of A∗
could be interpreted as the maximal weight of any path from i to j in G(A), provided we
recognise the additional concept of an empty circuit of length 0 and weight e from every vertex
to itself.
Using Lemma 1.17, it is immediate from the definition of A∗ that if all the circuits in G(A) have
non-positive average weight, then A∗ = A⊗0 ⊕ A⊗1 ⊕ · · · ⊕ A⊗n. However, as the lemma below
shows, thanks to the addition of the identity matrix (i.e. the A⊗0 term) in A∗, we are able to
refine this result slightly by dropping the final term in the sum.
Lemma 1.20. Let A ∈ Rn×n
max be such that any circuit in G(A) has non-positive average weight.
Then we have
A∗
= A⊗0
⊕ A⊗1
⊕ A⊗2
⊕ · · · ⊕ A⊗(n−1)
∈ Rn×n
max
Proof. The same argument applies as in the proof of Lemma 1.17. Note that any path p in G(A)
from i to j of length n or greater necessarily contains at least one circuit, and so removing the
circuit(s) yields a path from i to j of length at most n − 1 and with greater average weight. For
the special case when i = j and p is an elementary circuit of length n (so visiting each vertex
in G(A) exactly once), the i-th entry on the diagonal of A⊗0 (which equals e by definition) will
always be greater than the corresponding entry in A⊗n, since e is the maximum possible weight
of any circuit. This is why we can drop the A⊗n term.
Note that we also have a direct analogue of Lemma 1.18 for the matrix A∗, and this will be
useful in the analysis that follows:
Proposition 1.21. For A ∈ Rn×n
max , we have that A∗ ⊗ A∗ = A∗.
14](https://image.slidesharecdn.com/10d08b3f-967c-4123-bade-f0ef8267a94d-151012204735-lva1-app6891/85/Thesis_JR-19-320.jpg)
![Proof. From Lemma 1.18 we have that A+ = A+ ⊗A+. Recalling the definition of A∗ and using
idempotency of matrix addition, we have
A∗
⊗ A∗
= (A+
⊕ E) ⊗ (A+
⊕ E)
= (A+
⊗ A+
) ⊕ (A+
⊗ E) ⊕ (E ⊗ A+
) ⊕ E
= A+
⊕ A+
⊕ A+
⊕ E
= A+
⊕ E = A∗
as required.
To finish this section, we introduce one more important property of square matrices over max-
plus known as irreducibility. The definition comes in three parts:
Definitions 1.22. (i) In a graph G, a vertex j is reachable from vertex i if there exists a
path from i to j.
(ii) A graph is strongly connected if every vertex is reachable from every other vertex.
(iii) A matrix A ∈ Rn×n
max is irreducible if G(A) is strongly connected.
The class of irreducible matrices over max-plus will turn out to be of real significance in Section
1.4. From a practical point of view it is not obvious how to determine whether a given matrix
A ∈ Rn×n
max is irreducible, but as the proposition below shows, one option is to examine the matrix
A+. Combined with Lemma 1.17 (when A has the appropriate properties), this provides us with
a handy (and computationally quick) way to check for matrix irreducibility over max-plus.
Proposition 1.23. A matrix A ∈ Rn×n
max is irreducible if and only if all the entries of A+ are
different from ε.
Proof. A matrix is irreducible if there is a path between any two vertices i and j in G(A), which
by Theorem 1.16 occurs exactly when the entry [A+]ji is not equal to ε.
1.4 Spectral Theory
1.4.1 Eigenvalues and Eigenvectors
Given a matrix A ∈ Rn×n
max , we consider the problem of existence of eigenvalues and eigenvectors.
The main result in max-plus spectral theory is that, under mild conditions, A has a unique
eigenvalue with a simple graph-theoretic interpretation. As can be seen below, the definition of
max-plus eigenvalues and eigenvectors is a direct translation from conventional linear algebra,
with the × operator replaced with ⊗:
Definition 1.24. Let A ∈ Rn×n
max . If there exists a scalar µ ∈ Rmax and a vector v ∈ Rn
max
(containing at least one finite element) such that
A ⊗ v = µ ⊗ v
15](https://image.slidesharecdn.com/10d08b3f-967c-4123-bade-f0ef8267a94d-151012204735-lva1-app6891/85/Thesis_JR-20-320.jpg)
![then µ is an eigenvalue of A and v is an eigenvector of A associated with the eigenvalue µ.
Note that Definition 1.24 allows an eigenvalue to be µ = ε. However, the proposition below says
that this can only happen when A has a column in which all entries are ε. In graph-theoretic
terms this means that G(A) has a vertex which, once visited, can never be left (sometimes called
a sink). This is uninteresting from an analytical point of view, so it is reasonable to consider
the case µ = ε to be trivial. Before we prove this result, we introduce some simple notation.
Notation. Let A ∈ Rn×n
max . For i ∈ {1, . . . , n}, we denote the i-th row of A by [A]i·. Similarly,
for j ∈ {1, . . . , n}, we denote the j-th column of A by [A]·j.
Proposition 1.25. ε is an eigenvalue of A ∈ Rn×n
max iff A has at least one column in which all
entries are ε.
Proof. Let A ∈ Rn×n
max be such that [A]·j = (ε, . . . , ε) for some j ∈ {1, . . . , n}. Let v ∈ Rn
max be
such that [v]i = ε ∀i = j and [v]j = α = ε. Then it is easy to verify that [A ⊗ v]i = ε for all
i = 1, . . . , n; that is, ε is an eigenvalue of A with an associated eigenvector v.
Conversely, suppose A ∈ Rn×n
max has eigenvalue ε with an associated eigenvector v. let J = {j :
vj = ε}, which is non-empty by definition. Then for each i = 1, . . . , n we have
ε = [A ⊗ v]i =
n
j=1
aij ⊗ vj =
j∈J
aij ⊗ vj
=⇒ aij = ε ∀j ∈ J
So every column j of A for which vj = ε has all its entries equal to ε. In particular, A contains
at least one column in which all entries are ε.
Corollary 1.26. If A ∈ Rn×n
max is irreducible then ε is not an eigenvalue of A.
Proof. If A is irreducible then it cannot have a column in which all entries are ε. Thus by
Proposition 1.25, ε is not an eigenvalue of A.
Note that eigenvectors are not unique: any scalar multiple of an eigenvector is also an eigen-
vector, and more generally, if µ is an eigenvalue of A, v1, v2 are associated eigenvectors and
α1, α2 ∈ Rmax{ε}, then we have
A ⊗ (α1 ⊗ v1) ⊕ (α2 ⊗ v2) = A ⊗ (α1 ⊗ v1) ⊕ A ⊗ (α2 ⊗ v2)
= α1 ⊗ (A ⊗ v1) ⊕ α2 ⊗ (A ⊗ v2)
= α1 ⊗ (µ ⊗ v1) ⊕ α2 ⊗ (µ ⊗ v2)
= µ ⊗ (α1 ⊗ v1) ⊕ (α2 ⊗ v2)
So (α1 ⊗ v1) ⊕ (α2 ⊗ v2) is also an eigenvector associated with the eigenvalue µ. In fact, the
eigenvectors associated with a given eigenvalue form a vector space in max-plus called the
eigenspace which we shall explore in depth later.
16](https://image.slidesharecdn.com/10d08b3f-967c-4123-bade-f0ef8267a94d-151012204735-lva1-app6891/85/Thesis_JR-21-320.jpg)
![As we mentioned at the beginning of Section 1.3, many of the results in the area of max-plus
spectral theory can be interpreted graphically, and the next key lemma constitutes the first step
in doing just that.
Lemma 1.27. Let A ∈ Rn×n
max have finite eigenvalue µ. Then µ is the average weight of some
elementary circuit in G(A).
Proof. Let v be an associated eigenvector of µ. Then by definition not all the entries of v equal
ε, i.e. there exists a vertex/index j1 ∈ {1, . . . , n} such that vj1 = ε. Now v is an eigenvector
and so we have [A ⊗ v]j1 = µ ⊗ vj1 = ε. But [A ⊗ v]j1 = n
k=1 aj1k ⊗ vk, and therefore there
exists a vertex j2 such that
aj1j2 ⊗ vj2 = [A ⊗ v]j1 = ε (1.1)
which implies aj1j2 = ε, i.e. (j2, j1) is an edge in G(A). (1.1) also implies that vj2 = ε, so we
can continue in the same fashion to find a vertex j3 with (j3, j2) an edge in G(A) and vj3 = ε.
Proceeding in this way, eventually some vertex, say, vertex jh, must be encountered for a second
time since the number of vertices is finite. Thus by ignoring the edges prior to encountering jh
for the first time, we have found an elementary circuit
c = ((jh, jh+l−1), (jh+l−1, jh+l−2), . . . , (jh+1, jh))
of length |c|l = l, and with weight
|c|w =
l−1
k=0
ajh+kjh+k+1
(1.2)
where jh = jh+l. By construction, we have that
l−1
k=0
(ajh+kjh+k+1
⊗ vjh+k+1
) = µ⊗l
⊗
l−1
k=0
vjh+k
or equivalently in conventional algebra (for ease of manipulation):
l−1
k=0
ajh+kjh+k+1
+ vjh+k+1
= (l × µ) +
l−1
k=0
vjh+k
Now, because jh = jh+l it follows that l−1
k=0 vjh+k+1
= l−1
k=0 vjh+k
, so subtracting l−1
k=0 vjh+k
from both sides yields
l−1
k=0
ajh+k
jh+k+1 = l × µ
and translated back into max-plus, we can substitute this into (1.2) to see that |c|w = µ⊗l.
17](https://image.slidesharecdn.com/10d08b3f-967c-4123-bade-f0ef8267a94d-151012204735-lva1-app6891/85/Thesis_JR-22-320.jpg)
![Thus we have that the average weight of the circuit c is equal to
|c|w
|c|l
=
µ⊗l
l
= µ
as required.
Lemma 1.27 tells us that the only candidates for eigenvalues are the average weights of circuits
in G(A). However, it does not tell us which circuits actually define an eigenvalue and which
do not. Fortunately, when A is irreducible the answer to this question is very simple: only
the maximal average circuit weight defines an eigenvalue. This result is established in the two
theorems below, but first we require some additional definitions and notation.
Definitions 1.28. (i) A circuit c ∈ C(A) is critical if its average weight is maximal.
(ii) For A ∈ Rn×n
max , the critical graph of A, denoted Gc(A), is the graph containing the vertices
and edges which belong to the critical circuits in G(A). We write Gc(A) = (Vc(A), Ec(A)),
and refer to the vertices in Vc(A) as critical vertices.
(iii) The critical classes of A ∈ Rn×n
max are the maximal strongly connected components of Gc(A).
Notation. Let A ∈ Rn×n
max . For β ∈ Rmax{ε}, define the matrix Aβ by [Aβ]ij = aij − β.
Note that the ‘−’ operator is to be interpreted in conventional algebra, where we adopt the
convention ε − x = ε ∀x ∈ R. If β is an eigenvalue of A, the matrix Aβ is sometimes called the
normalised matrix.
Note that the communication graphs G(A) and G(Aβ) are identical except for their edge weights,
and if a circuit c in G(A) has average weight w then the same circuit in G(Aβ) has average weight
w − β. In particular, if G(A) has finite maximal average circuit weight λ then the maximal
average circuit weight in G(Aλ) is λ − λ = 0. Furthermore, a circuit in G(A) is critical if and
only if it is critical in G(Aλ), and therefore Gc(A) and Gc(Aλ) are identical (again, except for
their edge weights).
Consider the matrix A+
λ , which is to be read (Aλ)+
. By Theorem 1.16, the element [A+
λ ]ij gives
the maximal weight of any path from j to i in G(Aλ). In particular, since all circuits in G(Aλ)
have non-positive average weight, we must have [A+
λ ]ii ≤ e for all i ∈ {1, . . . , n}. Furthermore,
for the matrix A∗
λ (also to be read (Aλ)∗
) we obtain [A∗
λ]ii = e⊕[A+
λ ]ii = e for all i ∈ {1, . . . , n}.
Theorem 1.29. Let the communication graph G(A) of a matrix A ∈ Rn×n
max have finite maximal
average circuit weight λ. Then λ is an eigenvalue of A, with an associated eigenvector [A∗
λ]·j
for any vertex j ∈ Vc(A).
Proof. Firstly note that all the circuits in G(Aλ) have non-positive average weight, and therefore
A+
λ is well-defined by Lemma 1.17. Now, every vertex in Gc(Aλ) is contained in a non-empty
circuit which has weight e, i.e.
∀j ∈ Vc
(A) : [A+
λ ]jj = e (1.3)
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![Next, write
[A∗
λ]ij = [E ⊕ A+
λ ]ij =
ε ⊕ [A+
λ ]ij for i = j
e ⊕ [A+
λ ]ij for i = j
Then from (1.3), for j ∈ Vc(A) it follows that
[A+
λ ]·j = [A∗
λ]·j (1.4)
Now, note that we have
A+
λ = A⊗1
λ ⊕ A⊗2
λ ⊕ . . .
= Aλ ⊗ (A⊗0
λ ⊕ A⊗1
λ ⊕ . . . ) = Aλ ⊗ A∗
λ
So substituting this into (1.4) gives for j ∈ Vc(A)
[Aλ ⊗ A∗
λ]·j = [A∗
λ]·j ⇐⇒ Aλ ⊗ [A∗
λ]·j = [A∗
λ]·j
⇐⇒ A ⊗ [A∗
λ]·j = λ ⊗ [A∗
λ]·j
Therefore λ is an eigenvalue of A and the j-th column of A∗
λ is an associated eigenvector for
any j ∈ Vc(A).
Theorem 1.30. Let A ∈ Rn×n
max be irreducible. Then A has a unique eigenvalue, denoted λ(A),
which is finite and equal to the maximal average circuit weight in G(A).
Proof. Let the maximal average circuit weight in G(A) be denoted by λ. Since A is irreducible,
G(A) must contain a circuit and therefore λ is necessarily finite. Thus by Theorem 1.29 we
know that λ is an eigenvalue of A, and it remains to show uniqueness.
Let c = (j1, . . . , jl+1) be an arbitrary circuit in C(A) of length l = |c|l, with jl+1 = j1. Then
ajk+1jk
= ε for all k ∈ {1, . . . , l}. Further, suppose that µ is an eigenvalue of A with an associated
eigenvector v. Note that A is irreducible, so by Corollary 1.26 we have that µ = ε. Now, since
A ⊗ v = µ ⊗ v, it follows that
ajk+1jk
⊗ vjk
≤ µ ⊗ vjk+1
, k ∈ {1, . . . , l}.
and arguing as in Lemma 1.27 (replacing equalities with the appropriate inequalities), we see
that the average weight of the circuit c satisfies
|c|w
|c|l
≤
µ⊗l
l
= µ (1.5)
That is, µ ≥ λ (since (1.5) holds for all c ∈ C(A), and we already have that the maximal
average circuit weight is λ). But by Lemma 1.27, µ is equal to the average weight of some
circuit c ∈ C(A), and so µ ≤ λ also. Hence µ = λ, i.e. λ is the unique eigenvalue of A.
19](https://image.slidesharecdn.com/10d08b3f-967c-4123-bade-f0ef8267a94d-151012204735-lva1-app6891/85/Thesis_JR-24-320.jpg)
![When A is large it is often difficult to identify the maximal average circuit weight in G(A). In
fact, there exist several numerical procedures used to determine the eigenvalue of an irreducible
matrix in max-plus, including Karp’s Algorithm and the Power Algorithm. However, none of
these has a particularly attractive order of complexity - for example, the complexity of Karp’s
Algorithm is of order n3, and the complexity of the Power Algorithm is not known precisely
(see [11]). We do not have space here to describe the methods in detail; for more information
readers are referred to chapter five of [13].
We end this section with a simple proposition that, while interesting in its own right, will come
in handy when we begin to look at the eigenspace.
Proposition 1.31. Let A ∈ Rn×n
max be an irreducible matrix with eigenvalue λ and associated
eigenvector v. We have that vi > ε for all i ∈ {1, . . . , n}.
Proof. Call the set of vertices of G(A) corresponding to the finite entries of v the support of
v, denoted Z(v). Suppose that Z(v) does not contain all the elements of V(A). Since A is
irreducible, there must be edges from the vertices in Z(v) to vertices not belonging to Z(v).
Hence there exists vertices j ∈ Z(v), i /∈ Z(v) with aij = ε. Then
[A ⊗ v]i ≥ aij ⊗ vj > ε
That is, Z(A ⊗ v) is strictly bigger than Z(v). But A ⊗ v = λ ⊗ v (and λ is finite by Theorem
1.30), so Z(v) and Z(A⊗v) should be equal. This is a contradiction, and so Z(v) must contain
all the elements of V(A).
1.4.2 The Eigenspace
Let A ∈ Rn×n
max have finite eigenvalue λ. In this part of our analysis we let V (A, λ) denote the
set of all eigenvectors of A associated with the eigenvalue λ, which we call the eigenspace of A
w.r.t. λ. If A is irreducible then by Theorem 1.30 we know that it has a unique eigenvalue, so
we can drop the dependence on λ and denote the eigenspace of A simply by V (A).
The main aim of this section is to find an expression that completely characterises the eigenspace
of A. In Theorem 1.29 we established that [A∗
λ]·j is an eigenvector of A for any j ∈ Vc(A),
but are these the only eigenvectors (of course, up to taking linear combinations, as discussed
above)? We will eventually see that the answer to this question is yes, but first we require some
intermediate steps.
Lemma 1.32. Let A ∈ Rn×n
max . We have that A∗
λ = (E ⊕ Aλ)⊗(n−1).
Proof. If n = 1 then the result is trivial. Otherwise, since E and Aλ commute, we can carry
out the iterated multiplication (E ⊕ Aλ) ⊗ · · · ⊗ (E ⊕ Aλ) to obtain
(E ⊕ Aλ)⊗(n−1)
= E ⊕
n−1
i=1
A⊗i
λ ⊕ · · · ⊕ A⊗i
λ
(n−1
i ) times
(1.6)
20](https://image.slidesharecdn.com/10d08b3f-967c-4123-bade-f0ef8267a94d-151012204735-lva1-app6891/85/Thesis_JR-25-320.jpg)
![Each power A⊗0
λ , . . . , A
⊗(n−1)
λ occurs at least once, so by idempotency of ⊕, (1.6) becomes
(E + Aλ)⊗(n−1)
= E ⊕ Aλ ⊕ A⊗2
λ ⊕ . . . A
⊗(n−1)
λ (1.7)
However, noting that every circuit in G(Aλ) must have non-positive weight, we can apply Lemma
1.20 to see that the right-hand side of (1.7) is equal to A∗
λ. This completes the proof.
Lemma 1.33. Let A ∈ Rn×n
max be an irreducible matrix, with eigenvalue λ and an associated
eigenvector v. Then the matrix A∗
λ has eigenvalue e, also with an associated eigenvector v.
Proof. Firstly, note that for any j ∈ {1, . . . , n}
[λ ⊗ v]j = [A ⊗ v]j ⇐⇒ vj = [A ⊗ v]j − λ ⇐⇒ e ⊗ vj = [Aλ ⊗ v]j
That is, e ⊗ v = Aλ ⊗ v, and v is also an eigenvector of Aλ (whose unique eigenvalue must be e
by Theorem 1.30). Thus the eigenspaces V (A) and V (Aλ) coincide. Next, note that
(E ⊕ Aλ) ⊗ v = (E ⊗ v) ⊕ (Aλ ⊗ v) = v ⊕ v = v
Therefore, using Lemma 1.32:
A∗
λ ⊗ v = (E ⊕ Aλ)⊗(n−1)
⊗ v = v = e ⊗ v
as required.
Definition 1.34. Let A ∈ Rn×n
max be a matrix with eigenvalue λ and associated eigenvector v.
The saturation graph of A with respect to λ, denoted Sλ(A, v), is the graph consisting of those
edges (j, i) ∈ E(A) such that aij ⊗ vj = λ ⊗ vi, with vi, vj = ε.
Recall that by definition, if v is an eigenvector of A then there exists at least one i ∈ {1, . . . , n}
such that vi = ε. Then, since A ⊗ v = λ ⊗ v we have that n
j=1 aij ⊗ vj = λ ⊗ vi, which implies
that there exists (at least one) j ∈ {1, . . . , n} such that aij ⊗ vj = λ ⊗ vi. This value is finite
(assuming λ = ε), so we must have (j, i) ∈ E Sλ(A, v) . That is, the saturation graph of A
w.r.t. λ is never empty. Indeed, if A is irreducible, by Proposition 1.31 we know that vi > ε for
all i ∈ {1, . . . , n}, and so by the same argument, Sλ(A, v) contains all the vertices in V(A). In
this case we know that the eigenvalue λ is unique, and therefore we drop the dependence on λ
and simply refer to the saturation graph of A.
Lemma 1.35. Let A ∈ Rn×n
max be an irreducible matrix, with eigenvalue λ and associated eigen-
vector v. We have:
(i) For each vertex i ∈ V(A), there exists a circuit in S(A, v) from which vertex i can be
reached in a finite number of steps.
(ii) Any circuit in S(A, v) belongs to Gc(A).
Proof. (i) A is irreducible, so by Proposition 1.31 we know that vi > ε for all i ∈ {1, . . . , n}. Let
21](https://image.slidesharecdn.com/10d08b3f-967c-4123-bade-f0ef8267a94d-151012204735-lva1-app6891/85/Thesis_JR-26-320.jpg)
![i ∈ V(A), which by the discussion above we know is a vertex of the saturation graph S(A, v).
Thus there is a vertex j such that λ ⊗ vi = aij ⊗ vj. Repeating this argument, we can identify
a vertex k such that λ ⊗ vj = ajk ⊗ vk. Repeating this argument an arbitrary number of times,
say, m, we get a path in S(A, v) of length m. If m > n, the constructed path must contain a
circuit.
(ii) Let c = (i1, i2, . . . , il+1) be a circuit of length l in S(A, v). By definition, for all k ∈ {1, . . . , n}
we have that
λ ⊗ vik+1
= aik+1ik
⊗ vik
which implies that
λ⊗l
⊗ vi1 =
l
k=1
aik+1ik
⊗ vi1
Hence, recalling that vi1 is finite:
λ⊗l
=
l
k=1
aik+1ik
But the right-hand side is simply equal to the weight of the circuit c, which thus has average
weight λ. But A is irreducible, so by Theorem 1.30 λ is equal to the maximal average circuit
weight in G(A). Thus c is critical, and belongs to Gc(A).
Lemma 1.36. Let A ∈ Rn×n
max be an irreducible matrix, with eigenvalue λ and associated eigen-
vector v. Then v can be written as
v =
j∈Vc(A)
αj ⊗ [A∗
λ]·j
for some αj ∈ Rmax, j ∈ Vc(A).
Proof. Consider two vertices i, j in S(Aλ, v) such that there exists a path from i to j, say,
(i1, i2, . . . , il+1), with i1 = i and il+1 = j. Then by definition of the saturation graph, this gives
[Aλ]ik+1ik
⊗ vik
= vik+1
, k ∈ {1, . . . , l}
Hence vj = a ⊗ vi, where a is given by
a =
l
k=1
[Aλ]ik+1ik
≤ [A⊗l
λ ]ji ≤ [A∗
λ]ji (1.8)
Now, using that vj = a ⊗ vi, for an arbitrary vertex ν ∈ {1, . . . , n}:
[A∗
λ]νj ⊗ vj = [A∗
λ]νj ⊗ a ⊗ vi
≤ [A∗
λ]νj ⊗ [A∗
λ]ji ⊗ vi (by (1.8))
22](https://image.slidesharecdn.com/10d08b3f-967c-4123-bade-f0ef8267a94d-151012204735-lva1-app6891/85/Thesis_JR-27-320.jpg)
![≤ [A∗
λ]νi ⊗ vi (1.9)
where the last inequality follows from Proposition 1.21. By applying Lemma 1.35, for any vertex
j in S(Aλ, v) there exists a vertex i = i(j) ∈ Vc(A). Inequality (1.9) therefore implies
j∈S(Aλ,v)
[A∗
λ]νj ⊗ vj ≤
i∈Vc(Aλ)
[A∗
λ]νi ⊗ vi (1.10)
and this holds for any ν ∈ {1, . . . , n}.
Now, by Lemma 1.33, A∗
λ has eigenvalue e with an associated eigenvector v, i.e. v = A∗
λ ⊗ v.
The value of vν is equal to [A∗
λ]νj ⊗vj for some j, which by definition has to be in the saturation
graph S(Aλ, v). Thus it holds for ν ∈ {1, . . . , n} that
vν =
j∈S(Aλ,v)
[A∗
λ]νj ⊗ vj
(1.10)
≤
j∈Vc(Aλ)
[A∗
λ]νj ⊗ vj
On the other hand, since v is an eigenvector of A∗
λ associated with the eigenvalue e,
vν = [A∗
λ ⊗ v]ν =
n
j=1
[A∗
λ]νj ⊗ vj ≥
i∈Vc(Aλ)
[A∗
λ]νi ⊗ vi
which also holds for any ν ∈ {1, . . . , n}. Thus we have shown
vν =
i∈Vc(Aλ)
[A∗
λ]νi ⊗ vi
and since Vc(Aλ) = Vc(A) (see the proof of Theorem 1.29), the proof is complete.
The lemma above shows that for an irreducible matrix A, the vectors [A∗
λ]·j, with j ∈ Vc(A),
constitute a generating set for the eigenspace of A. Notice that in the proof we have actually
identified the coefficients αi to which we referred in the statement of the lemma. If some of the
columns of A∗
λ are colinear then the αi’s are non-unique and some can be chosen to equal ε.
We have now done most of the work in characterising the eigenspace of an irreducible matrix.
We now require a small extension of our notation and one more lemma before we are able to
give a complete expression for the eigenspace, and we will end this section by referring to a
theorem which shows that it is not possible to simplify this expression any further.
Notation. Recall that the critical classes of a matrix A ∈ Rn×n
max are the maximal strongly
connected subgraphs of Gc(A). Let Nc(A) denote the number of critical classes of A, so Nc(A) ∈
N. For r ∈ {1, . . . , Nc(A)}, let Gc
r(A) = (Vc
r (A), Ec
r (A)) denote the r-th critical class of A and
let jc
r := min{j ∈ Vc
r (A)} be the smallest numbered vertex in the r-th critical class. We call
{jc
1, . . . , jc
Nc(A)} a set of representative vertices of the critical classes of A.
Note that in the way defined above, the set of representative vertices is unique. However, this
is not important - in general, a representative vertex jc
r of the rth critical class of A can be any
23](https://image.slidesharecdn.com/10d08b3f-967c-4123-bade-f0ef8267a94d-151012204735-lva1-app6891/85/Thesis_JR-28-320.jpg)
![j ∈ Vc
r (A).
Lemma 1.37. Let A ∈ Rn×n
max be an irreducible matrix with eigenvalue λ. Then for i, j ∈ Vc(A),
there exists α ∈ Rmax{ε} such that
α ⊗ [A∗
λ]·i = [A∗
λ]·j
iff i and j are members of the same critical class.
Proof. Suppose that i, j ∈ Vc(A) are members of the same critical class of Aλ. Then i and j
communicate with each other in the critical graph, i.e. (i, j, i) is an elementary circuit in Gc(Aλ).
As we have argued before (see Theorem 1.29), any circuit in Gc(Aλ) must have weight e, and
therefore in this case we have [Aλ]ji ⊗ [Aλ]ij = e. Then by definition of A∗
λ, we have that
[A∗
λ]ji ⊗ [A∗
λ]ij ≥ [Aλ]ji ⊗ [Aλ]ij = e (1.11)
Now by a previous observation we know that [A∗
λ]jj = e, and by Proposition 1.21 we have that
A∗
λ = A∗
λ ⊗ A∗
λ. Therefore we also have
[A∗
λ]ji ⊗ [A∗
λ]ij ≤
n
l=1
[A∗
λ]jl ⊗ [A∗
λ]lj = [A∗
λ ⊗ A∗
λ]jj = [A∗
λ]jj = e (1.12)
and from (1.11) and (1.12) we conclude that [A∗
λ]ji ⊗ [A∗
λ]ij = e. Thus for all l ∈ {1, . . . , n}:
[A∗
λ]li ⊗ [A∗
λ]ij ≤ [A∗
λ]lj
= [A∗
λ]lj ⊗ [A∗
λ]ji ⊗ [A∗
λ]ij
≤ [A∗
λ]li ⊗ [A∗
λ]ij
and therefore [A∗
λ]lj = [A∗
λ]li ⊗ [A∗
λ]ij. Hence the statement of the lemma has been proved, with
α = [A∗
λ]ij.
Conversely, suppose now that i, j ∈ Vc(A) do not belong to the same critical class, and suppose
for contradiction that we can find α ∈ Rmax{ε} such that α ⊗ [A∗
λ]·i = [A∗
λ]·j. The i-th and
j-th components of this equation read
α ⊗ e = [A∗
λ]ij and α ⊗ [A∗
λ]ji = e
respectively, from which it follows that
[A∗
λ]ij ⊗ [A∗
λ]ji = e
Therefore the elementary circuit (i, j, i) has average weight e, and therefore belongs to Gc(Aλ).
Thus vertices i and j are members of the same critical class (since they communicate with each
other), which is a contradiction.
Theorem 1.38. Let A ∈ Rn×n
max be an irreducible matrix with (unique) eigenvalue λ. The
24](https://image.slidesharecdn.com/10d08b3f-967c-4123-bade-f0ef8267a94d-151012204735-lva1-app6891/85/Thesis_JR-29-320.jpg)
![eigenspace of A is given by
V (A) =
Nc(A)
r=1
αr ⊗ [A∗
λ]·jc
r
: αr ∈ Rmax, at least one αr finite
for any set of representative vertices {jc
1, . . . , jc
Nc(A)} of the critical classes of A.
Proof. By Lemma 1.36 we know that any eigenvector of A is a linear combination of the columns
[A∗
λ]·j, for j ∈ Vc(A). However, by Lemma 1.37 we know that the columns [A∗
λ]·j for j in some
critical class Vc
r (A) are all colinear. Therefore to build any eigenvector we only need one column
corresponding to each critical class, and so it suffices to take the sum over a set of representative
vertices of the critical classes of A.
Theorem 1.39. No column [A∗
λ]·i, for i ∈ Vc(A), can be expressed as a linear combination of
columns [A∗
λ]·jc
r
, where jc
r varies over the representative vertices of critical classes distinct from
that of i.
Proof. The proof of this statement requires substantial groundwork which we do not have the
space to include. For all the details and a full proof, readers are referred to theorem 3.101 in
[2].
Theorem 1.39 above tells us that we cannot simplify any further the expression for V (A) given
in Theorem 1.38. It also tells us that for an irreducible matrix A, the columns [A∗
λ]·jc
r
, where
{jc
1, . . . , jc
Nc(A)} is a set of representative vertices of the critical classes of A, form a basis for
the eigenspace V (A).
1.4.3 A Worked Example
Consider the matrix
A =
ε −2 ε 6
1 ε 4 ε
ε 8 ε ε
ε 5 ε 6
Thus G(A) looks like
25](https://image.slidesharecdn.com/10d08b3f-967c-4123-bade-f0ef8267a94d-151012204735-lva1-app6891/85/Thesis_JR-30-320.jpg)
![1
4
2 3
1 8
-2
5
4
6
6
Figure 1.1: Communication graph of the matrix A given above. Vertices are represented as
circles and numbered 1-4 by convention. Edges are present only if the corresponding entry in
A is finite, in which case this value specifies the edge weight.
We can see that G(A) is strongly connected, so A is irreducible. Thus by Theorem 1.30, A has
a unique eigenvalue λ given by the maximal average circuit weight in G(A). The elementary
circuits and their average weights are
c1 = (1, 2, 1) |c1|w/|c1|l = (1 ⊗ −2)/2 = −0.5
c2 = (1, 2, 4, 1) |c2|w/|c2|l = (1 ⊗ 5 ⊗ 6)/3 = 4
c3 = (2, 3, 2) |c3|w/|c3|l = (8 ⊗ 4)/2 = 6
c4 = (4, 4) |c4|w/|c4|l = (6)/1 = 6
and therefore λ = max{−0.5, 4, 6, 6} = 6. Circuits c3 and c4 are critical, so the critical graph
Gc(A) looks like
2 3 4
8
4
6
Figure 1.2: Critical graph of the matrix A given above. Both the circuits have maximal average
weight of 6. The other circuits present in Figure 1.1 are no longer included because they are
not critical (their average weight is not maximal).
We can see that Vc(A) = {2, 3, 4}, and Gc(A) has two critical classes with vertex sets Vc
1(A) =
{2, 3} and Vc
2(A) = {4} respectively. Thus {jc
1 = 2, jc
2 = 4} is a set of representative vertices of
the critical classes of A. Now, using that [Aλ]ij = aij − λ, we have
Aλ =
ε −8 ε e
−5 ε −2 ε
ε 2 ε ε
ε −1 ε e
and either by inspection of G(Aλ), or by using Lemma 1.17 and computing A⊗1
λ , A⊗2
λ , A⊗3
λ and
26](https://image.slidesharecdn.com/10d08b3f-967c-4123-bade-f0ef8267a94d-151012204735-lva1-app6891/85/Thesis_JR-31-320.jpg)
![A⊗4
λ , we can see that
A+
λ =
−6 −1 −3 e
−5 e −2 −5
−3 2 e −3
−6 −1 −3 e
Similarly, by using Lemma 1.20 (or by simply replacing any non-zero diagonal values in A+
λ
above by e), we obtain
A∗
λ =
e −1 −3 e
−5 e −2 −5
−3 2 e −3
−6 −1 −3 e
Now by theorems 1.38 and 1.39, the columns [A∗
λ]·2 & [A∗
λ]·4 form a basis for the eigenspace of
A, i.e.
V (A) =
α1 ⊗
−1
e
2
−1
α2 ⊗
e
−5
−3
e
: α1, α2 ∈ Rmax, at least one αr finite
For example, if we take α1 = −2, α2 = 1 we get
v := −2 ⊗
−1
e
2
−1
1 ⊗
e
−5
−3
e
=
−3
−2
e
−3
1
−4
−2
1
=
1
−2
e
1
and we can easily verify that this is indeed an eigenvector of A, associated with the unique
eigenvalue λ = 6:
A ⊗ v =
ε −2 ε 6
1 ε 4 ε
ε 8 ε ε
ε 5 ε 6
1
−2
e
1
=
7
4
6
7
= 6 ⊗
1
−2
e
1
= λ ⊗ v
Finally, we can observe that
[A∗
λ]·3 =
−3
−2
e
−3
= −2 ⊗
−1
e
2
−1
= −2 ⊗ [A∗
λ]·2
That is, columns [A∗
λ]·2 and [A∗
λ]·3 are scalar multiples of each other, which we would expect
27](https://image.slidesharecdn.com/10d08b3f-967c-4123-bade-f0ef8267a94d-151012204735-lva1-app6891/85/Thesis_JR-32-320.jpg)
![If c is a critical circuit of length l in G(A⊗k) then there is a corresponding critical circuit c of
length k ×l in G(A). This circuit must be in Gc(A) (because it is critical), which in turn implies
that c is a critical circuit in G((Ac)⊗k). Hence, it follows that Gc((Ac)⊗k) ⊇ Gc(A⊗k). The other
inclusion is proved in the same way.
Lemma 1.46. Let A ∈ Rn×n
max be an irreducible matrix with cyclicity σ = σ(A). Then the
cyclicity of the matrix A⊗σ is equal to one.
Proof. Firstly, suppose the critical matrix Ac is irreducible. By the remarks prior to Lemma 1.45
we know that the cyclicity of Ac and that of its communication graph is equal to σ, so by Lemma
1.43, after a suitable relabelling of vertices, (Ac)⊗σ corresponds to a block diagonal matrix with
square diagonal blocks that are irreducible and have graph cyclicity one. However, by Lemma
1.45 with k = σ, we have that Gc((Ac)⊗σ) = G((Ac)⊗σ), and therefore the communication graph
of each of the diagonal blocks of (Ac)⊗σ coincides with its critical graph. Thus for each diagonal
block both cyclicities coincide, and therefore both are one.
If Ac is reducible then the same process can be done for each of the critical classes of Gc(A) with
their individual cyclicities. According to Definition 1.41, the least common multiple of these
cyclicities equals σ, the matrix cyclicity of A. Noting that σ is a multiple of σG(A), it follows
from Corollary 1.44 that each diagonal block of (Ac)⊗σ corresponds to a block diagonal matrix
with square diagonal blocks that are irreducible and have cyclicity one. Note that if Gc(A) does
not cover all the vertices of G(A) then we must augment the overall block diagonal matrix with
a square block with entries equal to ε in order to keep it the same size as the original matrix A.
In both cases it follows that each diagonal block of the block diagonal matrix corresponding to
(Ac)⊗σ is irreducible and has cyclicity one. Taking the least common multiple of all cyclicities,
this means that the cyclicity of the whole matrix (Ac)⊗σ is equal to one, and therefore the graph
cyclicity of Gc((Ac)⊗σ) is also equal to one. But by Lemma 1.45 with k = σ, this graph is the
same as Gc(A⊗σ), which therefore must also have cyclicity one. Thus A⊗σ has matrix cyclicity
one, which completes the proof.
We now state a fundamental theorem, the proof of which can be found in [4].
Theorem 1.47. Let β1, . . . , βq ∈ N be such that gcd{β1, . . . , βq} = 1. Then there exists N ∈ N
such that for all k ≥ N there exist n1, . . . , nq ∈ N0 such that k = (n1 × β1) + · · · + (nq × βq).
We finally state and prove one last prerequisite result which is essentially a special case of the
theorem that follows. It turns out that the generalisation is relatively straightforward, so in
proving this lemma we will have done most of the work in proving the main result.
Lemma 1.48. Let A ∈ Rn×n
max be an irreducible matrix with unique eigenvalue e and cyclicity
one. Then there exists N ∈ N such that
A⊗(k+1)
= A⊗k
33](https://image.slidesharecdn.com/10d08b3f-967c-4123-bade-f0ef8267a94d-151012204735-lva1-app6891/85/Thesis_JR-38-320.jpg)
![for all k ≥ N.
Proof. The proof comes in three stages. We show that there exists N ∈ N such that for all
k ≥ N:
1. [A⊗k]ii = [A+]ii = e for all i ∈ Vc(A),
2. [A⊗k]ij = [A+]ij for all i ∈ Vc(A) and j ∈ {1, . . . , n},
3. [A⊗k]ij = l∈Vc(A)[A+]il ⊗ [A+]lj for all i, j ∈ {1, . . . , n}.
The result then follows immediately from statement 3 since the right hand side does not depend
on k.
Statement 1. Consider i ∈ Vc(A). Then there is a critical class of Gc(A), say Gc
1(A) =
(Vc
1(A), Ec
1(A)), such that i ∈ Vc
1(A). Since the cyclicity of matrix A is one, it follows that
the cyclicity of graph Gc
1(A) is equal to one too. Hence there exist circuits in Gc
1(A), say
c1, . . . , cq, whose lengths have a greatest common divisor equal to one. Since Gc
1(A) is a critical
class it must be strongly connected, and therefore there exists a circuit α in Gc
1(A) that passes
through i and through all circuits c1, . . . , cq (i.e. α ∩ cj = ∅ ∀j = 1, . . . , q).
Now, by Theorem 1.47, there exists N ∈ N such that for each k ≥ N, there exist n1, . . . , nq ∈ N0
such that
k = |α|l + (n1 × |c1|l) + · · · + (nq × |cq|l).
For these n1, . . . , nq, we can construct a circuit passing through i, built from circuit α, n1 copies
of circuit c1, n2 copies of circuit c2 and so on, up to nq copies of circuit cq. Clearly this circuit
is in Gc
1(A), so it must be critical with weight e. Since the maximal average circuit weight in
G(A) is e, it follows that [A⊗k]ii = e for all k ≥ N, which, by the definition of A+, also implies
that [A+]ii = e, as required.
Statement 2. By the definition of A+ there exists l ∈ N such that [A⊗l]ij = [A+]ij. In fact, since
the eigenvalue of A is e, it follows from Lemma 1.17 that l ≤ n. From statement 1, for k large
enough, i ∈ Vc(A) and j ∈ {1, . . . , n}, we then have
[A⊗(k+l)
]ij ≥ [A⊗k
]ii ⊗ [A⊗l
]ij = [A⊗l
]ij = [A+
]ij.
In addition, clearly we also have
[A+
]ij =
∞
m=1
[A⊗m
]ij ≥ [A⊗(k+l)
]ij ≥ [A+
]ij,
so by replacing k + l with k, it therefore follows that [A⊗k]ij = [A+]ij for all i ∈ Vc(A) and
j ∈ {1, . . . , n}, with k large enough. This is what we wanted to prove.
Statement 3. Following the same lines as in the proof of statement 2, we can also show that
[A⊗m]ij = [A+]ij for all i ∈ {1, . . . , n}, j ∈ Vc(A) and with m large enough. Together, take k
34](https://image.slidesharecdn.com/10d08b3f-967c-4123-bade-f0ef8267a94d-151012204735-lva1-app6891/85/Thesis_JR-39-320.jpg)
![and m large enough such that [A⊗k]il = [A+]il and [A⊗m]lj = [A+]lj for all l ∈ Vc(A). Then
[A⊗(k+m)
]ij ≥ [A⊗k
]il ⊗ [A⊗m
]lj = [A+
]il ⊗ [A+
]lj,
for all l ∈ Vc(A). By replacing k + m with k, it follows that for k large enough
[A⊗k
]ij ≥
l∈Vc(A)
[A+
]il ⊗ [A+
]lj.
Now let the maximal average weight of a non-critical circuit (i.e. a circuit not passing through
any vertex in Vc(A)) be δ. Then the weight of a path from j to i of length k + 1 in G(A) not
passing through any vertex in Vc(A) can be bounded above by [A+]ij + (k × δ) = [A+]ij ⊗ δ⊗k,
since such a path consists of an elementary path from j to i (whose weight is bounded above by
[A+]ij) and at most k non-critical circuits (whose weights are each bounded above by δ). Since
the maximal average circuit weight in G(A) is e we must have δ < e, and so for k large enough
[A+
]ij ⊗ δ⊗k
≤
l∈Vc(A)
[A+
]il ⊗ [A+
]lj.
Indeed, the right-hand side is fixed, while the left-hand side tends to ε as k → ∞. Hence for k
large enough we have that
[A⊗k
]ij =
l∈V(A)
[A+
]il ⊗ [A+
]lj =
l∈Vc(A)
[A+
]il ⊗ [A+
]lj,
for all i, j = 1, . . . , n.
We can now state and prove the main theorem of this section.
Theorem 1.49. Let A ∈ Rn×n
max be an irreducible matrix with unique eigenvalue λ and cyclicity
σ := σ(A). Then there exists N ∈ N such that
A⊗(k+σ)
= λ⊗σ
⊗ A⊗k
for all k ≥ N.
Proof. Consider the matrix B := (Aλ)⊗σ. Recall that σ is the cyclicity of the critical graph of
A, which is a multiple of the cyclicity of the communication graph G(A). By Corollary 1.44,
after a suitable relabelling of the vertices of G(A), matrix B is a block diagonal matrix with
square diagonal blocks whose communication graphs are strongly connected and have cyclicity
one. By Lemma 1.46 we have that the cyclicity of B is one, which implies that the cyclicity of
each of its diagonal blocks is one. Hence by applying Lemma 1.48 to each diagonal block, it
ultimately follows that there exists M ∈ N such that B⊗(l+1) = B⊗l for all l ≥ M. That is,
(Aλ)⊗σ
⊗(l+1)
= (Aλ)⊗σ
⊗l
,
35](https://image.slidesharecdn.com/10d08b3f-967c-4123-bade-f0ef8267a94d-151012204735-lva1-app6891/85/Thesis_JR-40-320.jpg)
![which can further be written as (Aλ)⊗(l×σ+σ) = (Aλ)⊗(l×σ), or
A⊗(l×σ+σ)
= λ⊗σ
⊗ A⊗(l×σ)
,
for all l ≥ M. Finally, note that A⊗(l×σ+j+σ) = λ⊗σ ⊗A⊗(l×σ+j) for any 0 ≤ j ≤ σ−1, implying
that for all k ≥ N := M × σ it follows that
A⊗(k+σ)
= λ⊗σ
⊗ A⊗k
,
as required.
Theorem 1.49 can be seen as the max-plus analogue of the Perron-Frobenius theorem in con-
ventional linear algebra. Strictly speaking it is the normalised matrix Aλ that exhibits periodic
behaviour, since the unique eigenvalue of Aλ is e = 0, and then A
⊗(k+σ)
λ = A⊗k
λ for k sufficiently
large. However, we use the term ‘periodic’ to describe the more general behaviour seen here.
Note that the cyclicity of A is the smallest possible length of such periodic behaviour (see [2] for
the proof of this). For our purposes, we now move on to applying this result to the recurrence
relations studied in Section 1.5.1.
Recall the form of the basic first-order recurrence relation
x(k + 1) = A ⊗ x(k), k ≥ 0, (1.20)
which has the solution
x(k) = A⊗k
⊗ x(0).
We can apply Theorem 1.49 in this context to give us that for k sufficiently large:
x(k + σ(A)) = A⊗(k+σ(A))
⊗ x(0)
= λ⊗σ(A)
⊗ A⊗k
⊗ x(0)
= λ⊗σ(A)
⊗ x(k).
That is, the solution x(k) is periodic with period σ(A). If we interpret k as a time index,
then also by Theorem 1.49, the solution enters this periodic regime after N =: t(A) time steps,
where we call t(A) is the transient time of A. In particular, if A has cyclicity equal to 1 then
x(k+1) = A⊗x(k) = λ⊗x(k) ∀k ≥ t(A), and so for k sufficiently large x(k) effectively becomes
an eigenvector of A. In other words, after t(A) time steps, x(k) behaves like an eigenvector, and
the effect of the initial condition x(0) has died out.
Note that the transient time of a matrix can be large even for systems of small dimension. For
example, the matrix A defined by
A =
−1 −N
e e
36](https://image.slidesharecdn.com/10d08b3f-967c-4123-bade-f0ef8267a94d-151012204735-lva1-app6891/85/Thesis_JR-41-320.jpg)
![where N ∈ {2, 3, . . . } has transient time t(A) = N, while its cyclicity is clearly 1.
Finally, we make some observations regarding the growth rate of the solution x(k). Note that
if we take x(0) = v in (1.20), where v is an eigenvector of A, then we immediately obtain that
for all j = 1, . . . , n:
lim
k→∞
xj(k)
k
= λ,
where λ is the unique eigenvalue of A. By applying Theorem 1.49 it should be clear that this
holds true for any initial value x(0) and not just for eigenvectors; indeed this result is proved
in [13]. We therefore say that the solution has an asymptotic growth rate of λ. Assuming
irreducibility, all recurrence relations over max-plus exhibit this behaviour, regardless of the
choice of the matrix A!
37](https://image.slidesharecdn.com/10d08b3f-967c-4123-bade-f0ef8267a94d-151012204735-lva1-app6891/85/Thesis_JR-42-320.jpg)
![Chapter 2
Petri Nets and Timed Event Graphs
2.1 A Motivating Example
The following example is adapted from chapter 1 of [2]. Consider a manufacturing system
consisting of three machines M1, M2 and M3, which produces three kinds of parts P1, P2 and
P3 according to different product mixes. The manufacturing process for each part is depicted
below.
M1 M2 M3
P2
P3
P1
Figure 2.1: Manufacturing Process for each part. Grey boxes represent the three machines;
arrows represent the routes that the different parts must take in their respective manufacture.
Processing times are different for each machine and each part, and are given in the following
table:
P1 P2 P3
M1 - 1 5
M2 3 2 3
M3 4 3 -
Table 2.1: Processing times for each part at each machine (arbitrary time units). Blank entries
correspond to combinations of machine & part that do not form part of the manufacturing
process.
Parts are carried through the manufacturing process on a limited number of pallets. We make
38](https://image.slidesharecdn.com/10d08b3f-967c-4123-bade-f0ef8267a94d-151012204735-lva1-app6891/85/Thesis_JR-43-320.jpg)
![the following assumptions:
1. Only one pallet is available for each part type.
2. Once production of a part is completed, it is removed from its respective pallet and the
pallet returns to the beginning of the production line.
3. There are no set-up times or traveling times between machines.
4. The sequencing of part types on the machines is fixed, and for M1 is (P2, P3), for M2
(P1, P2, P3) and for M3 (P1, P2).
Assumption (3) gives no loss of generality since if set-up times or traveling times did exist,
we could combine them with the processing time at the appropriate machine. Assumption (4)
means that machines have to wait for the appropriate part rather than starting work on any
part that arrives first (see below for an example). This may or may not be realistic; extensions
to the theory presented below in which this assumption is dropped are discussed in chapter 9
of [2].
We can model the time evolution of this system by considering the time that each machine starts
working on the k-th part of type i, for i = 1, 2, 3 and k ∈ N. There are seven combinations of
machines and parts, so we define x(k) = (x1(k), . . . , x7(k)) as follows:
Variable xi(k) Definition
x1(k) time that M1 starts working on the k-th unit of P2
x2(k) time that M1 starts working on the k-th unit of P3
x3(k) time that M2 starts working on the k-th unit of P1
x4(k) time that M2 starts working on the k-th unit of P2
x5(k) time that M2 starts working on the k-th unit of P3
x6(k) time that M3 starts working on the k-th unit of P1
x7(k) time that M3 starts working on the k-th unit of P2
Table 2.2: Definitions of each entry of the state vector x(k), for k ∈ N.
By examining the production process, work by each machine on the (k+1)-st part is constrained
in the following way:
x1(k + 1) ≥ max x7(k) + 3, x2(k) + 5
x2(k + 1) ≥ max x5(k) + 3, x1(k + 1) + 1
x3(k + 1) ≥ max x6(k) + 4, x5(k) + 3
x4(k + 1) ≥ max x3(k + 1) + 3, x1(k + 1) + 1
x5(k + 1) ≥ max x2(k + 1) + 5, x4(k + 1) + 2
x6(k + 1) ≥ max x3(k + 1) + 3, x7(k) + 3
x7(k + 1) ≥ max x6(k + 1) + 4, x4(k + 1) + 2
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![and let A(k), k ∈ Z, be the (|Q | × M) × (|Q | × M) matrix defined by the relation
A(k) :=
A(k + 1, k) A(k + 1, k − 1) . . . . . . A(k + 1, k + 1 − M)
E E . . . E E
E E
...
... E
...
... E E E
E . . . E E E
(2.14)
where E denotes the |Q | × |Q | matrix of ε. Finally, let v(k) be the (|Q | × M)-dimensional
vector
v(k) :=
v(k + 1)
ε
...
ε
.
Then we can state the following corollary, which gives the standard form of the basic autonomous
equation:
Corollary 2.20. The extended state space vector x(k) satisfies the (M × |Q |)-dimensional
first-order evolution equation
x(k + 1) = A(k) ⊗ x(k) ⊕ v(k), (2.15)
for k ∈ Z.
Proof. This technique is standard and the result follows immediately from conducting the ap-
propriate matrix multiplications.
Once again, in the particular case of a compatible initial condition, these equations read
x(k + 1) = A(k) ⊗ x(k)
provided one uses the appropriate continuation of xj(k) for k ≤ 0.
Note that we can associate the standard form basic autonomous equation given in Corollary
2.20 with an event graph which is once again equivalent to the original event graph, but this
time with initial marking equal to 1 everywhere. An algorithm for obtaining the derived graph
can be found in [2].
2.3.4 Behaviour of the Solution
Assume for a moment that the holding times of our event graph are constant and that we forgo
the state space reduction described above. We have that the evolution of a live autonomous
FIFO timed event graph (G, µ), supplemented with a compatible initial condition, is described
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![by the recurrence relation
x(k) =
M
m=0
A(m) ⊗ x(k − m)
and we have just shown that the same equation in standard form
x(k + 1) = A ⊗ x(k)
equivalently describes the evolution of the system, and can be associated with an event graph G
which is equivalent to G. Since we assume that G is strongly connected it can also be assumed
that G is strongly connected, provided unnecessary transitions (not involved in circuits) be
cancelled. Hence we can assume that A is irreducible. We can therefore apply Theorem 1.49 to
see that after a finite period of time, the solution will enter a periodic regime with period t(A)
and asymptotic growth rate λ, where λ is the unique eigenvalue of A. An interpretation of this
is that on average, each transition fires once every λ units of time, i.e. the long-run throughput
of each transition is 1/λ, thus achieving some sort of stationarity.
A nice result given in [2] is that the value λ can be obtained directly from the original event
graph G, provided we modify slightly our definition of path lengths and weights. To be specific,
in the case of an event graph we can think of places as edges with weight equal to their holding
time, and length equal to the number of tokens in their initial marking. Using this definition,
λ is then equal to the maximal average circuit weight in G.
The behaviour of the solution x(k) in the case of variable holding times is more difficult to
describe. It can be shown that if there are sufficiently long sub-intervals for which the holding
times are constant then the solution exhibits periodicity within each of these intervals. This
can be seen in our simple example to which we return below.
2.4 A Simple Example Revisited
2.4.1 General Solution
Consider the example given in Section 2.2.2, and suppose that we extend to the model to allow
for variable travel times between stations. We denote the travel time of the k-th train between
Sj and Sl by αlj(k). Similarly, wlj(k) denotes the remaining travel time of the k-th train
between Sj and Sl present at time 0 (and is equal to ε if no such train exists). We assume that
the initial marking is the same as that given in Figure 2.3. The aim of this section is to apply
the theory developed above to find a timetable describing the optimal departure times of trains
at each station.
Under the assumption that the holding times are such that the event graph is FIFO, according
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![Departure Times
S1 - 0 5 12 17 24 29 37 43 51 57
S2 - 0 7 12 19 24 31 37 45 51 59
S3 - 2 7 14 19 26 31 39 45 53 59
S4 2 4 9 16 21 28 33 41 47 55 61
Table 2.3: Optimal Timetable of the Train Network. Recall that synchronisation results in
departures from stations coinciding, so the times given here refer to departures from each
station in all directions (for example, at time 5 one train departs on the inner-city loop at S1,
a second departs for S3 and a third departs for S3).
Note that we have shifted the variable k by 1 and 2 for the third and fourth rows respectively
so that the departure times line up nicely. Specifically, the columns in Table 2.3 represent the
vectors (x1(k), x2(k), x3(k + 1), x4(k + 2)) for k = −1, 0, 1, . . . , 9. This results in a sequence
of departure times that is increasing (when read down and then across). This is purely for
presentational reasons - we opt to make the departure timetable easy to read for someone
travelling through the network starting at station S1.
Notice that as per Section 2.3.4, the solution settles into a temporary periodic regime for k ≤ 5,
with asymptotic growth rate λ1 = 6 and transient time 2. λ1 here corresponds to the maximal
average weight of all the circuits of the event graph given in Figure 2.3 (where we think of the
holding times as weights and the number of tokens in the initial marking as lengths; see Section
2.3.4). For k > 5, the solution enters a second periodic regime with asymptotic growth rate
λ2 = 7. λ2 once again corresponds to the maximal average weight of all the circuits of the
original event graph but with the holding times updated to reflect the adjusted travel times.
There is an additional transient period of 2 time units (k = 6 to 8) during which the solution is
temporarily aperiodic.
In both cases the period is equal to 2, which therefore must be the cyclicity of the matrix A (k)
for which the whole system is described by the evolution equation x(k + 1) = A (k) ⊗ x(k) (i.e.
if we had not reduced the state space). We know that this matrix must be irreducible (again
see the discussion in Section 2.3.4), but it is not immediate that the support of its critical graph
(i.e. the set of edges (i, j) ∈ Ec(A (k)) for which [A (k)]ji = ε) is invariant under changes in k.
In fact, in this case we can show that it is, which explains why the cyclicity does not change for
the second periodic regime.
2.5 The Non-autonomous Case
This section focuses on FIFO timed event graphs that are non-autonomous; that is, event graphs
that include source transitions. Whilst we require one or two additional concepts, a lot of the
theory is analogous to the autonomous case and we therefore omit proofs whenever appropriate.
Firstly, we define a new class of transitions known as input transitions.
Definition 2.21. An input transition consists of a source transition and a non-decreasing
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![Chapter 3
Stochastic Event Systems Over
Max-plus
3.1 Introduction & Stochastic Background
In this chapter we consider stochastic FIFO event graphs; that is, FIFO event graphs in which
the holding times and lag times are random variables rather than predetermined constants. We
are interested in the conditions under which these graphs enter a stationary regime. Most of the
material in this chapter is taken from [1], [2] and [12]. We begin by laying some foundations of
Ergodic theory and describing the statistical assumptions we will use throughout. The manner
in which we introduce the concepts below may be surprising - specifically, what we state as
definitions are actually theorems which follows from the classical definition of ergodicity. Whilst
the reader may be more familiar with the latter approach, we find that in the context of max-plus
stochastic event systems it is intuitively helpful to proceed in this way.
Definition 3.1. Let (Ω, F, P) be a probability space. The mapping θ : Ω → Ω is a shift operator
on (Ω, F, P) if it is bijective and measurable, and if it is such that P is left invariant by θ, namely
E[f] = E[f ◦ θ] for all measurable and integrable functions f : Ω → R (where E represents the
expectation w.r.t. P).
Definitions 3.2. Let θ be a shift operator on (Ω, F, P).
(i) A sequence of R-valued random variables a(k, ω) k∈Z defined on (Ω, F, P) is θ-stationary
if a(k, ω)
d
= a(0; θk(ω)) ∀k ≥ 0, where θk is the composition of θ by itself k times.
(ii) θ is ergodic if the almost sure (a.s.) limit
lim
k→∞
1
k
k
l=1
f ◦ θ l
(ω) = E[f]
holds for all measurable and integrable functions f : Ω → R.
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![A simple example of an ergodic sequence of random variables is a sequence of independent
identically distributed (i.i.d.) random variables, and this will often be the obvious choice for
the sequence of holding times in the modelling of real-life systems such as rail networks.
Notation. Recall from Definition 1.3 that for a, b ∈ Rmax, we say a ≤ b if a ⊕ b = b. In this
chapter we introduce the notation a ∧ b to mean the greatest lower bound of a and b, i.e.
a ≤ b ⇐⇒ a ⊕ b = b ⇐⇒ a ∧ b = a. (3.1)
The ∧ operation is associative, commutative and idempotent but it does not have an identity
element, and does not always distribute over ⊗. For our purposes, we can think of a∧b to mean
min {a, b}.
We must take extra care to ensure that some fundamental concepts of probability are applicable
in the max-plus setting. For example, if X is a random variable taking values in Rmax defined
on a probability space (Ω, F, P) then it may take the value ε with positive probability, which
confuses the idea of integrability. We therefore call X ∈ Rmax integrable if X ⊕ e and
X ∧ e are both integrable, and if E[X ⊕ e] is finite. The expected value of X is then given by
E[X] = E[X ⊕e] ⊗ E[X ∧e]. Moreover, a random matrix A ∈ Rn×m
max (i.e. a matrix whose entries
are random variables taking values in Rmax) is called integrable if the entries aij are integrable
∀i, j, and its expected value is given by the matrix E[A] with entries E[A] ij = E[aij].
3.2 Statistical Assumptions
In this chapter we consider the max-plus evolution equation
x(k + 1) = A(k) ⊗ x(k), k ∈ Z (3.2)
with initial condition x(0) = x0. We will often stress the dependence of x(k) on the initial
condition by writing x(k; x0). We assume that A(k) k∈Z is a sequence of random matrices
defined on a common probability space (Ω, F, P) endowed with an ergodic shift operator θ, and
that x0 is also a random variable defined on this space. If we let
A := A(0) (3.3)
then θ-stationarity gives us that A(k)
d
= A ◦ θ k ∀k ∈ Z. We assume in addition that each entry
of A is either a.s. equal to ε or non-negative and integrable, and that each diagonal of A is
non-negative.
Throughout the rest of this chapter, what we have described above will be known as the SENI
framework (stationary, ergodic, non-negative and integrable). Note that the assumptions of the
SENI framework mean that the sequence of matrices A(k) has fixed support (i.e. the topologies
of the communication graphs G(A(k)) are non-random and do not vary with k). A formal
definition is given below.
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![Definition 3.3. The sequence of matrices A(k) ∈ Rn×n
max has fixed support if the probability that
[A(k)]ij equals ε is either 0 or 1 and does not depend on k.
The remainder of this section is devoted to showing that both autonomous and non-autonomous
FIFO stochastic event graphs are described by an evolution equation which falls into the SENI
framework.
The Autonomous Case By the remark that follows Corollary 2.20, for an autonomous FIFO
timed event graph with a compatible initial condition, the extended state space vector x(k)
satisfies the evolution equation
x(k) = A(k) ⊗ x(k)
which is of the form (3.2). Suppose in addition that the holding times αi(k) and the lag times
wi(k), i = 1, . . . , |P|, are random variables defined on a common probability space endowed
with an ergodic shift operator θ, and that the sequence αi(k) k∈Z is θ-stationary. Then it is
easily verified that the matrices A(k) satisfy the θ-stationarity property and that each entry of
A := A(0) is either a.s. equal to ε or non-negative and integrable. Furthermore, by the FIFO
assumption we have that xj(k + 1) ≥ xj(k) ∀k, and therefore Ajj(k) ≥ e for all j. Therefore,
under the appropriate statistical assumptions, any autonomous FIFO stochastic event graph
with compatible initial condition falls into the SENI framework.
The Non-autonomous Case By the remark that follows Corollary 2.26, the evolution of a non-
autonomous FIFO timed event graph with a compatible initial condition is described by the
equation
x(k) = A(k) ⊗ x(k)
which is also of the form (3.2). If the holding times αi(k) and the inter-input times Ujj(k)
are sequences of non-negative and integrable random variables satisfying θ-stationarity, then it
is again straightforward to check that the matrices A(k) are also θ-stationary. The additional
conditions mentioned above also follow in the same way as the autonomous case. Therefore the
SENI framework also covers the non-autonomous case, provided we make additional assumptions
on the inter-input times.
3.3 Asymptotic Firing Rates
In Section 2.3.4 we showed that in the case of constant holding times, any strongly connected
timed event graph will eventually reach a periodic regime in which each transition fires every
λ units of time (where λ is a constant determined by the event graph in question). The aim
of this section is to develop a corresponding theory for stochastic event graphs, i.e. to examine
whether the firing rates settle into a steady state distribution after a long period of time. We
will consider the case when the matrix A defined in (3.3) (herein assumed to be of dimension
n × n) is both a.s. irreducible and a.s. reducible, which is a well-defined distinction owing to
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![the assumption of fixed support within the SENI framework. This corresponds to stochastic
event graphs which are a.s. strongly connected and a.s. not strongly connected respectively.
We begin with some new notation.
Notation. For A ∈ Rn×m
max , let
|A|⊕
:=
n
i=1
m
j=1
aij
and
|A|∧
:=
n
i=1
m
j=1
aij.
Thus |A|⊕ gives the maximal entry of A and |A|∧ gives the minimal entry of A.
Lemma 3.4. For all pairs of matrices A, B ∈ Rn×m
max such that the product A ⊗ B is well
defined, we have
|A ⊗ B|⊕ ≤ |A|⊕ ⊗ |B|⊕
|A ⊗ B|⊕ ≥ |A|∧ ⊗ |B|⊕
|A ⊗ B|⊕ ≥ |A|⊕ ⊗ |B|∧
and
|A ⊗ B|∧ ≥ |A|∧ ⊗ |B|∧
|A ⊗ B|∧ ≤ |A|∧ ⊗ |B|⊕
|A ⊗ B|∧ ≤ |A|⊕ ⊗ |B|∧
Proof. Since Aik ≤ |A|⊕ for all i, k,
i,j k
Aik ⊗ Bkj ≤ |A|⊕ ⊗
j,k
Bkj
= |A|⊕ ⊗ |B|⊕
The proofs of the other formulae follow similarly.
We now recall an important theorem from Ergodic theory known as Kingman’s Theorem on
subadditive ergodic processes. This will be used in the material below. We do not offer a proof
here; readers are referred to [14] for the full working.
Theorem 3.5 (Kingman’s subadditive ergodic theorem). Let ξm,n, m > n ∈ Z, be an integrable
random process defined on the probability space (Ω, F, P) such that
ξm,m+k
d
= ξ0,k ◦ θm
∀m ∈ Z, k > 0 (stationarity)
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![and
ξm,n ≤ ξm,k + ξk,n a.s. ∀ m < k < n (subadditivity).
Assume in addition that there exists a positive constant c such that E[ξ0,k] ≥ −ck for all k > 0.
Then there exists a constant γ such that the following two equations hold:
lim
k→∞
ξ0,k
k
= γ a.s., lim
k→∞
E[ξ0,k]
k
= γ = inf
k
E[ξ0,k]
k
.
We now able to begin our task of characterising the asymptotic behaviour of the firing rates in
a stochastic event graph.
Theorem 3.6. Within the SENI framework, there exists a non-negative constant a ∈ Rmax
(known as the maximal Lyapunov exponent) such that, for all finite initial conditions x0, the
limit
lim
k→∞
|x(k; x0)|⊗1/k
⊕
= a (3.4)
holds almost surely. If the initial condition is integrable, in addition we have
lim
k→∞
E |x(k; x0)|⊗1/k
⊕
= lim
k→∞
E |x(k; x0)|⊕
⊗1/k
= a. (3.5)
Proof. Using the integrability assumptions of the SENI framework, we obtain by induction that
|x(k; e)|⊕ is integrable for all k ≥ 0. We therefore have
e ≤ E |x(k; e)|⊕ < ∞ ∀k ≥ 0.
Now let
ξm,m+k = |x(k; e)|⊕ ◦ θm
, m ∈ Z, k ≥ 0.
Since
|x(k, e)|⊕ = |A ◦ θk−1
⊗ · · · ⊗ A ⊗ e|⊕ = |A ◦ θk−1
⊗ · · · ⊗ A|⊕ ,
we obtain from Lemma 3.4 that for any k ≥ 1 and for all 0 ≤ p ≤ k:
|A ◦ θk−1
⊗ · · · ⊗ A ◦ θp
⊗ A ◦ θp−1
⊗ · · · ⊗ A|⊕ ◦ θm
≤ |A ◦ θk−1
⊗ · · · ⊗ A ◦ θp
|⊕ ◦ θm
⊗ |A ◦ θp−1
⊗ · · · ⊗ A|⊕ ◦ θm
;
that is, ξm,m+k ≤ ξm,m+p +ξm+p,m+k. Thus ξm,m+k is a non-negative and integrable subadditive
process. Using Theorem 3.5, we obtain
lim
k→∞
(ξ0,k)⊗1/k
= lim
k→∞
E (ξ0,k)⊗1/k
= a a.s.,
for some constant a < ∞, which concludes the proof for |x(k; e)|⊕ . Now, using that x(k) =
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![A(k − 1) ⊗ · · · ⊗ A(0) ⊗ x0, by Lemma 3.4 we have that for all finite initial conditions x0 and
∀k ≥ 0:
|x(k; e)|⊕ ⊗ |x0|∧ ≤ |x(k; x0)|⊕ ≤ |x(k; e)|⊕ ⊗ |x0|⊕ .
Therefore
|x(k; e)|⊗1/k
⊕
⊗ |x0|⊗1/k
∧
≤ |x(k; x0)|⊗1/k
⊕
≤ |x(k; e)|⊗1/k
⊕
⊗ |x0|⊗1/k
⊕
(3.6)
for all k ≥ 0. We then immediately obtain property (3.4) by letting k tend to ∞. Similarly, if x0
is integrable then we can take expectations in (3.6) and use the fact that limk→∞ E[(ξ0,k)⊗1/k]
= a to obtain (3.5).
Note that we have only shown the existence of the constant a here, whereas in the deterministic
case we were able to specify its value. Computing exactly the maximal Lyapunov exponent of
products of matrices over the max-plus semiring is a long-standing problem, and only for a few
special cases are explicit formulae known [12]. However, it can be shown (see [2]) that if the
communication graph G(A) contains at least one circuit with two vertices i0, j0 in this circuit
such that E[Aj0i0 (k)] > e, then the maximal Lyapunov exponent a is strictly positive. This
corresponds to having a circuit with at least one place with positive mean holding time in the
stochastic event graph.
Theorem 3.6 tells us that |x(k)|⊕ grows like a⊗k. Using the same techniques we could prove an
analogous result for the growth rate of |x(k)|∧ which is also a constant b, called the minimal
Lyapunov exponent, and with b ≤ a. In the following section we examine the growth rate of the
individual state variables xj(k) within the SENI framework. In the irreducible case we show
that all state variables have the same asymptotic growth rate equal to the maximal Lyapunov
exponent defined above, and a similar result holds in the reducible case. This matches our
intuition that the speed with which a system operates is determined by its slowest component.
3.3.1 The Strongly Connected Case
We assume that A is a.s. irreducible. Recall that within the SENI framework we also assume
that the diagonal entries of A are a.s. non-negative. It is then easy to see that the matrices
G(k) := A(k + n − 1) ⊗ A(k + n − 2) ⊗ · · · ⊗ A(k), k ∈ Z
are such that Gij(k) ≥ e for all i, j = 1, . . . , n. This allows us to state the following corollary.
Corollary 3.7. Within the SENI framework, if the matrix A is irreducible, then for all finite
initial conditions x0 and for all j = 1, . . . , n, we have that
lim
k→∞
xj(k; x0)
⊗1/k
= a a.s.,
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![where a is the maximal Lyapunov exponent of Theorem 3.6. If x0 is integrable, we also have
lim
k→∞
E xj(k; x0)
⊗1/k
= a.
Proof. From the remark above we obtain that xj(k; x0) ≥ xi(k − n; x0) for all i, j = 1, . . . , n
and k > n. This gives that
|x(k − n; x0)|⊕ ≤ xj(k; x0) ≤ |x(k; x0)|⊕ ∀j = 1, . . . , n
and the result follows by letting k tend to ∞ and applying Theorem 3.6. In the integrable case,
the proof of the convergence of the expectations in then immediate.
Corollary 3.7 tells us that in the case of a strongly connected stochastic event graph, all tran-
sitions have the same asymptotic firing rate. In this case the maximal Lyapunov exponent a is
also called the cycle time of the stochastic event graph, and its inverse a−1 is often referred to
as the throughput. By viewing the deterministic case of Chapter 2 as a special case, if A(k) = A
for all k ∈ N then the maximal Lyapunov exponent is simply the unique eigenvalue λ of A.
3.3.2 The General Case
If G(A) is not strongly connected then it can be decomposed into a number N(A) > 1 of
maximal strongly connected subgraphs (m.s.c.s.’s), in the same way as we did in Section 1.4 for
the critical graph Gc(A). Note that due to the assumptions of the SENI framework, the number
N(A) of m.s.c.s.’s and their topologies are non-random. Let Gr(A) = (Vr(A), Er(A)) denote the
r-th m.s.c.s. of G(A) and let jr := min j ∈ Vr(A) be the smallest numbered vertex in the r-th
m.s.c.s. We call {j1, . . . , jN(A)} a set of representative vertices of the m.s.c.s’s of G(A). We now
introduce some new notation and a simple definition.
Notation. For i ∈ V(A), let [i] denote the subset of vertices of the m.s.c.s. containing vertex i.
We then let x(i)(k) denote the subvector of x(k) associated with the vertex set [i]; that is, the
vector x(k) restricted to the entries corresponding to vertices in the m.s.c.s. containing vertex
i. Similarly, A(i)(j)(k) denotes the block extracted from A(k) by keeping rows associated with
the vertex set [i] and the columns associated with the vertex set [j].
Consistent with the notation used in Section 2.2, we let π+(i) be the set of ascendants of vertex
i; that is, the set of all vertices j such that a path exists from j to i (which by convention does
not include i itself). In addition, we let π∗(i) := {i} ∪ π+(i). We then let x(<i)(k) denote the
subvector of x(k) associated with the vertex set j∈π+(i)[j] (so x(<i)(k) denotes the restriction
of x(k) to the entries corresponding to vertices in the m.s.c.s’s of all the ascendants of vertex i,
not including the m.s.c.s. of i itself), and x(≤i)(k) denote the subvector of x(k) associated with
the vertex set j∈π∗(i)[j]. Finally, the matrices A(≤i)(≤j), A(<i)(<j) etc. are defined in the same
way.
Definition 3.8. The reduced graph of G(A) is the graph G(A) = (V(A), E(A)), with vertex set
V(A) := {j1, . . . , jN(A)} (one vertex per m.s.c.s.) and where (j1, j2) ∈ E(A) iff (k, l) ∈ E(A)
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![Definition 3.12. A queuing system satisfies condition (A) if it only has one class of items, no
state-dependent service times, all queues are FCFS with blocking after service, and blocking is
resolved according to FBFU.
Note that a queuing system satisfying condition (A) is not necessarily max-plus linear (and so
cannot necessarily be represented by a stochastic event graph). However, as was our aim, it
turns out that (A) does specify the class of queuing system for which we can derive necessary
and sufficient conditions for max-plus linearity. In other words, reducing our analysis to queuing
systems of this type imposes no restriction in the context of stochastic event graphs.
The main reason for requiring the assumption (A) is that in a max-plus linear model we have no
information about the physical state of the system (in terms of queue lengths), and this means
that any dependence of the service times on the physical state cannot be incorporated. The
intuition should be reasonably clear and we choose to omit the details here; a full discussion
can be found in [12].
It is possible to derive a general recursive formula (in max-plus) describing the departure times
from queues in a queuing system satisfying (A). Once again, the full working can be found
in [12]. Having done this, it remains to identify conditions under which this formula can be
reduced to the form of the basic non-autonomous equation of Theorem 2.23 (or its autonomous
counterpart), which, as we have remarked above, will mean that the queuing system is max-plus
linear, and can be represented by a stochastic event graph. It boils down to whether the general
recursive formula is of finite order. A simple set of sufficient conditions for max-plus linearity
are summarised in the following result:
Theorem 3.13. A queuing system satisfying (A) is max-plus linear if it admits no internal
overtaking, and if all resequencing queues have a finite buffer.
The proof is not difficult but of considerable length, and can be found in [12].
3.4.2 Example: The G/G/1 Queue
The event graph in Figure 3.1 below represents the FIFO G/G/1 queue with an infinite buffer.
The G/G part states that both the interarrival times and holding times form general stationary
sequences. The 1 indicates that there is a single server (we assume that there is an infinite
buffer in front of the server). The input u1 into transition q1 features the external input stream
of items, p1 is the infinite buffer which stores the items to be served, q2 represents the single
server and the holding times in p2 represent the service times.
76](https://image.slidesharecdn.com/10d08b3f-967c-4123-bade-f0ef8267a94d-151012204735-lva1-app6891/85/Thesis_JR-81-320.jpg)
![q1 p1 q2
p2
Figure 3.1: Stochastic event graph representation of the G/G/1 queue with an infinite buffer.
Items enter the system via q1 and are stored in p1. Departures occur whenever q2 fires since this
will remove one token from the system. Note that p2 will never contain more than one token;
i.e. there can never be more than one person in service at any one time.
Notice that transition q2 has two upstream places but only one downstream place, so whenever
q2 fires (i.e. whenever the service of an item is completed) the total number of tokens in the
system reduces by one. Thus departures from the queue are modelled without the use of a sink
transition. By construction there will always be one token in place p2 (corresponding to the
item in service), whereas p1 could hold any number of tokens. The holding times in p1 are taken
to be e, so in particular they are non-random.
Recalling the notation of Chapter 2, notice that M = 1 and that both |Q | and |I| are equal
to 1. Hence the matrices A(k, k − 1) and B(k, k − 1) in the basic non-autonomous equation of
Theorem 2.23 are one-dimensional:
A(k, k − 1) = α(k) , B(k, k − 1) = (e).
Let A(k) = α(k+1). If the initial condition is weakly compatible (which will happen if u(1) ≥ e,
and the lag times w1, w2 satisfy w1 ≤ e, w2 ≤ α(1) and w1 ⊕ w2 ≥ e), then it is also compatible
(since each transition is followed by at most one place with non-zero initial marking), so the
basic non-autonomous equation gives us that the firing times satisfy
x(k + 1) = A(k) ⊗ x(k) ⊕ u(k), k ≥ 0,
provided we now take x(0) = w2 ⊗ α(1)⊗−1 and u(0) = w1.
3.4.3 Stability Analysis of Waiting Times
In this final section we present a celebrated result in the area of stochastic max-plus theory.
Recall that if in a G/G/1 queue the expected interarrival time is larger than the expected service
time, then the sequence of waiting times converges, independent of the initial condition, to a
unique stationary regime (see [16]). Our aim is to generalise this result to the waiting times in
open max-plus linear queuing systems.
Consider an open queuing system (i.e. one in which items can arrive from the outside world)
with one input transition and J queues, satisfying the conditions of Theorem 3.13. It is therefore
max-plus linear, and so the vector of departure times from each queue, denoted x(k), satisfies
77](https://image.slidesharecdn.com/10d08b3f-967c-4123-bade-f0ef8267a94d-151012204735-lva1-app6891/85/Thesis_JR-82-320.jpg)
![Letting k tend to ∞ in (3.24) shows that
M = A ◦ θ−1
⊗ C(U) ⊗ M ⊕ B ◦ θ−1
;
that is, M is the stationary solution of (3.17), and it remains to show uniqueness. To do this,
let M(k, w) denote the vector of sojourn times at time 0 provided that the sequence is started
at −k and with initial vector w ∈ RJ
max. This gives
M(k, w) =
k
i=1
A(−i) ⊗ C(U(−i + 1)) ⊗ w ⊕
k−1
j=0
C(−u(−j)) ⊗ D(j).
Assuming that w has at least one finite element, we have that |w|⊕ < ∞. Following the same
argument as in step 2 above, we obtain that for ν > a:
lim
k→∞
k
i=1
A(−i) ⊗ C(U(−i + 1)) ⊗ w
⊕
= ε a.s.,
and so
lim
k→∞
k
i=1
A(−i) ⊗ C(U(−i + 1)) ⊗ w ⊕
k−1
j=0
C(−u(−j)) ⊗ D(j) = M a.s.
Thus for any initial value w, M(k, w) has the same limit as M(k), and uniqueness has been
established.
If W(k, w) is the vector of k-th sojourn times initiated at w, then M(k, w) and W(k, w) are
equal in distribution, and so M is the unique (weak) limit of W(k, w) , independent of the w
chosen. Finally, we refer to [12] for the proof that W(k) converges with strong coupling to
the stationary regime M. We can summarise this result in the following theorem:
Theorem 3.14. Assume we are working under the (modified) assumptions of the SENI frame-
work, and denote the maximal Lyapunov exponent of {A(k)} by a. If ν > a, then the sequence
W(k) of sojourn times at each queue converges with strong coupling to a unique stationary
regime M (defined in (3.23)).
This result is essentially an example of a max-plus multiplicative ergodic theorem. Whilst we
have worked in the context of queuing systems, the same theory applies for general stochastic
event graphs. From what we have proved above, it is straightforward to show that other
increments of the form xj(k + 1) − xi(k), for i, j ∈ {1, . . . , J}, also couple with a stationary
and ergodic process [1]. The autonomous case turns out to be considerably more involved and
is dealt with in [2].
82](https://image.slidesharecdn.com/10d08b3f-967c-4123-bade-f0ef8267a94d-151012204735-lva1-app6891/85/Thesis_JR-87-320.jpg)
![Bibliography
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![[15] G. L. Litvinov, V. P. Maslov, A. G. Kushner, and S. N. Sergeev. Tropical and Idempotent
Mathematics. Institute for Information Transmission Problems of RAS, Moscow, Russia,
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[16] R. Loynes. The stability of queues with non-independent inter-arrival and service times.
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Thesis_JR
- 1. Max-Plus Theory andApplications Jeremy Rolph August 10, 2015
- 2. Abstract In this thesiswe consider the ‘max-plus’ algebra; that is, the set Rmax = R ∪ {−∞} endowed with the operations a ⊕ b = max{a, b} and a ⊗ b = a + b. It is shown that Rmax has the structure of a semiring with several additional useful properties. We introduce the idea of matrices over the max-plus semiring and develop max-plus variants of several familiar concepts from classical linear algebra; most notably the theory of eigenvalues and eigenvectors. In Chapter 2 we introduce the theory of event graphs which are used to model dynamical systems which admit a degree of synchronisation such as rail networks or automated manufacturing processes. We use the theory of max-plus algebra developed in Chapter 1 to derive results concerning the time evolution of such systems and also consider their long-term behaviour. Finally, in Chapter 3 we consider event graphs in which the timed elements form sequences of random variables. We look for steady state distributions and conditions for their existence, and attempt to characterise the asymptotic behaviour of the event timings concerned. We conclude by exploring how we can represent certain types of queuing systems by stochastic event graphs and present a key theorem regarding the stability of their waiting times. i
- 3. Contents Abstract i Table ofContents ii 0 Introduction 1 1 Max-Plus Algebra 3 1.1 The Max-Plus Semiring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.1.1 Basic Definitions and Properties . . . . . . . . . . . . . . . . . . . . . . . 3 1.1.2 Other Algebraic Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.2 Vectors and Matrices over Rmax . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.2.1 Definitions and Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.2.2 Matrix Inversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.2.3 Determinants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.3 Graph-theoretic Interpretations in Max-Plus . . . . . . . . . . . . . . . . . . . . . 11 1.4 Spectral Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.4.1 Eigenvalues and Eigenvectors . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.4.2 The Eigenspace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 1.4.3 A Worked Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 1.5 Recurrence Relations & Periodicity . . . . . . . . . . . . . . . . . . . . . . . . . . 28 1.5.1 Solving Max-Plus Recurrence Relations . . . . . . . . . . . . . . . . . . . 28 ii
- 4. 1.5.2 Limiting Behaviour. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 2 Petri Nets and Timed Event Graphs 38 2.1 A Motivating Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 2.2 Preliminaries of Event Graph Theory . . . . . . . . . . . . . . . . . . . . . . . . . 41 2.2.1 Definitions and Set-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 2.2.2 A Simple Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 2.3 The Basic Autonomous Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 2.3.1 Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 2.3.2 Extensions to the Initial Condition . . . . . . . . . . . . . . . . . . . . . . 49 2.3.3 Solving the Basic Autonomous Equation . . . . . . . . . . . . . . . . . . . 51 2.3.4 Behaviour of the Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 2.4 A Simple Example Revisited . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 2.4.1 General Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 2.4.2 An Optimal Timetable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 2.5 The Non-autonomous Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 3 Stochastic Event Systems Over Max-plus 64 3.1 Introduction & Stochastic Background . . . . . . . . . . . . . . . . . . . . . . . . 64 3.2 Statistical Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 3.3 Asymptotic Firing Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 3.3.1 The Strongly Connected Case . . . . . . . . . . . . . . . . . . . . . . . . . 69 3.3.2 The General Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 3.4 Queuing Systems and Timed Event Graphs . . . . . . . . . . . . . . . . . . . . . 74 3.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 3.4.2 Example: The G/G/1 Queue . . . . . . . . . . . . . . . . . . . . . . . . . 76 3.4.3 Stability Analysis of Waiting Times . . . . . . . . . . . . . . . . . . . . . 77 iii
- 5.
- 6. Chapter 0 Introduction Exotic semiringssuch as (R ∪ {−∞}, max, +) and (R ∪ {+∞}, min, +) have been studied at length since the 1950s, beginning primarily in the area of operational research. Nowadays the term ‘tropical mathematics’ is often used to describe their study, though this term originally referred to one particular discrete version of the max-plus algebra introduced by I. Simon in 1988 [15]. Their applications span a wide range of fields including optimisation & control, mathematical physics, algebraic geometry, dynamic programming and mathematical biology [10, 15]. In particular, the study of such algebras in relation to discrete event system theory (both deterministic and stochastic), graph theory, Markov decision processes, asymptotic analysis and language theory has lead to some significant progress in these areas over the last 30 years [8]. Many of the concepts developed in conventional linear algebra have been ‘translated’ into the world of max-plus, including solutions to linear and non-linear systems (both analytical and numerical), linear dependence and independence, determinants, eigenvalues and eigenvectors [9]. In 1979 Cuninghame-Green authored the first comprehensive unified account of these results entitled “Minimax Algebra” [7], building on many papers published over the preceding 20 years from various disciplines within mathematics, economics and computer science. As recently as 2006, Heidergott, Olsder and Woude published what they consider the first ‘textbook’ in the area of max-plus algebra [13], and many of the ideas explored below can be found in this publication. In the first chapter of this thesis, we aim to give an overview of max-plus linear algebra and to build the necessary groundwork required for the applications discussed in the chapters that follow. In particular, we present two celebrated theorems in the area of max-plus theory. The first, which can be found in [7], concerns spectral theory and says that under mild conditions, a matrix over the max-plus algebra has a unique eigenvalue with a simple graph-theoretic interpretation. The second, originally proved by M. Viot in 1983 [2, 6], relates to the asymptotic behaviour of sequential powers of max-plus matrices, which turns out to be essentially periodic and has great implications for the material explored in Chapters 2 & 3. In chapter 2 we introduce the concept of timed Petri nets & event graphs. For a thorough 1
- 7. discussion on thescope of their application readers are referred to [18]; in this thesis we fo- cus solely on their use in the modelling of the time behaviour of a class of dynamic systems known as ‘discrete event dynamic systems’. In simple terms, these are systems in which a finite number of resources (e.g. processors or machines) are shared by several users (e.g. packets or manufactured objects) which all contribute to the achievement of some common goal (e.g. a parallel computation or the assembly of a product) [2]. We will see that under certain conditions these systems, while highly non-linear in the conventional sense, can be ‘linearised’ by using the max-plus algebra. This observation, first made in [5], is of vital importance and constitutes one of the main reasons for the continued study of max-plus algebra today. The main content of Chapter 2 concerns the ‘basic autonomous equation’ which governs the time evolution of discrete event systems, and the steps towards its solution. We are then able to apply some ideas from Chapter 1 to explore the long-term behaviour of such systems. Chapter 3 concerns stochastic event graphs, which can be thought of as a natural extension to the concepts introduced in Chapter 2. As the name suggests, we now assume a degree of randomness in the event timings of the systems we are trying to model. Amongst other things, stochastic event graphs can be used to model many types of queuing systems [3], the most simple of which being the G/G/1 queue. We introduce several key ‘first order’ theorems which establish the nature of stationary regimes in terms of the inverse throughput, and explore the conditions under which such regimes are reached. We end by presenting a ‘second order’ theorem concerning the stability of inter-event timings (for example, waiting times) in the context of queuing systems. 2
- 8. Chapter 1 Max-Plus Algebra 1.1The Max-Plus Semiring 1.1.1 Basic Definitions and Properties In this thesis we work exclusively with the max-plus algebra (Rmax, ⊕, ⊗), where Rmax = R ∪ {−∞}, and for a, b ∈ Rmax: a ⊕ b := max{a, b} a ⊗ b := a + b We begin by examining its algebraic structure, and we will then move on to vectors and matrices over Rmax. We start by defining the term semiring. Definition 1.1. A semiring is a triple (R, +, ×) where R is a non-empty set and +, × are binary operations on R (referred to as addition and multiplication respectively) such that (i) (R, +) is commutative and associative, with zero element εR: (a) a + b = b + a (b) (a + b) + c = a + (b + c) (c) εR + a = a + εR = a (ii) (R, ×) is associative, with unit element eR: (a) (a × b) × c = a × (b × c) (b) eR × a = a × eR = a (iii) Multiplication distributes over addition: (a) a × (b + c) = (a × b) + (a × c) (b) (a + b) × c = (a × c) + (b × c) (iv) Multiplication by εR annihilates R: 3
- 9. (a) εR ×a = a × εR = εR Note that the final axiom is not required in the definition of a standard ring since it follows from the others, but it is needed here. As the title of this section suggests, the max-plus algebra is a semiring with additive identity ε := −∞ and multiplicative identity e := 0. It is straightforward to verify that all the axioms of Definition 1.1 hold in the case of (Rmax, ⊕, ⊗). For example, the first distributive law holds since a ⊗ (b ⊕ c) = a + max{b, c} = max{a + b, a + c} = (a ⊗ b) ⊕ (a ⊗ c) and the others follow similarly. For the sake of simplicity we will write Rmax for (Rmax, ⊕, ⊗) when the context is clear. Below we list three additional algebraic properties of Rmax which do not form part of the definition of a semiring: (i) Commutativity of ⊗: ∀a, b ∈ Rmax : a ⊗ b = b ⊗ a (ii) Existence of multiplicative inverses: ∀a ∈ Rmax{ε} ∃ b ∈ Rmax such that a ⊗ b = e (iii) Idempotency of ⊕: ∀a ∈ Rmax : a ⊕ a = a The first two properties follow directly from the fact that (R, +) forms an abelian group, and the third property is easily proved: a ⊕ a = max{a, a} = a. Properties (i) and (ii) mean that we could refer to (Rmax, ⊕, ⊗) as a semifield (i.e. a field without additive inverses), though this term can be ambiguous and is seldom used in mathematical literature. Note also that in general, any semiring in which addition is idempotent we call an idempotent semiring. The term dioid (originating from the phrase double monoid) was introduced by Baccelli et al. in 1992 to mean idempotent semiring [2], but we do not use this word here. The crucial difference between a semiring and a ring in general is that an element of the former need not have an additive inverse. Note that this does not say that additive inverses can never exist - there may be a non-empty subset of R containing elements which do have additive inverses (which could be thought of as the additive analogue to the set of units in a standard ring). However, the following lemma immediately tells us that no elements of Rmax (apart from 4
- 10. ε) have additiveinverses. Lemma 1.2. Let (R, +, ×) be a semiring. If + is idempotent then additive inverses do not exist. Proof. Suppose that εR = a ∈ R has an additive inverse b. Then a + b = εR Adding a to both sides of the equation yields a + a + b = a + εR By idempotency of +, the left-hand side is equal to a + b, whereas the right-hand side is equal to a. Hence we have a + b = a which contradicts a + b = εR. Thus a does not have an additive inverse. 1.1.2 Other Algebraic Definitions For a ∈ Rmax, n ∈ N, define a⊗n := a ⊗ a ⊗ · · · ⊗ a n times Thus exponentiation in max-plus is equivalent to conventional multiplication a⊗n = n×a. Some of the laws of exponentiation are therefore different to what we are used to. For a, b ∈ Rmax, m, n ∈ N: (i) a⊗m ⊗ a⊗n = ma + na = (m + n)a = a⊗(m⊗n) (ii) (a⊗m)⊗n = (ma)⊗n = nma = a⊗(m⊗n) (iii) a⊗1 = 1a = a (iv) a⊗m ⊗ b⊗m = ma + mb = m(a + b) = (a ⊗ b)⊗m and we also adopt the natural conventions a⊗ε := ε and a⊗e := e. For negative exponents we can take a⊗−n := (a⊗n )⊗−1 where the outer exponent on the right-hand side denotes the max-plus multiplicative inverse, which was shown to exist in the previous section. Finally, we can extend the concept of ex- ponentiation in Rmax to non-integer exponents using conventional notation in the following 5
- 11. way: a⊗ n m := n m ×a which is well-defined, assuming m = ε. Next, we can equip the max-plus algebra with a natural order relation as follows: Definition 1.3. For a, b ∈ Rmax, we say a ≤ b if a ⊕ b = b. It is easily verified that the max-plus operations ⊕ and ⊗ preserve this order, i.e. ∀a, b, c ∈ Rmax, a ≤ b ⇒ a ⊕ c ≤ b ⊕ c and a ⊗ c ≤ b ⊗ c. Finally, infinite sums in max-plus are defined by i∈I xi := sup{xi : i ∈ I} for any possibly infinite (even uncountable) family {xi}i∈I of elements of Rmax, when the supremum exists. In general, we say that an idempotent semiring is complete if any such family has a supremum, and if the product distributes over infinite sums. The max-plus semiring Rmax is not complete (a complete idempotent semiring must have a maximal element), but it can be embedded in the complete semiring (Rmax, ⊕, ⊗), where Rmax := Rmax ∪ {+∞}. 1.2 Vectors and Matrices over Rmax 1.2.1 Definitions and Structure Let n, m ∈ N. We denote the set of n × m matrices over Rmax by Rn×m max . For i ∈ {1, . . . , n}, j ∈ {1, . . . , m}, the element of a matrix A ∈ Rn×m max in row i and column j is denoted by [A]ij, or simply aij for notational convenience. Thus A ∈ Rn×m max can be written as a11 a12 · · · a1m a21 a22 · · · a2m ... ... ... ... an1 an2 · · · anm where a11, . . . , anm ∈ Rmax. In a similar vein, the elements of Rn max := Rn×1 max are called max-plus vectors, and we write the i-th element of a vector x ∈ Rn max as [x]i, or simply xi. Typical concepts and operations from conventional algebra are defined for max-plus matrices in the usual way (replacing + and × with ⊕ and ⊗ respectively), as outlined in the following definitions. Definition 1.4. The n × n max-plus identity matrix, denoted En, is defined by [En]ij = 0 i = j ε i = j We will write E := En whenever the context is clear. 6
- 12. Definitions 1.5. (i)For A, B ∈ Rn×m max , their sum A ⊕ B is defined by [A ⊕ B]ij = aij ⊕ bij = max aij, bij (ii) For A ∈ Rn×k max and B ∈ Rk×m max , their product A ⊗ B is defined by [A ⊗ B]il = k j=1 (aij ⊗ bjl) = max j=1,...,k (aij + bjl) (iii) The transpose of a matrix A ∈ Rn×m max is denoted by A and is defined as usual by [A ]ij = [A]ji (iv) For A ∈ Rn×n max and k ∈ N, the k-th power of A, denoted A⊗k, is defined by A⊗k = A ⊗ A ⊗ · · · ⊗ A k times For k = 0, A⊗0 := En. (v) For A ∈ Rn×m max and α ∈ Rmax, α ⊗ A is defined by [α ⊗ A]ij = α ⊗ [A]ij We now look at a crucial result concerning the algebraic structure of square matrices over Rmax. Proposition 1.6. (Rn×n max , ⊕, ⊗) is an idempotent semiring with multiplicative identity En. Proof. The axioms of Definition 1.1 all follow from the semiring structure of Rmax, and are readily verified. For example, for A, B, C ∈ Rn×n max we have that [A ⊗ (B ⊕ C)]il = n j=1 (aij ⊗ (bjl ⊕ cjl)) = n j=1 (aij ⊗ bjl) ⊕ (aij ⊗ cjl) = n j=1 (aij ⊗ bjl) ⊕ n j=1 (aij ⊗ cjl) = [(A ⊗ B) ⊕ (A ⊗ C)]il and so A ⊗ (B ⊕ C) = (A ⊗ B) ⊕ (A ⊗ C). The other axioms follow similarly. Note that since addition in (Rn×n max , ⊕, ⊗) is idempotent, we can apply Lemma 1.2 once again to see that no element of Rn×n max has an additive inverse. However, unlike in Rmax, multiplication 7
- 13. of matrices overRmax is not commutative. For example 1 e ε −2 2 −1 3 ε = 3 e 1 ε = 3 2 4 3 = 2 −1 3 ε 1 e ε −2 Also unlike Rmax, matrices over Rmax do not necessarily have multiplicative inverses (i.e. they are not necessarily invertible). We explore this in the next section. 1.2.2 Matrix Inversion Definition 1.7. Let A, B ∈ Rn×n max . B is a right inverse of A if A ⊗ B = E, and B is a left inverse of A if B ⊗ A = E. Definition 1.8. A max-plus permutation matrix is a matrix A ∈ Rn×n max with each row and each column containing exactly one entry equal to e, with all other entries equal to ε. If σ : {1, . . . , n} → {1, . . . , n} is a permutation, the max plus permutation matrix Pσ is defined by [Pσ]ij := e i = σ(j) ε i = σ(j) As the name suggests, left multiplication by Pσ permutes the rows of a matrix: the i-th row of a matrix A ∈ Rn×n max will appear as the σ(i)-th row of Pσ ⊗ A. For example, if n = 2 and σ is defined by σ(1) = 2, σ(2) = 1: ε e e ε 1 2 3 4 = 3 4 1 2 Similarly, it is straightforward to see that right multiplication by Pσ permutes the columns of a matrix. Definition 1.9. A matrix A ∈ Rn×n max is diagonal if [A]ij = ε for all i = j. If a1, . . . , an ∈ Rmax{ε}, the diagonal matrix D(a1, . . . , an) is defined by [D(a1, . . . , an)]ij := ai i = j ε i = j Combining these two definitions, if σ is a permutation and a1, . . . , an ∈ Rmax {ε}, Pσ ⊗ D(a1, . . . , an) gives a matrix in which each row and each column contains exactly one finite entry. This class of matrices (sometimes referred to as generalised permutation matrices) in max-plus turns out to be of some significance, as the theorem below shows. Theorem 1.10. A matrix A ∈ Rn×n max has a right inverse if and only if A = Pσ ⊗ D(a1, . . . , an) for some permutation σ and a1, . . . , an ∈ Rmax{ε}. Proof. Suppose A = Pσ ⊗ D(a1, . . . , an) for some permutation σ and a1, . . . , an ∈ Rmax{ε}. 8
- 14. Recalling from Section1.1.1 that multiplicative inverses exist in Rmax, define B ∈ Rn×n max by [B]ij = [A]⊗−1 ji if [A]ji = ε ε otherwise Then for i, j = 1, . . . , n we have that [A ⊗ B]ij = max k=1,...,n aik ⊗ bkj = e j = i ε j = i Since if j = i, at least one of aik, bkj is equal to ε for each k = 1, . . . , n (since A only has one finite element per column and row). Thus A ⊗ B = E, and B is a right inverse of A. Conversely, suppose A has inverse B ∈ Rn×n max . For i, j = 1, . . . , n we have n k=1 [A]ik ⊗ [B]kj = [E]ij and therefore for each i = 1, . . . , n there is a (least) index c(i) (1 ≤ c(i) ≤ n) such that [A]ic(i) and [B]c(i)i are both finite, since [E]ii = e. Moreover we cannot have [A]hc(i) finite with h = i, since then [A ⊗ B]hi ≥ [A]hc(i) ⊗ [B]c(i)i > ε = [E]hi which contradicts our assumption that B is a right inverse of A. It follows that the mapping i → c(i) is a bijection, i.e. each column of A is labelled c(i) for some i and contains exactly one finite element, and each row of A contains exactly one finite element. That is, A = Pσ ⊗D(a1, . . . , an) for some permutation σ and a1, . . . , an ∈ Rmax{ε}. Theorem 1.11. For A, B ∈ Rn×n max , A ⊗ B = E if and only if B ⊗ A = E (i.e. right and left inverses are equivalent), and A uniquely determines B. Proof. Suppose that A has right inverse BR ∈ Rn×n max . Then by Theorem 1.10, we know that A = Pσ ⊗ D(a1, . . . , an) for some permutation σ and a1, . . . , an ∈ Rmax{ε}. Now, as before, define BL ∈ Rn×n max by [BL]ij = [A]⊗−1 ji if [A]ji = ε ε otherwise and using the same reasoning as before we observe that BL is a left inverse of A. Finally, note that BR = E ⊗ BR = (BL ∗ A) ⊗ BR = BL ⊗ (A ⊗ BR) = BL ⊗ E = BL showing that BR is uniquely determined, and is also a left inverse. 9
- 15. Theorem 1.11 tellsus that we do not need to make a distinction between right and left inverses, as we did in Definition 1.7. Before moving on we show one last result which says that the product of two invertible matrices is also invertible. Proposition 1.12. If A, B ∈ Rn×n max are invertible then A ⊗ B is also invertible. Proof. This proof uses some simple results regarding diagonal and permutation matrices in conventional algebra, whose analogues are easily proved in max-plus. To start, recall that for a permutation matrix Pσ, we have that P−1 σ = Pσ−1 . Thus if D(a1, . . . , an) is a diagonal matrix: D(a1, . . . , an) ⊗ Pσ = (Pσ ⊗ Pσ−1 ) ⊗ D(a1, . . . , an) ⊗ Pσ = Pσ ⊗ (Pσ−1 ⊗ D(a1, . . . , an) ⊗ Pσ) = Pσ ⊗ D(aσ(1), . . . , aσ(n)) Now from Theorem 1.10 we can write A = PσA ⊗D(a1, . . . , an), B = PσB ⊗D(b1, . . . , bn). Then using the above A ⊗ B = PσA ⊗ D(a1, . . . , an) ⊗ PσB ⊗ D(b1, . . . , bn) = PσA ⊗ PσB ⊗ D(aσA(1), . . . , aσA(n)) ⊗ D(b1, . . . , bn) = PσB ◦ σA ⊗ D(aσA(1) ⊗ b1, . . . , aσA(n) ⊗ bn) and therefore A ⊗ B is invertible by Theorem 1.10. 1.2.3 Determinants Recall that in conventional algebra, the determinant of a matrix A ∈ Rn×n is defined as det(A) = σ∈Sn sgn(σ) n i=1 aiσi where Sn is the symmetric group on n elements (so an element of Sn is a permutation σ : {1, . . . , n} → {1, . . . , n}), and the sign of a permutation σ ∈ Sn, denoted sgn(σ), is defined by sgn(σ) = 1 σ even −1 σ odd Unfortunately this definition cannot be immediately translated into max-plus (i.e. by replacing + and × with ⊕ and ⊗ respectively) because the use of the sign function requires that we have additive inverses. Instead, two related concepts are introduced below which offer alternatives to the notion of the determinant in the case of the max-plus algebra. Definition 1.13. Let A ∈ Rn×n max . The permanent of A, denoted perm(A), is defined as perm(A) = σ∈Sn n i=1 aiσi 10
- 16. Note that, crudelyput, the permanent is the max-plus analogue of the determinant with the minuses simply removed. We can understand the formula to give the maximal sum of the diagonal values for all permutations of the columns of A. The permanent has been studied at length both in the case of conventional algebra (see [17]) and in max-plus & related semirings (see [19]). Note that if A ∈ Rn×n max is invertible then by Theorem 1.10, A = Pσ ⊗ D(a1, . . . , an) and so perm(A) = n i=1 ai = ε. However, unlike in the case of determinants in conventional matrix algebra, the converse is not necessarily true. The second concept in max-plus related to the determinant, known as the dominant, can be thought of as a refinement of the permanent. It is defined below. Definition 1.14. Let A ∈ Rn×n max and let the matrix zA be defined by [zA]ij = zaij . The dominant of A, denoted dom(A), is defined as dom(A) = highest exponent in det(zA) if det(zA) = 0 ε otherwise The dominant can be used to prove max-plus analogues of major results such as Cramér’s Theorem and the Cayley-Hamilton Theorem. We do not have the space to include these here; for a comprehensive discussion readers are again referred to [19]. 1.3 Graph-theoretic Interpretations in Max-Plus As in conventional linear algebra, when working with vectors and matrices it is often natural to interpret definitions and theorems graphically. It turns out that in the case of max-plus algebra, it is not only natural to do so but also rather insightful. We will only really be able to appreciate this when we come to look at the eigenvalue problem in the next section, but firstly we must define all of the graph-theoretic concepts that we will require. Definitions 1.15. (i) A directed graph G is a pair (V, E) where V is the set of vertices (or nodes) and E ⊆ V × V is the set of edges (or arcs). (ii) A path from vertex i to vertex j is a sequence of edges p = (i1, . . . , is+1) with i1 = i and is+1 = j, such that (ik, ik+1) ∈ E for all k ∈ {1, . . . , s}. (iii) The length of a path p = (i1, . . . , is+1), denoted |p|l, is equal to s. The set of paths from vertex i to vertex j of length k is denoted Pk(i, j). (iv) The weight of a path p from vertex i to vertex j of length d is given by |p|w = d k=1 aik+1,ik where i1 = i and id+1 = j. 11
- 17. (v) The averageweight of a path p is given by |p|w |p|l . (vi) A circuit of length s is a path of length s which starts and finishes at the same vertex, i.e. a path c = (i1, . . . , is+1) such that i1 = is+1. (vii) A circuit c = (i1, . . . , is+1) is elementary if i1, . . . , is are distinct, and s ≥ 1. We denote the set of elementary circuits in G(A) by C(A). (viii) For A ∈ Rn×n max , the communication graph (or the precedence graph) of A, denoted G(A), is the graph with vertex set V(A) = {1, . . . , n} and edge set E(A) = {(i, j) : aji = ε}. The weight of the edge (i, j) ∈ E(A) is given by the entry aji. Note that the (i, j)-th entry of the matrix A specifies the weight of the edge in G(A) from vertex j to vertex i. This is common practice in the area of max-plus and graph theory but may not appear intuitive to those new to the subject. We now move on to looking at two particular matrices that play a vital role in relating graph theory to max-plus linear algebra. For A ∈ Rn×n max , let A+ := ∞ k=1 A⊗k The element [A+]ji gives the maximal weight of any path from i to j in G(A). This statement is non-trivial, but follows directly from the theorem below. Theorem 1.16. Let A ∈ Rn×n max . Then ∀k ∈ N: [A⊗k ]ji = max |p|w : p ∈ Pk(i, j) if Pk(i, j) = ∅ ε if Pk(i, j) = ∅ Proof. We use induction on k. Let i, j ∈ {1, . . . , n}. When k = 1, P1(i, j) either contains a single path of length 1, namely the edge (i, j), or is empty if no such edge exists. In the first case, the weight of the path is by definition [A]ji, and in the second case max |p|w : p ∈ Pk(i, j) = ε, which is again equal to the value [A]ji (since there is no edge from i to j). Now suppose the result holds for some k. Firstly, assume that Pk+1(i, j) = ∅. A path p ∈ Pk+1(i, j) can be split up into a subpath of length k running from i to some vertex l, and a path consisting of a single edge from l to j. More formally: p = ˆp ◦ (l, j) with ˆp ∈ Pk(i, l) The maximal weight of any path in Pk+1(i, j) can thus be obtained from max l=1,...,n [A]jl + max{|ˆp|w : ˆp ∈ Pk(i, l)} = max l=1,...,n [A]jl + [A⊗k ]li (Inductive hypothesis) 12
- 18. = n l=1 [A]jl ⊗ [A⊗k ]li =[A ⊗ A⊗k ]ji = [A⊗(k+1) ]ji which is what we wanted to prove. Finally, consider the case when Pk+1(i, j) = ∅; i.e. when there exists no path of length k + 1 from i to j. This implies that for any vertex l, either there is no path of length k from i to l or there is no edge from l to j (or possibly both). Hence for any l, at least one of the values [A]jl, [A⊗k]li equals ε. Therefore [A⊗(k+1)]ji = ε, and this completes the proof. Note that Theorem 1.16 immediately tells us that A+ is not necessarily well-defined. For example, if there exists a circuit c = (i1, . . . , is+1) in G(A) in which every edge has positive weight, then [A⊗k]ji diverges (i.e. tends to +∞) as k → ∞ for any i, j ∈ {i1, . . . , is+1} (since we can loop around the circuit c as many times a we like, creating a path of higher and higher weight). The next lemma provides us with a sufficient condition for A+ to be well-defined, and also reduces the complexity of the infinite sum. Lemma 1.17. Let A ∈ Rn×n max be such that any circuit in G(A) has non-positive average weight (i.e. less than or equal to e). Then we have A+ = A⊗1 ⊕ A⊗2 ⊕ A⊗3 ⊕ · · · ⊕ A⊗n ∈ Rn×n max Proof. Since A is of dimension n, any path p in G(A) from i to j of length greater than n necessarily contains at least one circuit. We have assumed that all of the circuits in G(A) have non-positive weights, so removing the circuits in p yields a path from i to j of length at most n, and of greater average weight. It follows that [A+ ]ji ≤ max [A⊗k ]ji : k ∈ {0, . . . , n} and the reverse inequality is immediate from the definition of A+. This concludes the proof. Before moving on, we prove one simple property of A+ that will come in handy later on. Proposition 1.18. For A ∈ Rn×n max , we have that A+ ⊗ A+ = A+. Proof. Consider two vertices i, l ∈ {1, . . . , n}. A path of maximal weight from i to l can be split up as a path of maximal weight from i to j plus a path of maximal weight from j to l, for any j ∈ {1, . . . , n} for which the sum of the two path weights is maximal. Indeed this relationship holds if and only if j is in the path of maximal weight from i to l, but for our purposes we can simply take the maximum over all vertices. By Theorem 1.16, the weight of such a path is given by [A+]li. Thus in max-plus notation 13
- 19. (recalling that ⊗is commutative for scalars α ∈ Rmax), we can write [A+ ]li = n j=1 [A+ ]ji ⊗ [A+ ]lj = n j=1 [A+ ]lj ⊗ [A+ ]ji = [A+ ⊗ A+ ]li and therefore A+ = A+ ⊗ A+ as required. We now introduce one more definition which is closely related to the object A+ defined above. This will prove to be an integral concept throughout the rest of this chapter and beyond, and as such, this is one of the most important definitions in this thesis. Definition 1.19. For A ∈ Rn×n max , let A∗ := ∞ k=0 A⊗k = E ⊕ A+ Clearly, A∗ and A+ only differ on the leading diagonal. By Theorem 1.16, the (j, i)-th of A∗ could be interpreted as the maximal weight of any path from i to j in G(A), provided we recognise the additional concept of an empty circuit of length 0 and weight e from every vertex to itself. Using Lemma 1.17, it is immediate from the definition of A∗ that if all the circuits in G(A) have non-positive average weight, then A∗ = A⊗0 ⊕ A⊗1 ⊕ · · · ⊕ A⊗n. However, as the lemma below shows, thanks to the addition of the identity matrix (i.e. the A⊗0 term) in A∗, we are able to refine this result slightly by dropping the final term in the sum. Lemma 1.20. Let A ∈ Rn×n max be such that any circuit in G(A) has non-positive average weight. Then we have A∗ = A⊗0 ⊕ A⊗1 ⊕ A⊗2 ⊕ · · · ⊕ A⊗(n−1) ∈ Rn×n max Proof. The same argument applies as in the proof of Lemma 1.17. Note that any path p in G(A) from i to j of length n or greater necessarily contains at least one circuit, and so removing the circuit(s) yields a path from i to j of length at most n − 1 and with greater average weight. For the special case when i = j and p is an elementary circuit of length n (so visiting each vertex in G(A) exactly once), the i-th entry on the diagonal of A⊗0 (which equals e by definition) will always be greater than the corresponding entry in A⊗n, since e is the maximum possible weight of any circuit. This is why we can drop the A⊗n term. Note that we also have a direct analogue of Lemma 1.18 for the matrix A∗, and this will be useful in the analysis that follows: Proposition 1.21. For A ∈ Rn×n max , we have that A∗ ⊗ A∗ = A∗. 14
- 20. Proof. From Lemma1.18 we have that A+ = A+ ⊗A+. Recalling the definition of A∗ and using idempotency of matrix addition, we have A∗ ⊗ A∗ = (A+ ⊕ E) ⊗ (A+ ⊕ E) = (A+ ⊗ A+ ) ⊕ (A+ ⊗ E) ⊕ (E ⊗ A+ ) ⊕ E = A+ ⊕ A+ ⊕ A+ ⊕ E = A+ ⊕ E = A∗ as required. To finish this section, we introduce one more important property of square matrices over max- plus known as irreducibility. The definition comes in three parts: Definitions 1.22. (i) In a graph G, a vertex j is reachable from vertex i if there exists a path from i to j. (ii) A graph is strongly connected if every vertex is reachable from every other vertex. (iii) A matrix A ∈ Rn×n max is irreducible if G(A) is strongly connected. The class of irreducible matrices over max-plus will turn out to be of real significance in Section 1.4. From a practical point of view it is not obvious how to determine whether a given matrix A ∈ Rn×n max is irreducible, but as the proposition below shows, one option is to examine the matrix A+. Combined with Lemma 1.17 (when A has the appropriate properties), this provides us with a handy (and computationally quick) way to check for matrix irreducibility over max-plus. Proposition 1.23. A matrix A ∈ Rn×n max is irreducible if and only if all the entries of A+ are different from ε. Proof. A matrix is irreducible if there is a path between any two vertices i and j in G(A), which by Theorem 1.16 occurs exactly when the entry [A+]ji is not equal to ε. 1.4 Spectral Theory 1.4.1 Eigenvalues and Eigenvectors Given a matrix A ∈ Rn×n max , we consider the problem of existence of eigenvalues and eigenvectors. The main result in max-plus spectral theory is that, under mild conditions, A has a unique eigenvalue with a simple graph-theoretic interpretation. As can be seen below, the definition of max-plus eigenvalues and eigenvectors is a direct translation from conventional linear algebra, with the × operator replaced with ⊗: Definition 1.24. Let A ∈ Rn×n max . If there exists a scalar µ ∈ Rmax and a vector v ∈ Rn max (containing at least one finite element) such that A ⊗ v = µ ⊗ v 15
- 21. then µ isan eigenvalue of A and v is an eigenvector of A associated with the eigenvalue µ. Note that Definition 1.24 allows an eigenvalue to be µ = ε. However, the proposition below says that this can only happen when A has a column in which all entries are ε. In graph-theoretic terms this means that G(A) has a vertex which, once visited, can never be left (sometimes called a sink). This is uninteresting from an analytical point of view, so it is reasonable to consider the case µ = ε to be trivial. Before we prove this result, we introduce some simple notation. Notation. Let A ∈ Rn×n max . For i ∈ {1, . . . , n}, we denote the i-th row of A by [A]i·. Similarly, for j ∈ {1, . . . , n}, we denote the j-th column of A by [A]·j. Proposition 1.25. ε is an eigenvalue of A ∈ Rn×n max iff A has at least one column in which all entries are ε. Proof. Let A ∈ Rn×n max be such that [A]·j = (ε, . . . , ε) for some j ∈ {1, . . . , n}. Let v ∈ Rn max be such that [v]i = ε ∀i = j and [v]j = α = ε. Then it is easy to verify that [A ⊗ v]i = ε for all i = 1, . . . , n; that is, ε is an eigenvalue of A with an associated eigenvector v. Conversely, suppose A ∈ Rn×n max has eigenvalue ε with an associated eigenvector v. let J = {j : vj = ε}, which is non-empty by definition. Then for each i = 1, . . . , n we have ε = [A ⊗ v]i = n j=1 aij ⊗ vj = j∈J aij ⊗ vj =⇒ aij = ε ∀j ∈ J So every column j of A for which vj = ε has all its entries equal to ε. In particular, A contains at least one column in which all entries are ε. Corollary 1.26. If A ∈ Rn×n max is irreducible then ε is not an eigenvalue of A. Proof. If A is irreducible then it cannot have a column in which all entries are ε. Thus by Proposition 1.25, ε is not an eigenvalue of A. Note that eigenvectors are not unique: any scalar multiple of an eigenvector is also an eigen- vector, and more generally, if µ is an eigenvalue of A, v1, v2 are associated eigenvectors and α1, α2 ∈ Rmax{ε}, then we have A ⊗ (α1 ⊗ v1) ⊕ (α2 ⊗ v2) = A ⊗ (α1 ⊗ v1) ⊕ A ⊗ (α2 ⊗ v2) = α1 ⊗ (A ⊗ v1) ⊕ α2 ⊗ (A ⊗ v2) = α1 ⊗ (µ ⊗ v1) ⊕ α2 ⊗ (µ ⊗ v2) = µ ⊗ (α1 ⊗ v1) ⊕ (α2 ⊗ v2) So (α1 ⊗ v1) ⊕ (α2 ⊗ v2) is also an eigenvector associated with the eigenvalue µ. In fact, the eigenvectors associated with a given eigenvalue form a vector space in max-plus called the eigenspace which we shall explore in depth later. 16
- 22. As we mentionedat the beginning of Section 1.3, many of the results in the area of max-plus spectral theory can be interpreted graphically, and the next key lemma constitutes the first step in doing just that. Lemma 1.27. Let A ∈ Rn×n max have finite eigenvalue µ. Then µ is the average weight of some elementary circuit in G(A). Proof. Let v be an associated eigenvector of µ. Then by definition not all the entries of v equal ε, i.e. there exists a vertex/index j1 ∈ {1, . . . , n} such that vj1 = ε. Now v is an eigenvector and so we have [A ⊗ v]j1 = µ ⊗ vj1 = ε. But [A ⊗ v]j1 = n k=1 aj1k ⊗ vk, and therefore there exists a vertex j2 such that aj1j2 ⊗ vj2 = [A ⊗ v]j1 = ε (1.1) which implies aj1j2 = ε, i.e. (j2, j1) is an edge in G(A). (1.1) also implies that vj2 = ε, so we can continue in the same fashion to find a vertex j3 with (j3, j2) an edge in G(A) and vj3 = ε. Proceeding in this way, eventually some vertex, say, vertex jh, must be encountered for a second time since the number of vertices is finite. Thus by ignoring the edges prior to encountering jh for the first time, we have found an elementary circuit c = ((jh, jh+l−1), (jh+l−1, jh+l−2), . . . , (jh+1, jh)) of length |c|l = l, and with weight |c|w = l−1 k=0 ajh+kjh+k+1 (1.2) where jh = jh+l. By construction, we have that l−1 k=0 (ajh+kjh+k+1 ⊗ vjh+k+1 ) = µ⊗l ⊗ l−1 k=0 vjh+k or equivalently in conventional algebra (for ease of manipulation): l−1 k=0 ajh+kjh+k+1 + vjh+k+1 = (l × µ) + l−1 k=0 vjh+k Now, because jh = jh+l it follows that l−1 k=0 vjh+k+1 = l−1 k=0 vjh+k , so subtracting l−1 k=0 vjh+k from both sides yields l−1 k=0 ajh+k jh+k+1 = l × µ and translated back into max-plus, we can substitute this into (1.2) to see that |c|w = µ⊗l. 17
- 23. Thus we havethat the average weight of the circuit c is equal to |c|w |c|l = µ⊗l l = µ as required. Lemma 1.27 tells us that the only candidates for eigenvalues are the average weights of circuits in G(A). However, it does not tell us which circuits actually define an eigenvalue and which do not. Fortunately, when A is irreducible the answer to this question is very simple: only the maximal average circuit weight defines an eigenvalue. This result is established in the two theorems below, but first we require some additional definitions and notation. Definitions 1.28. (i) A circuit c ∈ C(A) is critical if its average weight is maximal. (ii) For A ∈ Rn×n max , the critical graph of A, denoted Gc(A), is the graph containing the vertices and edges which belong to the critical circuits in G(A). We write Gc(A) = (Vc(A), Ec(A)), and refer to the vertices in Vc(A) as critical vertices. (iii) The critical classes of A ∈ Rn×n max are the maximal strongly connected components of Gc(A). Notation. Let A ∈ Rn×n max . For β ∈ Rmax{ε}, define the matrix Aβ by [Aβ]ij = aij − β. Note that the ‘−’ operator is to be interpreted in conventional algebra, where we adopt the convention ε − x = ε ∀x ∈ R. If β is an eigenvalue of A, the matrix Aβ is sometimes called the normalised matrix. Note that the communication graphs G(A) and G(Aβ) are identical except for their edge weights, and if a circuit c in G(A) has average weight w then the same circuit in G(Aβ) has average weight w − β. In particular, if G(A) has finite maximal average circuit weight λ then the maximal average circuit weight in G(Aλ) is λ − λ = 0. Furthermore, a circuit in G(A) is critical if and only if it is critical in G(Aλ), and therefore Gc(A) and Gc(Aλ) are identical (again, except for their edge weights). Consider the matrix A+ λ , which is to be read (Aλ)+ . By Theorem 1.16, the element [A+ λ ]ij gives the maximal weight of any path from j to i in G(Aλ). In particular, since all circuits in G(Aλ) have non-positive average weight, we must have [A+ λ ]ii ≤ e for all i ∈ {1, . . . , n}. Furthermore, for the matrix A∗ λ (also to be read (Aλ)∗ ) we obtain [A∗ λ]ii = e⊕[A+ λ ]ii = e for all i ∈ {1, . . . , n}. Theorem 1.29. Let the communication graph G(A) of a matrix A ∈ Rn×n max have finite maximal average circuit weight λ. Then λ is an eigenvalue of A, with an associated eigenvector [A∗ λ]·j for any vertex j ∈ Vc(A). Proof. Firstly note that all the circuits in G(Aλ) have non-positive average weight, and therefore A+ λ is well-defined by Lemma 1.17. Now, every vertex in Gc(Aλ) is contained in a non-empty circuit which has weight e, i.e. ∀j ∈ Vc (A) : [A+ λ ]jj = e (1.3) 18
- 24. Next, write [A∗ λ]ij =[E ⊕ A+ λ ]ij = ε ⊕ [A+ λ ]ij for i = j e ⊕ [A+ λ ]ij for i = j Then from (1.3), for j ∈ Vc(A) it follows that [A+ λ ]·j = [A∗ λ]·j (1.4) Now, note that we have A+ λ = A⊗1 λ ⊕ A⊗2 λ ⊕ . . . = Aλ ⊗ (A⊗0 λ ⊕ A⊗1 λ ⊕ . . . ) = Aλ ⊗ A∗ λ So substituting this into (1.4) gives for j ∈ Vc(A) [Aλ ⊗ A∗ λ]·j = [A∗ λ]·j ⇐⇒ Aλ ⊗ [A∗ λ]·j = [A∗ λ]·j ⇐⇒ A ⊗ [A∗ λ]·j = λ ⊗ [A∗ λ]·j Therefore λ is an eigenvalue of A and the j-th column of A∗ λ is an associated eigenvector for any j ∈ Vc(A). Theorem 1.30. Let A ∈ Rn×n max be irreducible. Then A has a unique eigenvalue, denoted λ(A), which is finite and equal to the maximal average circuit weight in G(A). Proof. Let the maximal average circuit weight in G(A) be denoted by λ. Since A is irreducible, G(A) must contain a circuit and therefore λ is necessarily finite. Thus by Theorem 1.29 we know that λ is an eigenvalue of A, and it remains to show uniqueness. Let c = (j1, . . . , jl+1) be an arbitrary circuit in C(A) of length l = |c|l, with jl+1 = j1. Then ajk+1jk = ε for all k ∈ {1, . . . , l}. Further, suppose that µ is an eigenvalue of A with an associated eigenvector v. Note that A is irreducible, so by Corollary 1.26 we have that µ = ε. Now, since A ⊗ v = µ ⊗ v, it follows that ajk+1jk ⊗ vjk ≤ µ ⊗ vjk+1 , k ∈ {1, . . . , l}. and arguing as in Lemma 1.27 (replacing equalities with the appropriate inequalities), we see that the average weight of the circuit c satisfies |c|w |c|l ≤ µ⊗l l = µ (1.5) That is, µ ≥ λ (since (1.5) holds for all c ∈ C(A), and we already have that the maximal average circuit weight is λ). But by Lemma 1.27, µ is equal to the average weight of some circuit c ∈ C(A), and so µ ≤ λ also. Hence µ = λ, i.e. λ is the unique eigenvalue of A. 19
- 25. When A islarge it is often difficult to identify the maximal average circuit weight in G(A). In fact, there exist several numerical procedures used to determine the eigenvalue of an irreducible matrix in max-plus, including Karp’s Algorithm and the Power Algorithm. However, none of these has a particularly attractive order of complexity - for example, the complexity of Karp’s Algorithm is of order n3, and the complexity of the Power Algorithm is not known precisely (see [11]). We do not have space here to describe the methods in detail; for more information readers are referred to chapter five of [13]. We end this section with a simple proposition that, while interesting in its own right, will come in handy when we begin to look at the eigenspace. Proposition 1.31. Let A ∈ Rn×n max be an irreducible matrix with eigenvalue λ and associated eigenvector v. We have that vi > ε for all i ∈ {1, . . . , n}. Proof. Call the set of vertices of G(A) corresponding to the finite entries of v the support of v, denoted Z(v). Suppose that Z(v) does not contain all the elements of V(A). Since A is irreducible, there must be edges from the vertices in Z(v) to vertices not belonging to Z(v). Hence there exists vertices j ∈ Z(v), i /∈ Z(v) with aij = ε. Then [A ⊗ v]i ≥ aij ⊗ vj > ε That is, Z(A ⊗ v) is strictly bigger than Z(v). But A ⊗ v = λ ⊗ v (and λ is finite by Theorem 1.30), so Z(v) and Z(A⊗v) should be equal. This is a contradiction, and so Z(v) must contain all the elements of V(A). 1.4.2 The Eigenspace Let A ∈ Rn×n max have finite eigenvalue λ. In this part of our analysis we let V (A, λ) denote the set of all eigenvectors of A associated with the eigenvalue λ, which we call the eigenspace of A w.r.t. λ. If A is irreducible then by Theorem 1.30 we know that it has a unique eigenvalue, so we can drop the dependence on λ and denote the eigenspace of A simply by V (A). The main aim of this section is to find an expression that completely characterises the eigenspace of A. In Theorem 1.29 we established that [A∗ λ]·j is an eigenvector of A for any j ∈ Vc(A), but are these the only eigenvectors (of course, up to taking linear combinations, as discussed above)? We will eventually see that the answer to this question is yes, but first we require some intermediate steps. Lemma 1.32. Let A ∈ Rn×n max . We have that A∗ λ = (E ⊕ Aλ)⊗(n−1). Proof. If n = 1 then the result is trivial. Otherwise, since E and Aλ commute, we can carry out the iterated multiplication (E ⊕ Aλ) ⊗ · · · ⊗ (E ⊕ Aλ) to obtain (E ⊕ Aλ)⊗(n−1) = E ⊕ n−1 i=1 A⊗i λ ⊕ · · · ⊕ A⊗i λ (n−1 i ) times (1.6) 20
- 26. Each power A⊗0 λ, . . . , A ⊗(n−1) λ occurs at least once, so by idempotency of ⊕, (1.6) becomes (E + Aλ)⊗(n−1) = E ⊕ Aλ ⊕ A⊗2 λ ⊕ . . . A ⊗(n−1) λ (1.7) However, noting that every circuit in G(Aλ) must have non-positive weight, we can apply Lemma 1.20 to see that the right-hand side of (1.7) is equal to A∗ λ. This completes the proof. Lemma 1.33. Let A ∈ Rn×n max be an irreducible matrix, with eigenvalue λ and an associated eigenvector v. Then the matrix A∗ λ has eigenvalue e, also with an associated eigenvector v. Proof. Firstly, note that for any j ∈ {1, . . . , n} [λ ⊗ v]j = [A ⊗ v]j ⇐⇒ vj = [A ⊗ v]j − λ ⇐⇒ e ⊗ vj = [Aλ ⊗ v]j That is, e ⊗ v = Aλ ⊗ v, and v is also an eigenvector of Aλ (whose unique eigenvalue must be e by Theorem 1.30). Thus the eigenspaces V (A) and V (Aλ) coincide. Next, note that (E ⊕ Aλ) ⊗ v = (E ⊗ v) ⊕ (Aλ ⊗ v) = v ⊕ v = v Therefore, using Lemma 1.32: A∗ λ ⊗ v = (E ⊕ Aλ)⊗(n−1) ⊗ v = v = e ⊗ v as required. Definition 1.34. Let A ∈ Rn×n max be a matrix with eigenvalue λ and associated eigenvector v. The saturation graph of A with respect to λ, denoted Sλ(A, v), is the graph consisting of those edges (j, i) ∈ E(A) such that aij ⊗ vj = λ ⊗ vi, with vi, vj = ε. Recall that by definition, if v is an eigenvector of A then there exists at least one i ∈ {1, . . . , n} such that vi = ε. Then, since A ⊗ v = λ ⊗ v we have that n j=1 aij ⊗ vj = λ ⊗ vi, which implies that there exists (at least one) j ∈ {1, . . . , n} such that aij ⊗ vj = λ ⊗ vi. This value is finite (assuming λ = ε), so we must have (j, i) ∈ E Sλ(A, v) . That is, the saturation graph of A w.r.t. λ is never empty. Indeed, if A is irreducible, by Proposition 1.31 we know that vi > ε for all i ∈ {1, . . . , n}, and so by the same argument, Sλ(A, v) contains all the vertices in V(A). In this case we know that the eigenvalue λ is unique, and therefore we drop the dependence on λ and simply refer to the saturation graph of A. Lemma 1.35. Let A ∈ Rn×n max be an irreducible matrix, with eigenvalue λ and associated eigen- vector v. We have: (i) For each vertex i ∈ V(A), there exists a circuit in S(A, v) from which vertex i can be reached in a finite number of steps. (ii) Any circuit in S(A, v) belongs to Gc(A). Proof. (i) A is irreducible, so by Proposition 1.31 we know that vi > ε for all i ∈ {1, . . . , n}. Let 21
- 27. i ∈ V(A),which by the discussion above we know is a vertex of the saturation graph S(A, v). Thus there is a vertex j such that λ ⊗ vi = aij ⊗ vj. Repeating this argument, we can identify a vertex k such that λ ⊗ vj = ajk ⊗ vk. Repeating this argument an arbitrary number of times, say, m, we get a path in S(A, v) of length m. If m > n, the constructed path must contain a circuit. (ii) Let c = (i1, i2, . . . , il+1) be a circuit of length l in S(A, v). By definition, for all k ∈ {1, . . . , n} we have that λ ⊗ vik+1 = aik+1ik ⊗ vik which implies that λ⊗l ⊗ vi1 = l k=1 aik+1ik ⊗ vi1 Hence, recalling that vi1 is finite: λ⊗l = l k=1 aik+1ik But the right-hand side is simply equal to the weight of the circuit c, which thus has average weight λ. But A is irreducible, so by Theorem 1.30 λ is equal to the maximal average circuit weight in G(A). Thus c is critical, and belongs to Gc(A). Lemma 1.36. Let A ∈ Rn×n max be an irreducible matrix, with eigenvalue λ and associated eigen- vector v. Then v can be written as v = j∈Vc(A) αj ⊗ [A∗ λ]·j for some αj ∈ Rmax, j ∈ Vc(A). Proof. Consider two vertices i, j in S(Aλ, v) such that there exists a path from i to j, say, (i1, i2, . . . , il+1), with i1 = i and il+1 = j. Then by definition of the saturation graph, this gives [Aλ]ik+1ik ⊗ vik = vik+1 , k ∈ {1, . . . , l} Hence vj = a ⊗ vi, where a is given by a = l k=1 [Aλ]ik+1ik ≤ [A⊗l λ ]ji ≤ [A∗ λ]ji (1.8) Now, using that vj = a ⊗ vi, for an arbitrary vertex ν ∈ {1, . . . , n}: [A∗ λ]νj ⊗ vj = [A∗ λ]νj ⊗ a ⊗ vi ≤ [A∗ λ]νj ⊗ [A∗ λ]ji ⊗ vi (by (1.8)) 22
- 28. ≤ [A∗ λ]νi ⊗vi (1.9) where the last inequality follows from Proposition 1.21. By applying Lemma 1.35, for any vertex j in S(Aλ, v) there exists a vertex i = i(j) ∈ Vc(A). Inequality (1.9) therefore implies j∈S(Aλ,v) [A∗ λ]νj ⊗ vj ≤ i∈Vc(Aλ) [A∗ λ]νi ⊗ vi (1.10) and this holds for any ν ∈ {1, . . . , n}. Now, by Lemma 1.33, A∗ λ has eigenvalue e with an associated eigenvector v, i.e. v = A∗ λ ⊗ v. The value of vν is equal to [A∗ λ]νj ⊗vj for some j, which by definition has to be in the saturation graph S(Aλ, v). Thus it holds for ν ∈ {1, . . . , n} that vν = j∈S(Aλ,v) [A∗ λ]νj ⊗ vj (1.10) ≤ j∈Vc(Aλ) [A∗ λ]νj ⊗ vj On the other hand, since v is an eigenvector of A∗ λ associated with the eigenvalue e, vν = [A∗ λ ⊗ v]ν = n j=1 [A∗ λ]νj ⊗ vj ≥ i∈Vc(Aλ) [A∗ λ]νi ⊗ vi which also holds for any ν ∈ {1, . . . , n}. Thus we have shown vν = i∈Vc(Aλ) [A∗ λ]νi ⊗ vi and since Vc(Aλ) = Vc(A) (see the proof of Theorem 1.29), the proof is complete. The lemma above shows that for an irreducible matrix A, the vectors [A∗ λ]·j, with j ∈ Vc(A), constitute a generating set for the eigenspace of A. Notice that in the proof we have actually identified the coefficients αi to which we referred in the statement of the lemma. If some of the columns of A∗ λ are colinear then the αi’s are non-unique and some can be chosen to equal ε. We have now done most of the work in characterising the eigenspace of an irreducible matrix. We now require a small extension of our notation and one more lemma before we are able to give a complete expression for the eigenspace, and we will end this section by referring to a theorem which shows that it is not possible to simplify this expression any further. Notation. Recall that the critical classes of a matrix A ∈ Rn×n max are the maximal strongly connected subgraphs of Gc(A). Let Nc(A) denote the number of critical classes of A, so Nc(A) ∈ N. For r ∈ {1, . . . , Nc(A)}, let Gc r(A) = (Vc r (A), Ec r (A)) denote the r-th critical class of A and let jc r := min{j ∈ Vc r (A)} be the smallest numbered vertex in the r-th critical class. We call {jc 1, . . . , jc Nc(A)} a set of representative vertices of the critical classes of A. Note that in the way defined above, the set of representative vertices is unique. However, this is not important - in general, a representative vertex jc r of the rth critical class of A can be any 23
- 29. j ∈ Vc r(A). Lemma 1.37. Let A ∈ Rn×n max be an irreducible matrix with eigenvalue λ. Then for i, j ∈ Vc(A), there exists α ∈ Rmax{ε} such that α ⊗ [A∗ λ]·i = [A∗ λ]·j iff i and j are members of the same critical class. Proof. Suppose that i, j ∈ Vc(A) are members of the same critical class of Aλ. Then i and j communicate with each other in the critical graph, i.e. (i, j, i) is an elementary circuit in Gc(Aλ). As we have argued before (see Theorem 1.29), any circuit in Gc(Aλ) must have weight e, and therefore in this case we have [Aλ]ji ⊗ [Aλ]ij = e. Then by definition of A∗ λ, we have that [A∗ λ]ji ⊗ [A∗ λ]ij ≥ [Aλ]ji ⊗ [Aλ]ij = e (1.11) Now by a previous observation we know that [A∗ λ]jj = e, and by Proposition 1.21 we have that A∗ λ = A∗ λ ⊗ A∗ λ. Therefore we also have [A∗ λ]ji ⊗ [A∗ λ]ij ≤ n l=1 [A∗ λ]jl ⊗ [A∗ λ]lj = [A∗ λ ⊗ A∗ λ]jj = [A∗ λ]jj = e (1.12) and from (1.11) and (1.12) we conclude that [A∗ λ]ji ⊗ [A∗ λ]ij = e. Thus for all l ∈ {1, . . . , n}: [A∗ λ]li ⊗ [A∗ λ]ij ≤ [A∗ λ]lj = [A∗ λ]lj ⊗ [A∗ λ]ji ⊗ [A∗ λ]ij ≤ [A∗ λ]li ⊗ [A∗ λ]ij and therefore [A∗ λ]lj = [A∗ λ]li ⊗ [A∗ λ]ij. Hence the statement of the lemma has been proved, with α = [A∗ λ]ij. Conversely, suppose now that i, j ∈ Vc(A) do not belong to the same critical class, and suppose for contradiction that we can find α ∈ Rmax{ε} such that α ⊗ [A∗ λ]·i = [A∗ λ]·j. The i-th and j-th components of this equation read α ⊗ e = [A∗ λ]ij and α ⊗ [A∗ λ]ji = e respectively, from which it follows that [A∗ λ]ij ⊗ [A∗ λ]ji = e Therefore the elementary circuit (i, j, i) has average weight e, and therefore belongs to Gc(Aλ). Thus vertices i and j are members of the same critical class (since they communicate with each other), which is a contradiction. Theorem 1.38. Let A ∈ Rn×n max be an irreducible matrix with (unique) eigenvalue λ. The 24
- 30. eigenspace of Ais given by V (A) = Nc(A) r=1 αr ⊗ [A∗ λ]·jc r : αr ∈ Rmax, at least one αr finite for any set of representative vertices {jc 1, . . . , jc Nc(A)} of the critical classes of A. Proof. By Lemma 1.36 we know that any eigenvector of A is a linear combination of the columns [A∗ λ]·j, for j ∈ Vc(A). However, by Lemma 1.37 we know that the columns [A∗ λ]·j for j in some critical class Vc r (A) are all colinear. Therefore to build any eigenvector we only need one column corresponding to each critical class, and so it suffices to take the sum over a set of representative vertices of the critical classes of A. Theorem 1.39. No column [A∗ λ]·i, for i ∈ Vc(A), can be expressed as a linear combination of columns [A∗ λ]·jc r , where jc r varies over the representative vertices of critical classes distinct from that of i. Proof. The proof of this statement requires substantial groundwork which we do not have the space to include. For all the details and a full proof, readers are referred to theorem 3.101 in [2]. Theorem 1.39 above tells us that we cannot simplify any further the expression for V (A) given in Theorem 1.38. It also tells us that for an irreducible matrix A, the columns [A∗ λ]·jc r , where {jc 1, . . . , jc Nc(A)} is a set of representative vertices of the critical classes of A, form a basis for the eigenspace V (A). 1.4.3 A Worked Example Consider the matrix A = ε −2 ε 6 1 ε 4 ε ε 8 ε ε ε 5 ε 6 Thus G(A) looks like 25
- 31. 1 4 2 3 1 8 -2 5 4 6 6 Figure1.1: Communication graph of the matrix A given above. Vertices are represented as circles and numbered 1-4 by convention. Edges are present only if the corresponding entry in A is finite, in which case this value specifies the edge weight. We can see that G(A) is strongly connected, so A is irreducible. Thus by Theorem 1.30, A has a unique eigenvalue λ given by the maximal average circuit weight in G(A). The elementary circuits and their average weights are c1 = (1, 2, 1) |c1|w/|c1|l = (1 ⊗ −2)/2 = −0.5 c2 = (1, 2, 4, 1) |c2|w/|c2|l = (1 ⊗ 5 ⊗ 6)/3 = 4 c3 = (2, 3, 2) |c3|w/|c3|l = (8 ⊗ 4)/2 = 6 c4 = (4, 4) |c4|w/|c4|l = (6)/1 = 6 and therefore λ = max{−0.5, 4, 6, 6} = 6. Circuits c3 and c4 are critical, so the critical graph Gc(A) looks like 2 3 4 8 4 6 Figure 1.2: Critical graph of the matrix A given above. Both the circuits have maximal average weight of 6. The other circuits present in Figure 1.1 are no longer included because they are not critical (their average weight is not maximal). We can see that Vc(A) = {2, 3, 4}, and Gc(A) has two critical classes with vertex sets Vc 1(A) = {2, 3} and Vc 2(A) = {4} respectively. Thus {jc 1 = 2, jc 2 = 4} is a set of representative vertices of the critical classes of A. Now, using that [Aλ]ij = aij − λ, we have Aλ = ε −8 ε e −5 ε −2 ε ε 2 ε ε ε −1 ε e and either by inspection of G(Aλ), or by using Lemma 1.17 and computing A⊗1 λ , A⊗2 λ , A⊗3 λ and 26
- 32. A⊗4 λ , wecan see that A+ λ = −6 −1 −3 e −5 e −2 −5 −3 2 e −3 −6 −1 −3 e Similarly, by using Lemma 1.20 (or by simply replacing any non-zero diagonal values in A+ λ above by e), we obtain A∗ λ = e −1 −3 e −5 e −2 −5 −3 2 e −3 −6 −1 −3 e Now by theorems 1.38 and 1.39, the columns [A∗ λ]·2 & [A∗ λ]·4 form a basis for the eigenspace of A, i.e. V (A) = α1 ⊗ −1 e 2 −1 α2 ⊗ e −5 −3 e : α1, α2 ∈ Rmax, at least one αr finite For example, if we take α1 = −2, α2 = 1 we get v := −2 ⊗ −1 e 2 −1 1 ⊗ e −5 −3 e = −3 −2 e −3 1 −4 −2 1 = 1 −2 e 1 and we can easily verify that this is indeed an eigenvector of A, associated with the unique eigenvalue λ = 6: A ⊗ v = ε −2 ε 6 1 ε 4 ε ε 8 ε ε ε 5 ε 6 1 −2 e 1 = 7 4 6 7 = 6 ⊗ 1 −2 e 1 = λ ⊗ v Finally, we can observe that [A∗ λ]·3 = −3 −2 e −3 = −2 ⊗ −1 e 2 −1 = −2 ⊗ [A∗ λ]·2 That is, columns [A∗ λ]·2 and [A∗ λ]·3 are scalar multiples of each other, which we would expect 27
- 33. (see Lemma 1.37)since vertices 2 and 3 are in the same critical class. 1.5 Recurrence Relations & Periodicity 1.5.1 Solving Max-Plus Recurrence Relations In many of the applications discussed in Chapters 2 and 3 we will need to solve recurrence relations over the max-plus semiring. A key insight in doing this is to view implicit first-order recurrence relations of the form x(k + 1) = (A ⊗ x(k + 1)) ⊕ (B ⊗ x(k)) as a system of max-plus linear equations x = (A ⊗ x) ⊕ b. The result below uses the ∗ operator (see Definition 1.19) to solve systems of this form. Theorem 1.40. Let A ∈ Rn×n max and b ∈ Rn max. If the communication graph G(A) has no circuit with positive average weight, then the equation x = (A ⊗ x) ⊕ b (1.13) has the solution x = A∗ ⊗ b. Furthermore, if all the circuit weights in G(A) are negative, then this solution is unique. Proof. By Lemma 1.20 we know that A∗ exists. We therefore have A∗ ⊗ b = ∞ k=0 A⊗k ⊗ b = ∞ k=1 A⊗k ⊗ b ⊕ (E ⊗ b) = A ⊗ ∞ k=0 A⊗k ⊗ b ⊕ (E ⊗ b) = A ⊗ (A∗ ⊗ b) ⊕ b and therefore A∗ ⊗ b is indeed a solution of (1.13). To show uniqueness, suppose that x is a solution of x = b⊕(A⊗x); then we can substitute the expression for x back into the right-hand side of the equation to obtain x = b ⊕ (A ⊗ b) ⊕ (A⊗2 ⊗ x) Repeating this procedure yields x = b ⊕ (A ⊗ b) ⊕ (A⊗2 ⊗ b) ⊕ (A⊗3 ⊗ x) = . . . = b ⊕ (A ⊗ b) ⊕ · · · ⊕ (A⊗(k−1) ⊗ b) ⊕ (A⊗k ⊗ x) = k−1 l=0 (A⊗l ⊗ b) ⊕ (A⊗k ⊗ x) (1.14) 28
- 34. By Theorem 1.16,the entries A⊗k are the maximal weights of paths of length k. For k large enough, these paths necessarily contain elementary circuits, which have negative weight by assumption. Indeed, as k → ∞ the number of elementary circuits in these paths also necessarily tends to ∞, and so the elements of A⊗k tend to ε. Hence, letting k → ∞ in (1.14) gives that x = A∗ ⊗ b (where once again we have applied Lemma 1.20), as required. As we mentioned above, Theorem 1.40 above can be applied to the implicit recurrence relation x(k + 1) = A ⊗ x(k + 1) ⊕ B ⊗ x(k) to yield the explicit recurrence relation x(k + 1) = A∗ ⊗ B ⊗ x(k) and this technique will be used several times in Chapter 2. However, can we extend this theory? In many applications we will encounter systems whose dynamics follow a recurrence relation of order higher than one. Consider the most general (explicit) linear recurrence relation of order M ≥ 1: x(k) = M m=0 Am ⊗ x(k − m), k ≥ 0 (1.15) Here, A0, . . . , AM ∈ Rn×n max and x(m) ∈ Rn max, −M ≤ m ≤ −1 are given. We show below that we can transform (1.15) into a first-order recurrence relation of the form x(k + 1) = A ⊗ x(k), provided that A0 has no circuit of positive weight. To begin, set b(k) = M m=1 Am ⊗ x(k − m) Then (1.15) becomes x(k) = A0 ⊗ x(k) ⊕ b(k) (1.16) Then, since A0 has no circuit of positive weight by assumption, we can apply Theorem 1.40 to write (1.16) as x(k) = A∗ 0 ⊗ b(k) = A∗ 0 ⊗ A1 ⊗ x(k − 1) ⊕ · · · ⊕ A∗ 0 ⊗ AM ⊗ x(k − M) (1.17) Note that we have now changed the implicit M-th order recurrence relation (1.15) into the explicit M-th order recurrence relation (1.17) (the x(k) term does not feature on the right hand side). To finish the job, we set x(k) := (x (k − 1), x (k − 2), . . . , x (k − M)) 29
- 35. and (with Edenoting a matrix of all εs): A := A∗ 0 ⊗ A1 A∗ 0 ⊗ A2 . . . . . . A∗ 0 ⊗ AM E E . . . . . . E E E ... E ... ... ... E E . . . E E Then (1.15) can be written as x(k + 1) = A ⊗ x(k), k ≥ 0 (1.18) which is what we were aiming for. It will come as no surprise that problems of this form are closely related to the concept of eigenvalues and eigenvectors studied in the previous section. For example, if the recurrence relation x(k + 1) = A ⊗ x(k) is given the initial condition x(0), where x(0) is an eigenvector of A with corresponding eigenvalue λ, then the solution x(k) is given by x(k) = λ⊗k ⊗ x(0). It could then be said that the solution is periodic. The final section of this chapter explores the limiting behaviour of the solution x(k) when the system is initialised with an arbitrary vector x(0), and in particular whether we can say anything about its periodicity in general. 1.5.2 Limiting Behaviour In this section we state and prove a theorem which establishes an important result on the asymptotic behaviour of the powers of an irreducible matrix A in terms of its unique eigenvalue λ. In simple terms, this theorem says that sequential powers of A always exhibit periodic behaviour after a finite number of steps. We will then apply this result to the recurrence relations we studied in the previous section. It turns out that the periodicity depends on a quantity known as the cyclicity of A, which we define below in two steps. Definition 1.41. The cyclicity of a graph G, denoted σG, is defined as follows: • If G is strongly connected, then its cyclicity equals the greatest common divisor of the lengths of all the elementary circuits in G. If G consists of just one vertex without a self-loop, then its cyclicity is taken to be 1. • If G is not strongly connected, then its cyclicity equals the least common multiple of the cyclicities of all the maximal strongly connected subgraphs of G. Definition 1.42. The cyclicity of a matrix A ∈ Rn×n max , denoted σ(A), is equal to σGc(A), the cyclicity of the critical graph of A. If A is a square matrix over Rmax then we often talk of the graph cyclicity and matrix cyclicity of A, where the graph cyclicity refers to the cyclicity of the graph Gc(A). 30
- 36. It may seemstrange to define the cyclicity of a matrix A via its critical graph and not its communication graph. However, as we will see below, it turns out that the former quantity determines the periodic behaviour of the powers of A, so the reason for this choice should be clear. Before proving the main theorem of this section we require several preliminary results. The first one is an important lemma from graph theory, which we explore below. Lemma 1.43. Let A ∈ Rn×n max be an irreducible matrix, and let the cyclicity of its communica- tion graph be σG. Then, after a suitable relabelling of the vertices of G(A), the matrix A⊗σG corresponds to a block diagonal matrix with σG blocks on the diagonal. The communication graph of each diagonal block is strongly connected and has cyclicity one. Moreover, the eigenvalues of all diagonal blocks have the same value. Proof. For i, j ∈ V(A), define the relation i ∼ j ⇐⇒ the length of every path from i to j in G(A) is a multiple of σG. It is easy to show that this is an equivalence relation on V(A). Therefore if k0 ∈ V(A) is fixed, we can introduce equivalence classes C0, C1, . . . , CσG−1 as i ∈ Cl ⇐⇒ every path from k0 to i in G(A) has length (mod σG) equal to l, (1.19) for l = 0, 1, . . . , σG. Then for i, j ∈ V(A), we have that i ∼ j ⇐⇒ i, j ∈ Cl for some l = 0, 1, . . . , σG − 1. Assume that there is a path from i to j of length σG. By definition of cyclicity, the length of any circuit starting and finishing at i must be divisible by σG, so there must also be a path from j to i whose length is a multiple of σG. Therefore every path from i to j must have a length that is a multiple of σG (since if not, we could use such a path to create a circuit whose length is not divisible by σG). Hence, every path of length σG must start and end in the same equivalence class as defined in (1.19). Since A⊗σG can be computed by considering all paths of length σG in G(A) (see Theorem 1.16), it follows that A⊗σG is block-diagonal, possibly after an appropriate relabelling of the vertices according to the classes C1, . . . , CσG−1; for instance, by first labelling all vertices in C0, then all the vertices in C1, and so on. Now let l ∈ {0, 1, . . . , σG − 1}. From our remark above we know that if i, j ∈ Cl then i ∼ j, i.e. the length of every path from i to j is a multiple of σG. Since A is irreducible there must be at least one such path, which can be split up into a number of subpaths, all of length σG and going from one vertex in Cl to another vertex in Cl. It follows that the block of A⊗σG corresponding to class Cl is irreducible. Next, note that every circuit in G(A) must go through all the equivalence classes C1, . . . , CσG−1. To see this, suppose there is a circuit going through just τ of the classes, where τ < σG. Then there must be a class Cl and vertices i, j ∈ Cl such that there is a path from i to j of length 31
- 37. less than orequal to τ. This is a contradiction, since the length of a path between vertices in the same class must be a multiple of σG. Hence the number of circuits in G(A) is the same as the number of circuits going through any class Cl. Observe that circuits in G(A) of length κ × σG can be associated with circuits in G(A⊗σG ) of length κ. Since the greatest common divisor of all circuits in G(A) is σG, it follows that the communication graph of the block in A⊗σG corresponding to class Cl has cyclicity one. Finally, the fact that the eigenvalues of the diagonal blocks are identical follows immediately from the irreducibility of A. Corollary 1.44. Under the conditions of Lemma 1.43, let τ be a multiple of σG. Then, after a relabelling of the vertices of G(A), the matrix A⊗τ corresponds to a block diagonal matrix with σG blocks on the diagonal. The communication graph of each diagonal block is strongly connected and has cyclicity one. Proof. This follows along the same lines as the proof of Lemma 1.43. Let A ∈ Rn×n max be an irreducible matrix and let Gc(A) be its critical graph. Define the critical matrix of A, denoted Ac, to be the submatrix of A such that the communication graph of Ac is equal to the critical graph of A, i.e. G(Ac) = Gc(A). Matrix Ac can be obtained from matrix A by restricting A to those entries that correspond to edges in Gc(A). Clearly the critical graph of Ac is the same as its communication graph, i.e. Gc(Ac) = G(Ac), and therefore σGc(Ac) = σG(Ac). It then follows that the cyclicity of the matrix Ac is equal to the cyclicity of the communication graph G(Ac) (i.e. σ(Ac) = σG(Ac)); that is, for the critical matrix Ac both types of cyclicity coincide and are equal to σ(A). We know that G(Ac) = Gc(A) = Gc(Ac), but we can prove more: Lemma 1.45. Let A be an irreducible matrix, and let Ac be its corresponding critical matrix. Then, for all k ≥ 1 we have G((Ac )⊗k ) = Gc (A⊗k ) = Gc ((Ac )⊗k ). Proof. As we noted above, Ac is a submatrix of A, and therefore (Ac)⊗k is a submatrix of A⊗k. Furthermore, note that Gc(·) is a subgraph of G(·), which we shall denote Gc(·) ⊆ G(·). It follows that Gc((Ac)⊗k) ⊆ Gc(A⊗k) and Gc((Ac)⊗k) ⊆ G((Ac)⊗k). To prove the converse inclusions, note that any edge in G(A⊗k) from vertex i to vertex j corresponds to a path in G(A) of length k from vertex i to vertex j. Thus if a number of edges in G(A⊗k) form a circuit of length l, then the corresponding paths in G(A) form a circuit of length k×l. Conversely, consider a circuit in G(A), choose any vertex on the circuit and traverse the circuit with steps of length k until the chosen vertex is reached again. If l such steps are required then there is a corresponding circuit in G(A⊗k) of length l. In the same way, critical circuits in G(A⊗k) of length l correspond to critical circuits in G(A) of length k × l, and vice versa. 32
- 38. If c isa critical circuit of length l in G(A⊗k) then there is a corresponding critical circuit c of length k ×l in G(A). This circuit must be in Gc(A) (because it is critical), which in turn implies that c is a critical circuit in G((Ac)⊗k). Hence, it follows that Gc((Ac)⊗k) ⊇ Gc(A⊗k). The other inclusion is proved in the same way. Lemma 1.46. Let A ∈ Rn×n max be an irreducible matrix with cyclicity σ = σ(A). Then the cyclicity of the matrix A⊗σ is equal to one. Proof. Firstly, suppose the critical matrix Ac is irreducible. By the remarks prior to Lemma 1.45 we know that the cyclicity of Ac and that of its communication graph is equal to σ, so by Lemma 1.43, after a suitable relabelling of vertices, (Ac)⊗σ corresponds to a block diagonal matrix with square diagonal blocks that are irreducible and have graph cyclicity one. However, by Lemma 1.45 with k = σ, we have that Gc((Ac)⊗σ) = G((Ac)⊗σ), and therefore the communication graph of each of the diagonal blocks of (Ac)⊗σ coincides with its critical graph. Thus for each diagonal block both cyclicities coincide, and therefore both are one. If Ac is reducible then the same process can be done for each of the critical classes of Gc(A) with their individual cyclicities. According to Definition 1.41, the least common multiple of these cyclicities equals σ, the matrix cyclicity of A. Noting that σ is a multiple of σG(A), it follows from Corollary 1.44 that each diagonal block of (Ac)⊗σ corresponds to a block diagonal matrix with square diagonal blocks that are irreducible and have cyclicity one. Note that if Gc(A) does not cover all the vertices of G(A) then we must augment the overall block diagonal matrix with a square block with entries equal to ε in order to keep it the same size as the original matrix A. In both cases it follows that each diagonal block of the block diagonal matrix corresponding to (Ac)⊗σ is irreducible and has cyclicity one. Taking the least common multiple of all cyclicities, this means that the cyclicity of the whole matrix (Ac)⊗σ is equal to one, and therefore the graph cyclicity of Gc((Ac)⊗σ) is also equal to one. But by Lemma 1.45 with k = σ, this graph is the same as Gc(A⊗σ), which therefore must also have cyclicity one. Thus A⊗σ has matrix cyclicity one, which completes the proof. We now state a fundamental theorem, the proof of which can be found in [4]. Theorem 1.47. Let β1, . . . , βq ∈ N be such that gcd{β1, . . . , βq} = 1. Then there exists N ∈ N such that for all k ≥ N there exist n1, . . . , nq ∈ N0 such that k = (n1 × β1) + · · · + (nq × βq). We finally state and prove one last prerequisite result which is essentially a special case of the theorem that follows. It turns out that the generalisation is relatively straightforward, so in proving this lemma we will have done most of the work in proving the main result. Lemma 1.48. Let A ∈ Rn×n max be an irreducible matrix with unique eigenvalue e and cyclicity one. Then there exists N ∈ N such that A⊗(k+1) = A⊗k 33
- 39. for all k≥ N. Proof. The proof comes in three stages. We show that there exists N ∈ N such that for all k ≥ N: 1. [A⊗k]ii = [A+]ii = e for all i ∈ Vc(A), 2. [A⊗k]ij = [A+]ij for all i ∈ Vc(A) and j ∈ {1, . . . , n}, 3. [A⊗k]ij = l∈Vc(A)[A+]il ⊗ [A+]lj for all i, j ∈ {1, . . . , n}. The result then follows immediately from statement 3 since the right hand side does not depend on k. Statement 1. Consider i ∈ Vc(A). Then there is a critical class of Gc(A), say Gc 1(A) = (Vc 1(A), Ec 1(A)), such that i ∈ Vc 1(A). Since the cyclicity of matrix A is one, it follows that the cyclicity of graph Gc 1(A) is equal to one too. Hence there exist circuits in Gc 1(A), say c1, . . . , cq, whose lengths have a greatest common divisor equal to one. Since Gc 1(A) is a critical class it must be strongly connected, and therefore there exists a circuit α in Gc 1(A) that passes through i and through all circuits c1, . . . , cq (i.e. α ∩ cj = ∅ ∀j = 1, . . . , q). Now, by Theorem 1.47, there exists N ∈ N such that for each k ≥ N, there exist n1, . . . , nq ∈ N0 such that k = |α|l + (n1 × |c1|l) + · · · + (nq × |cq|l). For these n1, . . . , nq, we can construct a circuit passing through i, built from circuit α, n1 copies of circuit c1, n2 copies of circuit c2 and so on, up to nq copies of circuit cq. Clearly this circuit is in Gc 1(A), so it must be critical with weight e. Since the maximal average circuit weight in G(A) is e, it follows that [A⊗k]ii = e for all k ≥ N, which, by the definition of A+, also implies that [A+]ii = e, as required. Statement 2. By the definition of A+ there exists l ∈ N such that [A⊗l]ij = [A+]ij. In fact, since the eigenvalue of A is e, it follows from Lemma 1.17 that l ≤ n. From statement 1, for k large enough, i ∈ Vc(A) and j ∈ {1, . . . , n}, we then have [A⊗(k+l) ]ij ≥ [A⊗k ]ii ⊗ [A⊗l ]ij = [A⊗l ]ij = [A+ ]ij. In addition, clearly we also have [A+ ]ij = ∞ m=1 [A⊗m ]ij ≥ [A⊗(k+l) ]ij ≥ [A+ ]ij, so by replacing k + l with k, it therefore follows that [A⊗k]ij = [A+]ij for all i ∈ Vc(A) and j ∈ {1, . . . , n}, with k large enough. This is what we wanted to prove. Statement 3. Following the same lines as in the proof of statement 2, we can also show that [A⊗m]ij = [A+]ij for all i ∈ {1, . . . , n}, j ∈ Vc(A) and with m large enough. Together, take k 34
- 40. and m largeenough such that [A⊗k]il = [A+]il and [A⊗m]lj = [A+]lj for all l ∈ Vc(A). Then [A⊗(k+m) ]ij ≥ [A⊗k ]il ⊗ [A⊗m ]lj = [A+ ]il ⊗ [A+ ]lj, for all l ∈ Vc(A). By replacing k + m with k, it follows that for k large enough [A⊗k ]ij ≥ l∈Vc(A) [A+ ]il ⊗ [A+ ]lj. Now let the maximal average weight of a non-critical circuit (i.e. a circuit not passing through any vertex in Vc(A)) be δ. Then the weight of a path from j to i of length k + 1 in G(A) not passing through any vertex in Vc(A) can be bounded above by [A+]ij + (k × δ) = [A+]ij ⊗ δ⊗k, since such a path consists of an elementary path from j to i (whose weight is bounded above by [A+]ij) and at most k non-critical circuits (whose weights are each bounded above by δ). Since the maximal average circuit weight in G(A) is e we must have δ < e, and so for k large enough [A+ ]ij ⊗ δ⊗k ≤ l∈Vc(A) [A+ ]il ⊗ [A+ ]lj. Indeed, the right-hand side is fixed, while the left-hand side tends to ε as k → ∞. Hence for k large enough we have that [A⊗k ]ij = l∈V(A) [A+ ]il ⊗ [A+ ]lj = l∈Vc(A) [A+ ]il ⊗ [A+ ]lj, for all i, j = 1, . . . , n. We can now state and prove the main theorem of this section. Theorem 1.49. Let A ∈ Rn×n max be an irreducible matrix with unique eigenvalue λ and cyclicity σ := σ(A). Then there exists N ∈ N such that A⊗(k+σ) = λ⊗σ ⊗ A⊗k for all k ≥ N. Proof. Consider the matrix B := (Aλ)⊗σ. Recall that σ is the cyclicity of the critical graph of A, which is a multiple of the cyclicity of the communication graph G(A). By Corollary 1.44, after a suitable relabelling of the vertices of G(A), matrix B is a block diagonal matrix with square diagonal blocks whose communication graphs are strongly connected and have cyclicity one. By Lemma 1.46 we have that the cyclicity of B is one, which implies that the cyclicity of each of its diagonal blocks is one. Hence by applying Lemma 1.48 to each diagonal block, it ultimately follows that there exists M ∈ N such that B⊗(l+1) = B⊗l for all l ≥ M. That is, (Aλ)⊗σ ⊗(l+1) = (Aλ)⊗σ ⊗l , 35
- 41. which can furtherbe written as (Aλ)⊗(l×σ+σ) = (Aλ)⊗(l×σ), or A⊗(l×σ+σ) = λ⊗σ ⊗ A⊗(l×σ) , for all l ≥ M. Finally, note that A⊗(l×σ+j+σ) = λ⊗σ ⊗A⊗(l×σ+j) for any 0 ≤ j ≤ σ−1, implying that for all k ≥ N := M × σ it follows that A⊗(k+σ) = λ⊗σ ⊗ A⊗k , as required. Theorem 1.49 can be seen as the max-plus analogue of the Perron-Frobenius theorem in con- ventional linear algebra. Strictly speaking it is the normalised matrix Aλ that exhibits periodic behaviour, since the unique eigenvalue of Aλ is e = 0, and then A ⊗(k+σ) λ = A⊗k λ for k sufficiently large. However, we use the term ‘periodic’ to describe the more general behaviour seen here. Note that the cyclicity of A is the smallest possible length of such periodic behaviour (see [2] for the proof of this). For our purposes, we now move on to applying this result to the recurrence relations studied in Section 1.5.1. Recall the form of the basic first-order recurrence relation x(k + 1) = A ⊗ x(k), k ≥ 0, (1.20) which has the solution x(k) = A⊗k ⊗ x(0). We can apply Theorem 1.49 in this context to give us that for k sufficiently large: x(k + σ(A)) = A⊗(k+σ(A)) ⊗ x(0) = λ⊗σ(A) ⊗ A⊗k ⊗ x(0) = λ⊗σ(A) ⊗ x(k). That is, the solution x(k) is periodic with period σ(A). If we interpret k as a time index, then also by Theorem 1.49, the solution enters this periodic regime after N =: t(A) time steps, where we call t(A) is the transient time of A. In particular, if A has cyclicity equal to 1 then x(k+1) = A⊗x(k) = λ⊗x(k) ∀k ≥ t(A), and so for k sufficiently large x(k) effectively becomes an eigenvector of A. In other words, after t(A) time steps, x(k) behaves like an eigenvector, and the effect of the initial condition x(0) has died out. Note that the transient time of a matrix can be large even for systems of small dimension. For example, the matrix A defined by A = −1 −N e e 36
- 42. where N ∈{2, 3, . . . } has transient time t(A) = N, while its cyclicity is clearly 1. Finally, we make some observations regarding the growth rate of the solution x(k). Note that if we take x(0) = v in (1.20), where v is an eigenvector of A, then we immediately obtain that for all j = 1, . . . , n: lim k→∞ xj(k) k = λ, where λ is the unique eigenvalue of A. By applying Theorem 1.49 it should be clear that this holds true for any initial value x(0) and not just for eigenvectors; indeed this result is proved in [13]. We therefore say that the solution has an asymptotic growth rate of λ. Assuming irreducibility, all recurrence relations over max-plus exhibit this behaviour, regardless of the choice of the matrix A! 37
- 43. Chapter 2 Petri Netsand Timed Event Graphs 2.1 A Motivating Example The following example is adapted from chapter 1 of [2]. Consider a manufacturing system consisting of three machines M1, M2 and M3, which produces three kinds of parts P1, P2 and P3 according to different product mixes. The manufacturing process for each part is depicted below. M1 M2 M3 P2 P3 P1 Figure 2.1: Manufacturing Process for each part. Grey boxes represent the three machines; arrows represent the routes that the different parts must take in their respective manufacture. Processing times are different for each machine and each part, and are given in the following table: P1 P2 P3 M1 - 1 5 M2 3 2 3 M3 4 3 - Table 2.1: Processing times for each part at each machine (arbitrary time units). Blank entries correspond to combinations of machine & part that do not form part of the manufacturing process. Parts are carried through the manufacturing process on a limited number of pallets. We make 38
- 44. the following assumptions: 1.Only one pallet is available for each part type. 2. Once production of a part is completed, it is removed from its respective pallet and the pallet returns to the beginning of the production line. 3. There are no set-up times or traveling times between machines. 4. The sequencing of part types on the machines is fixed, and for M1 is (P2, P3), for M2 (P1, P2, P3) and for M3 (P1, P2). Assumption (3) gives no loss of generality since if set-up times or traveling times did exist, we could combine them with the processing time at the appropriate machine. Assumption (4) means that machines have to wait for the appropriate part rather than starting work on any part that arrives first (see below for an example). This may or may not be realistic; extensions to the theory presented below in which this assumption is dropped are discussed in chapter 9 of [2]. We can model the time evolution of this system by considering the time that each machine starts working on the k-th part of type i, for i = 1, 2, 3 and k ∈ N. There are seven combinations of machines and parts, so we define x(k) = (x1(k), . . . , x7(k)) as follows: Variable xi(k) Definition x1(k) time that M1 starts working on the k-th unit of P2 x2(k) time that M1 starts working on the k-th unit of P3 x3(k) time that M2 starts working on the k-th unit of P1 x4(k) time that M2 starts working on the k-th unit of P2 x5(k) time that M2 starts working on the k-th unit of P3 x6(k) time that M3 starts working on the k-th unit of P1 x7(k) time that M3 starts working on the k-th unit of P2 Table 2.2: Definitions of each entry of the state vector x(k), for k ∈ N. By examining the production process, work by each machine on the (k+1)-st part is constrained in the following way: x1(k + 1) ≥ max x7(k) + 3, x2(k) + 5 x2(k + 1) ≥ max x5(k) + 3, x1(k + 1) + 1 x3(k + 1) ≥ max x6(k) + 4, x5(k) + 3 x4(k + 1) ≥ max x3(k + 1) + 3, x1(k + 1) + 1 x5(k + 1) ≥ max x2(k + 1) + 5, x4(k + 1) + 2 x6(k + 1) ≥ max x3(k + 1) + 3, x7(k) + 3 x7(k + 1) ≥ max x6(k + 1) + 4, x4(k + 1) + 2 39
- 45. For example, theinequality for x6(k + 1) comes from the fact that M3 cannot start working on the (k + 1)-st unit of P1 until it has finished working on the k-th unit of P2, and until M2 has finished working on the (k + 1)-st unit of P1. If we are to optimise the system, the inequalities above will actually be equalities. This is where the theory of max-plus algebra comes to the fore. We can write the system in max-plus matrix form as x(k + 1) = A0 ⊗ x(k + 1) ⊕ A1 ⊗ x(k) where A0 = ε ε ε ε ε ε ε 1 ε ε ε ε ε ε ε ε ε ε ε ε ε 1 ε 3 ε ε ε ε ε 5 ε 2 ε ε ε ε ε 3 ε ε ε ε ε ε ε 2 ε 4 ε ; A1 = ε 5 ε ε ε ε 3 ε ε ε ε 3 ε ε ε ε ε ε 3 4 ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε 3 ε ε ε ε ε ε ε This is a first-order recurrence relation like we have seen in Section 1.5. A quick examination of G(A0) shows that it does not contain any circuits of positive weight (indeed it does not contain any circuits at all), and therefore we can apply Theorem 1.40 to find the unique solution x(k + 1) = A∗ 0 ⊗ A1 ⊗ x(k) = B ⊗ x(k) (2.1) where B := A∗ 0 ⊗ A1, or explicitly: B = ε 5 ε ε ε ε 3 ε 6 ε ε 3 ε 4 ε ε ε ε 3 4 ε ε 6 ε ε 6 7 4 ε 11 ε ε 8 9 9 ε ε ε ε 6 7 3 ε 8 ε ε 10 11 7 If numerical values of x1(1), . . . , x7(1) are given then these values constitute the initial condition, and the future evolution of the system is uniquely determined. There are no restrictions on x(1) from a mathematical point of view, but given the physical interpretation of the system, limitations do exist. For example, if we assume that all three pallets start at the beginning of their respective production lines (with M1 working on P2 first), we have x1(1) = x3(1) = 0, but x2(1) cannot be less than 1 since M1 has to finish working on P2 before it can start working on P3. 40
- 46. Note that ifwe had allowed more than one pallet on any of the three production lines then the system would have been of higher order (for example, if the production line of P1 had three pallets then work on the (k +1)-st unit could start once the (k −2)-th unit had been produced). This system would be solvable using the techniques developed at the end of Section 1.5. Another possible extension would be to incorporate variable processing times rather than the constant values given in table 2.1. The processing times could vary according to how many parts the machines have already processed (i.e. vary with k), or they could exhibit stochastic variability (i.e. following some specified probability distribution). The first type of variability will be introduced with the basic autonomous equation below; stochastic event graph theory will be discussed in Chapter 3. Note that since we can describe the evolution of the system by a recurrence relation of the form (2.1), we might expect that we can apply Theorem 1.49 to see that the system settles down into a periodic regime after a finite length of time. However, upon closer inspection we see that the matrix B has a column of ε, so it is not irreducible and thus Theorem 1.49 does not apply. Later on in this chapter we will discuss some techniques which ensure that the evolution equation does involve an irreducible matrix and therefore enables us to draw the relevant conclusions. To end this introductory example, note that the way we have modeled our system does not immediately give us the production times of the k-th unit of P1, P2 and P3. In order to find these we could introduce an output vector y(k) = (y1(k), y2(k), y3(k)) defined by y(k) = C ⊗ x(k) where C = ε ε ε ε ε 4 ε ε ε ε ε ε ε 3 ε ε ε ε 3 ε ε Left multiplication by C adds the appropriate processing time to the starting time at the last machine in the production line of each part. Thus yi(k) gives us the time of production of the k-th unit of part Pi. In the following section we will introduce the concept of timed event graphs, which are the tools we will use to model discrete event systems such as the production line we have considered here. 2.2 Preliminaries of Event Graph Theory 2.2.1 Definitions and Set-up As we have seen above, max-plus algebra allows us to describe the evolution of events on a network subject to synchronisation constraints. In our example, a part moving from one machine to the next is an event. An appropriate tool to model events on a certain class of 41
- 47. networks is knownas a Petri net. We will focus on certain type of Petri net called an event graph, which can be modeled by max-plus linear recurrence relations of the form discussed in Section 1.5. We start by defining the relevant terms and setting out some notation. In order to fully appreciate all the concepts we introduce, it may be helpful to read this section alongside the example that follows (Section 2.2.2). Definition 2.1. Let G = (V, E) be a graph and let i, j ∈ V. We say that i is a predecessor (or an upstream vertex) of j if (i, j) ∈ E, and that i is a successor (or a downstream vertex) of j if (j, i) ∈ E. Definition 2.2. A Petri net is a pair (G, µ) where G = (V, E) is a directed graph and µ is a vector, satisfying the following properties: (i) G is bipartite, i.e. V is partitioned into two disjoint sets P and Q (called places and transitions respectively) such that E only consists of edges of the form (pi, qj) and (qj, pi), with pi ∈ P and qj ∈ Q. (ii) µ is a |P|-vector of non-negative integers, known as the initial marking. Definition 2.3. An event graph is a Petri net in which every place has exactly one upstream and downstream transition. Notation. For general i ∈ V, we let π(i) denote the set of all predecessors of i and σ(i) denote the set of all successors of i. In the case of Petri nets and event graphs, when we want to work with indices we will sometimes use the following additional notation: if pi ∈ π(qj), we write i ∈ πq(j), and if qj ∈ π(pi), we write j ∈ πp(i). Similarly, if pi ∈ σ(qj), we write i ∈ σq(j), and if qj ∈ σ(pi), we write j ∈ σp(i). Note that in the case of an event graph, for any place pi we have that |πp(i)| = |σp(i)| = 1, so we often allow the abuse of notation πp(i) = j (as opposed to πp(i) = {j}). We can think of places as conditions and transitions as events. For example, a machine working on a part is a place, and a transition occurs when the part moves on to the next machine. Each place has an associated marking (given initially by the vector µ) which indicates whether or not the condition has been fulfilled, e.g. whether or not a machine is working on a given part type. Equivalently we say that each place has an associated number of tokens, which can be thought of as the number of data items or resources available at each place. In our example each place can have either 0 or 1 tokens, but in general there can be any amount (e.g. if machines are capable of working on more than one part at once). We say that a transition is enabled if each of its upstream places contains at least one token. When this is the case the transition fires, meaning that one token is removed from each of its upstream places and one token is added to each of its downstream places. If the initial marking 42
- 48. is µ, atransition firing gives a new marking µ, defined by µi = µi − 1 if pi ∈ π(qj) µi + 1 if pi ∈ σ(qj) µi otherwise In this case we say that the marking µ is reachable from µ. It is easy to see that for a general Petri net the total number of tokens can change when a transition fires; for example a transition may have one upstream place but two downstream places, in which case the transition firing causes the total number of tokens to increase by one. Furthermore, note that the definition of an event graph allows for input and output transitions (known and sources and sinks respectively), i.e. transitions that do not have any upstream or downstream places. Source transitions are enabled by the outside world and deliver tokens into the system; sink transitions remove tokens from the system completely. The following definition makes an important distinction between two types of event graph: Definition 2.4. An event graph is autonomous if it contains no source transitions, and non- autonomous otherwise. The important property of event graphs is that they do not allow for models conflicts; that is, a token in a given place can be consumed by only one predetermined transition. The ‘opposite’ to an event graph (i.e. a Petri net in which each transition has exactly one upstream place and one downstream place), known as a state machine, does allow for this competition element but does not admit synchronisation. It can be shown that state machines are equivalent to the automata studied in computer science, which shows that Petri nets in general have more modelling power than automata. Up until now, the theory we have introduced is only concerned with the ordering of events. If we wish to investigate network performance, it is necessary to introduce time. There are two ways in which this could be done: we can either associate durations with transition firings, or holding times with places. In fact, in many applications it could be that both times are present; for example the real-life manufacturing system in Section 2.1 would exhibit travel times as well as processing times. However, as we noted before, by incorporating the firing times into the holding times at places, in the case of event graphs it may be assumed without loss of generality that the firing times are equal to 0. We therefore introduce the concept of timed event graph below. Definition 2.5. A timed event graph is an event graph endowed with a |P|-vector α of holding times associated with each place. Note that the definition of a timed event graph does not uniquely determine all future firing times. This is because the initial marking does not specify how long each token has spent in its respective place. We will deal with this more fully when we come to look at the basic autonomous equation in the next section. 43
- 49. 2.2.2 A SimpleExample To consolidate all of this theory, consider this simple example. A train network connects the main stations of two cities. There are two routes from station S1 to station S2; one visiting an intermediate station S3 along the way and the other visiting a different intermediate station S4. Trains link up at S2 and return to S1 via a single fast track with no stops, where they then split up again and repeat their respective journeys. There are also two inner-city loops at S1 and S2 which visit the suburbs of their respective cities. The travel time from Sj to Sl is given as the (l, j)-th entry of the matrix A below: A = 2 5 ε ε ε 3 5 3 2 ε ε ε 4 ε ε ε . (2.2) We can represent this network as a standard graph as follows: S1 S3 S4 S22 2 4 3 5 5 3 Figure 2.2: Standard graph of the simple train network. Stations (the vertices) are represented by circles and tracks by weighted edges. The travel times are given by the edge weights. Similarly to before, we can assume that there are no waiting times at stations by incorporating them into the travel times. We want the system to be synchronised in the sense that trains arriving at a station should wait for each other to allow for the changeover of passengers. This means that departures from a given station will coincide (once the last train has arrived, all trains can then depart). We can model this system with a timed event graph, where ‘tracks’ are represented by places (the timed elements of the network); trains by tokens and departures at each station by transitions. Note that each transition has an upstream place so the event graph will be autonomous. In order to fully specify the event graph we need to state the positions of the trains in the network at time 0, which corresponds to the initial marking. We assume that at time 0 there is one train travelling from S1 to S3, two trains travelling from S1 to S4, one train travelling back from S2 to S1 and one train on each of the inner-city loops. This gives the following timed event graph, pictured at time 0: 44
- 50. q1 2 4 q4 3 q2 3 2 q3 5 5 Figure 2.3: Timedevent graph of the train network depicted in Figure 2.2. The transitions q1, q2, q3 and q4 represent departures from the four respective stations. The edges can be thought of as the tracks between stations, with the intermediate places (depicted as circles) specifying the travel times. Tokens inside the places represent trains on the tracks. Note that transitions are depicted by bars, places by circles and tokens by counters inside the circles. As we have noted before, we cannot tell which transition will fire first since we do not know how long each token of the initial marking has spent in its respective place (i.e. how close to their respective destinations the trains are at time 0). If transitions q3 and q4 both fire once, the token distribution changes to the following: q1 2 4 q4 3 q2 3 2 q3 5 5 Figure 2.4: Timed event graph of the train network after transitions q3 and q4 have fired. One token has been removed from each of their upstream places and one token has been added to each of their downstream places. This corresponds to the train on the track from S1 to S3 having reached S3 and departed for S2, and also one of the trains on the track from S1 to S4 having reached S4 and departed for S2. Once these trains both reach S2 they link up to form one train, and assuming the inner-city train at S2 is ready and waiting, transition q2 will fire and the token distribution of the event graph will change to: 45
- 51. q1 2 4 q4 3 q2 3 2 q3 5 5 Figure 2.5: Timedevent graph of the train network after transitions q3, q4 and then q2 have fired. The total number of tokens in the system has decreased by one since q2 has one more upstream place than it does downstream place. This corresponds to two trains joining together at S2 to form one fast train on the track from S2 to S1. Note that the total number of tokens in the system has decreased by one, corresponding to the trains linking up at S2. Similarly, whenever transition q1 fires (i.e. trains depart from station S1), the total number of tokens in the system will increase by one, which corresponds to the fast train splitting into two. 2.3 The Basic Autonomous Equation 2.3.1 Derivation In both of our examples thus far, the timings involved have been constant. We will now discuss event graphs in which the holding times are allowed to vary with k, and derive general evolution equations for the firing times of each transition. This section deals exclusively with autonomous event graphs; the non-autonomous case is discussed in Section 2.5. Also note that from now on, any autonomous event graphs with which we work are assumed to be strongly connected (see part (ii) of Definition 1.22). The main problem that arises when modelling event graphs with variable timing is that tokens can ‘overtake’ each other when traversing places. This results in the simple ordering of events breaking down and the system fails to be linear in the max-plus sense. We will therefore restrict our analysis to the case of event graphs with First In First Out (FIFO) places, which preserve linearity. Definition 2.6. A place pi is FIFO if the k-th token to enter pi is also the k-th token to start contributing to enabling the downstream transition at pi. Definition 2.7. A timed event graph is FIFO if all of its places are FIFO. Consider a FIFO timed event graph (G, µ) with set of places P and set of transitions Q. We now allow the holding times at each place to vary with k, so for i = 1, . . . , |P|, k ∈ Z, we denote the holding time of the k-th token at place pi by αi(k). We want to model the firing times of 46
- 52. each transition, sofor j = 1, . . . , |Q| and k ∈ Z, let xj(k) be the time when transition qj fires for the k-th time. These state variables are called daters. By convention, we continue the state variables to non-positive values of k using the relation xj(k) = ε ∀k ≤ 0. As we have noted before, in order to fully describe the system we will need to associate timings with each token specified in the initial marking µ. We do this using the concept of lag times: Definition 2.8. The lag time of the k-th token in the initial marking µi of pi, denoted wi(k), is the time at which the token starts contributing to enabling its downstream transition, σ(pi). Assuming that we start looking at the system evolution at time 0, these lag times should be compatible with the general rules that transitions fire as soon as they are enabled, and that tokens enable the downstream transition as soon as they have completed their holding times. In general, the initial condition of a timed event graph (consisting of the initial marking µ and a collection of lag times) must satisfy the condition of weak compatibility: Definition 2.9. The initial condition of a timed event graph is weakly compatible if 1. The lag time of each initial token does not exceed its holding time. 2. The time when the first transition fires is non-negative. Condition (1) means that tokens of the initial marking ‘enter’ pi before time 0. Condition (2) means that without loss of generality we can assume lag times to be non-negative, since negative lag times would only be relevant if the lag times of all the predecessors of a given transition were negative, in which case this transition would fire before time 0. By convention we order the lag times at each place pi in a non-decreasing fashion, i.e. wi(1) ≤ wi(2) ≤ · · · ≤ wi(µi). This amounts to choosing the ordering of the initial tokens in such a way that the initial token with lag time wi(k) is also the kth token of place pi (that is, the k-th token to enable σ(pi)). This immediately gives us the following two results (the proofs are straightforward): Lemma 2.10. The firing of qj that consumes the k-th token of pi (for all pi ∈ π(qj)) is the k-th firing of qj. Lemma 2.11. The k-th firing of qj, k ≥ 1, produces the (k+µi)-th token of pi, for all pi ∈ σ(qj). We can now begin to derive the evolution equations for our system. Firstly, let M := max i=1,...,|P| µi M gives the maximum number of tokens at any place in the initial marking, and indicates the order of the system. Define the |Q| × |Q| matrices A(k, k), A(k, k − 1), . . . , A(k, k − M) by Ajl(k, k − m) := {i∈πq(j) | πp(i)=l, µi=m} αi(k), (2.3) 47
- 53. and the |Q|-dimensionalvector v(k), k = 1, . . . , M, by vj(k) := {i∈πq(j) | µi≥k} wi(k). (2.4) The entry Ajl(k, k − m) gives the maximum holding time of the k-th token at all the places directly between transitions ql and qj with initial marking m. Similarly, the entry vj(k) gives the maximum lag time of the k-th token at all the upstream places of qj, assuming the k-th token at qj was present in the initial marking µ. We can now state the following important theorem, which gives what is known as the basic autonomous equation: Theorem 2.12. For an autonomous FIFO timed event graph, the state vector x(k) satisfies the evolution equation: x(k) = M m=0 A(k, k − m) ⊗ x(k − m) ⊕ v(k) (2.5) for k ∈ Z, where for all j = 1, . . . , |Q|, xj(k) := ε ∀k ≤ 0, and vj(k) = ε ∀k ≥ M. Proof. Let k ∈ Z and j ∈ {1, . . . , |Q|}. The k-th firing of transition qj starts as soon as the k-th token of pi contributes to enabling qj, for all i ∈ πq(j). By lemmas 2.10 and 2.11, for k > µi, this k-th token is produced by the (k −µi)-th firing of transition π(pi), so the time at which this token contributes to enabling σ(pi) is αi(k) ⊗ xπp(i)(k − µi). For k ≤ µi, this k-th token begins to contribute to the enabling of qj at time wi(k). We therefore have that the state variables xj(k), j = 1, . . . , |Q|, satisfy the evolution equations: xj(k) = {i∈πq(j) | k>µi} αi ⊗ xπp(i)(k − µi) ⊕ {i∈πq(j) | k≤µi} wi(k) . (2.6) We now use associativity and commutativity of ⊕ to rewrite (2.6) as xj(k) = M m=0 |Q| l=1 {i∈πq(j) | πp(i)=l, µi=m} αi(k) ⊗ xl(k − m) ⊕ {i∈πq(j) | k≤µi} wi(k) . The distributivity of ⊗ with respect to ⊕ now gives that xj(k) = M m=0 |Q| l=1 {i∈πq(j) | πp(i)=l, µi=m} αi(k) ⊗ xl(k − m) ⊕ {i∈πq(j) | k≤µi} wi(k) = M m=0 |Q| l=1 Ajl(k, k − m) ⊗ xl(k − m) ⊕ vj(k) 48
- 54. and therefore x(k) = M m=0 A(k,k − m) ⊗ x(k − m) ⊕ v(k) as required. 2.3.2 Extensions to the Initial Condition This section is devoted to a discussion on the initial condition of a general timed event graph. Note that the first weak compatibility condition given in Definition 2.9 can be translated into algebraic form as follows: wi(k) ≤ αi(k), i = 1, . . . , |P|, 1 ≤ k ≤ µi. (2.7) Furthermore, we know that the first transition to fire is necessarily contained in the set of transitions qj for which all pi ∈ π(qj) have at least one token in the initial marking. Since the set of tokens with the smallest lag times is the first to be consumed, the second weak compatibility condition can be translated as vj(1) ≥ e, for all j such that µi ≥ 1 ∀pi ∈ π(qj), where v is defined as in Theorem 2.12. Together, these form two linear constraints on the lag times wi(k). We can formulate an additional constraint on the initial condition which allows us to significantly simplify the basic autonomous equation of Theorem 2.12. Initial conditions that are weakly compatible and satisfy this extra constraint will be known as compatible. Firstly, for all i such that µi > 0, denote by yi(k), k ≤ 0, the entrance function associated with place pi, defined by yi(k − µi) := wi(k) ⊗ αi(k)⊗−1 if 1 ≤ k ≤ µi ε if k > µi . Note that max-plus multiplication by αi(k)⊗−1 is equivalent to the subtraction of αi(k) in conventional algebra. As the name suggests, the entrance function for a specified token of the initial marking at a given place gives the time at which that token ‘entered’ the system, which by the first condition of weak compatibility (see Definition 2.9) must be non-positive. The reason for using a non-positive argument in the functions yj(k) will become clear below. An initial condition is said to be compatible if it is weakly compatible and for any pair of places pi and pj which follow the same transition, the entrance times yi(k) and yj(k) coincide (provided k ≥ − min µi, µj ): Definition 2.13. An initial condition is compatible if it is weakly compatible and if there exist functions zj(k), j = 1, . . . , |Q|, k ≤ 0, such that yi(k) = zπp(i)(k), ∀i, k such that − µi + 1 ≤ k ≤ 0. 49
- 55. Note that thefunctions zj(k) are only defined for −Mj < k ≤ 0, where Mj := max µi | i ∈ σq (j) provided Mj ≥ 1. For other values of k, or if Mj = 0, we take zj(k) = ε. The simplest example of a compatible initial condition is to choose wi(k) = αi(k) for all 1 ≤ k ≤ µi. This means that all the initial tokens enter the system at time 0 and cannot enable downstream transitions until their full holding time has been spent. In this case, the functions zj(k) are simply defined by zj(k) = e if − Mj < k ≤ 0 ε if k ≤ Mj . Now, recall that our usual continuation of the state variables xj(k) to k ≤ 0 consisted of taking xj(k) = ε ∀k ≤ 0. If, when our initial condition is compatible, we instead take xj(k) = zj(k) ∀k ≤ 0, j = 1, . . . , |Q| (2.8) Then we can refine Theorem 2.12 to give the following: Theorem 2.14. For an autonomous FIFO timed event graph with a compatible initial condi- tion, the state vector x(k) satisfies the evolution equation: x(k) = M m=0 A(k, k − m) ⊗ x(k − m) for all k ∈ N. Proof. Using the definition of the entrance time function and the fact that the initial condition is compatible, we have that {i∈πq(j) | k≤µi} wi(k) = {i∈πq(j) | k≤µi} αi(k) ⊗ yi(k − µi) = {i∈πq(j) | k≤µi} αi(k) ⊗ zπp(i)(k − µi) for all k ∈ Z. We can substitute this into equation (2.6) to obtain xj(k) = {i∈πq(j)} xπp(i)(k − µi) ⊗ αi(k) for all k ∈ Z. Using the same reasoning as in Theorem 2.12, this implies that x(k) = M m=0 A(k, k − m) ⊗ x(k − m) 50
- 56. for all k∈ Z, as required. 2.3.3 Solving the Basic Autonomous Equation In this section we will aim to solve the evolution equations of an autonomous FIFO timed event graph supplemented with a weakly compatible initial condition. These equations are given in Theorem 2.12. We will restrict ourselves to the case of live event graphs; that is, graphs in which all the transitions remain active. For a general Petri net, the precise definition of ‘live’ is given below. Definition 2.15. A Petri net is live (with respect to the initial marking µ) if for any µ1, obtained after an arbitrary series of firings starting from µ, and for each transition qj, there exists another marking µ2 which can be obtained after a suitable series of firings starting from µ1, such that qj is enabled in µ2. We now state and prove two lemmas that will allow us to apply the theory developed in Section 1.5 to the general evolution equations of Section 2.3. Lemma 2.16. The following are equivalent: 1. An autonomous event graph G is live. 2. Every circuit in G contains at least one token with respect to the initial marking µ. Proof. Let (G, µ) be an autonomous event graph and consider a circuit in G. If there are no tokens at each place in the circuit with respect to the initial marking µ then this circuit will remain free of tokens and all of its transitions will never fire (since the only way tokens could enter the circuit is if one of its transitions fired, which will never happen because all the transitions in the circuit have an upstream place free of tokens). Conversely, since G is an autonomous event graph, if an arbitrary transition qj in G never fires then there exists at least one upstream transition qi that never fires also (since if all its upstream transitions did fire then all the predecessor places of qj would contain a token and qj would also fire). The intermediate place between qi and qj must therefore be token free. Since G is finite, by repeating this argument we will eventually return to the same transition qj, and so we have found a circuit free of tokens with respect to the initial marking µ. Lemma 2.17. An autonomous event graph is live if and only if for all k, the communication graph of the matrix A(k, k) contains no circuits. Proof. Let (G, µ) be an autonomous event graph. Recall that The entry Ajl(k, k) gives the maximum holding time of the k-th token at all the places directly between transitions ql and qj with no tokens in their initial marking. Thus an edge (ql, qj) exists in G(A(k, k)) if and only if there is a place in G directly between ql and qj with a zero initial marking. Therefore any circuit in G(A(k, k)) corresponds to an equivalent circuit in G, consisting of alternating places and transitions, such that all the places in the circuit have a zero initial marking. By Lemma 2.16 above, such a circuit exists if and only if the event graph (G, µ) is live. 51
- 57. Lemma 2.17 tellsus that in particular, the communication graph of the matrix A(k, k) contains no circuits of positive weight. Therefore, letting A(k, k − m) := A∗ (k, k) ⊗ A(k, k − m) ∀k ∈ Z, m = 1, . . . , M, (2.9) and v(k) := A∗ (k, k) ⊗ v(k) ∀k ∈ Z, (2.10) we can apply the technique described in Section 1.5 (whereby equation (1.15) was transformed into equation (1.18)) to obtain the following result: Theorem 2.18. For a live autonomous FIFO timed event graph with a weakly compatible initial condition, the evolution equation of the state vector x(k) can be written as x(k) = M m=1 A(k, k − m) ⊗ x(k − m) ⊕ v(k) (2.11) where xj(k) := ε ∀k ≤ 0. The significance of this result lies in the fact that the sum starts from m = 1, so the basic autonomous equation has been transformed into an explicit M-th order recurrence relation. Note that if the initial condition is compatible (and one takes the appropriate continuation of x(k) for k ≤ 0), the same argument shows that for k ∈ Z, the evolution equation can be written as x(k) = M m=1 A(k, k − m) ⊗ x(k − m). (2.12) As an aside, note that the matrix A(k, k−m), m ≥ 1, has a simple graph-theoretic interpretation. Let S(j , j, m) be the set of paths in the event graph G from transition qj to qj, such that the first two transitions are connected by a place with initial marking equal to m, while all the other places in the path have a zero initial marking. Then we have that Ajj (k, k − m) = {p=(j1,i1,j2,i2,...,ih−1,jh) ∈ S(j ,j,m)} h−1 n=1 αin (k). That is, the entry Ajj (k, k −m) gives the maximum holding time of all the paths in S(j , j, m). Note that so far, we have derived recurrence relations from event graphs. It is also possible to associate ‘new’ event graphs with the recurrence relations we obtain from our manipulations. For example, the relations (2.11) and (2.12) can be associated with an event graph derived from the original graph via the following transformation rules: 1. Take the same set of transitions as in the original graph. 52
- 58. 2. For eachpath of S(j , j, m), m ≥ 1, in the original event graph, create a place connecting j to j with m tokens and with a holding time given by the product of the holding times of the path in question. The question arises of what relation such derived event graphs have to the original event graph. We would expect them to be in some sense ‘equivalent’ (after all, they model the same system). The following definition makes this precise. Definition 2.19. Two event graphs (G1, µ1) and (G2, µ2) are equivalent if there exists a bijec- tion from a non-empty subset of the transitions of G1 to a non-empty subset of the transitions of G2 such that two corresponding transitions of G1 and G2 always fire at the same time. It is not difficult to see that the event graph defined by the recurrence relation (2.11) is equivalent to the original event graph, but with the additional property that all the places have a strictly positive initial marking. This is because there is no A(k, k) term in the equation. We now continue with our manipulation of the basic autonomous equation (2.5). Our aim is to reduce the size of the state space without losing any information on the evolution of the system. To do this, note that if there is an index j such that all the entries in the j-th column of A(k, k − m) are equal to ε for all m = 1, . . . , M, then this column makes no contribution to the equation (2.11). Equivalently, by definition of the matrices A(k, k − m), we can ignore any transitions whose downstream places all have a zero initial marking. We therefore define Q to be the set of transitions followed by at least one place with a positive initial marking, and consider the equivalent event graph obtained by ‘deleting’ all the transitions not in Q . Accordingly, our equations now only involve the reduced set of state variables xj(k), j ∈ Q , k ≥ 1. The remaining variables can be obtained as ‘output variables’ by using equation (2.11) once again: the values of xj(k), j ∈ Q Q , k ≥ 1, only require the values of the reduced state variables xj(k), j ∈ Q , k ≥ 1, and do not form part of the recurrence relation itself. To finish this section, we want to transform the Mth-order recurrence relation (2.11) into an equivalent recurrence relation of order 1. Once again we apply the techniques given in Section 1.5. We assume that the state space has been reduced according to the remark above, and relabel the transitions 1, . . . , |Q |. Define the (|Q | × M)-dimensional vector x(k) := x(k) x(k − 1) ... x(k + 1 − M) (2.13) 53
- 59. and let A(k),k ∈ Z, be the (|Q | × M) × (|Q | × M) matrix defined by the relation A(k) := A(k + 1, k) A(k + 1, k − 1) . . . . . . A(k + 1, k + 1 − M) E E . . . E E E E ... ... E ... ... E E E E . . . E E E (2.14) where E denotes the |Q | × |Q | matrix of ε. Finally, let v(k) be the (|Q | × M)-dimensional vector v(k) := v(k + 1) ε ... ε . Then we can state the following corollary, which gives the standard form of the basic autonomous equation: Corollary 2.20. The extended state space vector x(k) satisfies the (M × |Q |)-dimensional first-order evolution equation x(k + 1) = A(k) ⊗ x(k) ⊕ v(k), (2.15) for k ∈ Z. Proof. This technique is standard and the result follows immediately from conducting the ap- propriate matrix multiplications. Once again, in the particular case of a compatible initial condition, these equations read x(k + 1) = A(k) ⊗ x(k) provided one uses the appropriate continuation of xj(k) for k ≤ 0. Note that we can associate the standard form basic autonomous equation given in Corollary 2.20 with an event graph which is once again equivalent to the original event graph, but this time with initial marking equal to 1 everywhere. An algorithm for obtaining the derived graph can be found in [2]. 2.3.4 Behaviour of the Solution Assume for a moment that the holding times of our event graph are constant and that we forgo the state space reduction described above. We have that the evolution of a live autonomous FIFO timed event graph (G, µ), supplemented with a compatible initial condition, is described 54
- 60. by the recurrencerelation x(k) = M m=0 A(m) ⊗ x(k − m) and we have just shown that the same equation in standard form x(k + 1) = A ⊗ x(k) equivalently describes the evolution of the system, and can be associated with an event graph G which is equivalent to G. Since we assume that G is strongly connected it can also be assumed that G is strongly connected, provided unnecessary transitions (not involved in circuits) be cancelled. Hence we can assume that A is irreducible. We can therefore apply Theorem 1.49 to see that after a finite period of time, the solution will enter a periodic regime with period t(A) and asymptotic growth rate λ, where λ is the unique eigenvalue of A. An interpretation of this is that on average, each transition fires once every λ units of time, i.e. the long-run throughput of each transition is 1/λ, thus achieving some sort of stationarity. A nice result given in [2] is that the value λ can be obtained directly from the original event graph G, provided we modify slightly our definition of path lengths and weights. To be specific, in the case of an event graph we can think of places as edges with weight equal to their holding time, and length equal to the number of tokens in their initial marking. Using this definition, λ is then equal to the maximal average circuit weight in G. The behaviour of the solution x(k) in the case of variable holding times is more difficult to describe. It can be shown that if there are sufficiently long sub-intervals for which the holding times are constant then the solution exhibits periodicity within each of these intervals. This can be seen in our simple example to which we return below. 2.4 A Simple Example Revisited 2.4.1 General Solution Consider the example given in Section 2.2.2, and suppose that we extend to the model to allow for variable travel times between stations. We denote the travel time of the k-th train between Sj and Sl by αlj(k). Similarly, wlj(k) denotes the remaining travel time of the k-th train between Sj and Sl present at time 0 (and is equal to ε if no such train exists). We assume that the initial marking is the same as that given in Figure 2.3. The aim of this section is to apply the theory developed above to find a timetable describing the optimal departure times of trains at each station. Under the assumption that the holding times are such that the event graph is FIFO, according 55
- 61. to (2.3) wefirstly define the matrices A(k, k) = ε ε ε ε ε ε α23(k) α24(k) ε ε ε ε ε ε ε ε , A(k, k − 1) = α11(k) α12(k) ε ε ε α22(k) ε ε α31(k) ε ε ε ε ε ε ε , A(k, k − 2) = ε ε ε ε ε ε ε ε ε ε ε ε α41(k) ε ε ε , and according to (2.4), the vectors v(1) = w11(1) ⊕ w12(1) w22(1) w31(1) w41(1) , v(2) = ε ε ε w41(2) . In order for the initial condition to be weakly compatible, as per Section 2.3.2 the following algebraic constraints must hold true (see equation (2.7)): wlj(k) ≤ αlj(k) ∀l, j = 1, 2, 3, 4, w11(1) ⊕ w14(1) ≥ e, w21(1) ≥ e, w31(1) ≥ e. Then by Theorem 2.12, the departure times at each station given by the state vector x(k) = (x1(k), x2(k), x3(k), x4(k)) satisfy the basic autonomous equation (2.5): x(k) = A(k, k) ⊗ x(k) ⊕ A(k, k − 1) ⊗ x(k − 1) ⊕ A(k, k − 2) ⊗ x(k − 2) ⊕ v(k) If the initial condition is compatible (for example if wlj(k) = αlj(k) ∀l, j = 1, 2, 3, 4), then we set z1(0) := w11(1) ⊗ α⊗−1 11 (1) = w31(1) ⊗ α⊗−1 31 (1) = w41(2) ⊗ α⊗−1 41 (2), z1(−1) := w41(1) ⊗ α⊗−1 41 (1), z2(0) := w12(1) ⊗ α⊗−1 12 (1) = w22(1) ⊗ α⊗−1 22 56
- 62. and according to(2.8), define x(0) := z1(0) z2(0) ε ε , x(−1) := z1(−1) ε ε ε . Then the state vector x(k) satisfies the refined evolution equation as given in Theorem 2.14: x(k) = A(k, k) ⊗ x(k) ⊕ A(k, k − 1) ⊗ x(k − 1) ⊕ A(k, k − 2) ⊗ x(k − 2) (2.16) Next, we want to transform (2.16) into an explicit recurrence relation as per Theorem 2.18. By direct calculation, we have A∗ (k, k) = e ε ε ε ε e α23(k) α24(k) ε ε e ε ε ε ε e , and therefore, according to (2.9) we define the matrices A(k, k − 1) = A∗ (k, k) ⊗ A(k, k − 1) = α11(k) α12(k) ε ε α23(k) ⊗ α31(k) α22(k) ε ε α31(k) ε ε ε ε ε ε ε , A(k, k − 2) = A∗ (k, k) ⊗ A(k, k − 2) = ε ε ε ε α24(k) ⊗ α41(k) ε ε ε ε ε ε ε α41(k) ε ε ε , and according to (2.10), the vectors v(1) = A∗ (1, 1) ⊗ v(1) = w11(1) ⊕ w12(1) w22(1) ⊕ (α23(1) ⊗ w31(1)) ⊕ (α24(1) ⊗ w41(1)) w31(1) w41(1) , v(2) = A∗ (2, 2) ⊗ v(2) = ε α24(2) ⊗ w41(2) ε w41(2) . 57
- 63. Then by Theorem2.18, the evolution equation of the state vector x(k) can be written as x(k) = A(k, k − 1) ⊗ x(k − 1) ⊕ A(k, k − 2) ⊗ x(k − 2) ⊕ v(k). We now examine whether we can reduce the size of the state space. Note that the downstream places of transitions 3 and 4 all have a zero initial marking. We therefore let Q := {q1, q2}, and the evolution equation is reduced to x1(k) x2(k) = α11(k) α12(k) α23(k) ⊗ α31(k) α22(k) x1(k − 1) x2(k − 1) ⊕ ε ε α24(k) ⊗ α41(k) ε x1(k − 2) x2(k − 2) ⊕ v1(k) v2(k) . The other state variables are obtained from the reduced state variables by the relation x3(k) x4(k) = α31(k) ε ε ε x1(k − 1) x2(k − 1) ⊕ ε ε α41(k) ε x1(k − 2) x2(k − 2) ⊕ v3(k) v4(k) . (2.17) Finally, we apply Corollary 2.20 to transform the evolution equation into a first order system. According to (2.13), we set x(k) := x1(k) x2(k) x1(k − 1) x2(k − 1) , v(k) := v1(k + 1) v2(k + 1) ε ε , and according to (2.14) A(k) := α11(k + 1) α12(k + 1) ε ε α23(k + 1) ⊗ α31(k + 1) α22(k + 1) α24(k + 1) ⊗ α41(k + 1) ε e ε ε ε ε e ε ε , then the evolution equation becomes x(k + 1) = A(k) ⊗ x(k) ⊕ v(k). (2.18) Equation (2.18), supplemented with a weakly compatible or compatible initial condition, is sufficient to uniquely determine all future departure times of trains from each station in the network. 58
- 64. 2.4.2 An OptimalTimetable We now assume the travel times given in (2.2). In addition, suppose we know that all the tracks into and out of station S2 are becoming worn, so after five more journeys there will be speed restrictions imposed such that travel times along these tracks increases by one time unit. The initial distribution of trains in the network is given as in Figure 2.3. We assume that at time 0 all the trains have just departed from their respective stations, with the exception of the first train travelling from S1 to S4 which is half way through its journey. This means that all the lag times are equal to their corresponding holding times, apart from w41(1) which is equal to 2. This set-up defines a compatible initial condition, so substituting all this information into the solution we found above gives us that the system satisfies the evolution equation x(k + 1) = A(k) ⊗ x(k) where x(k) is defined as x(k) = x1(k) x2(k) x1(k − 1) x2(k − 1) and the matrix A(k) is defined as A(k) = 2 5 ε ε 5 3 4 ε e ε ε ε ε e ε ε for k ≤ 4, 2 6 ε ε 6 4 4 ε e ε ε ε ε e ε ε for k > 4, and with initial condition x(0) = e e −2 ε . Notice that in using the reduced state variable framework, this system only gives us the depar- ture times at stations S1 and S2. By recovering the values of the other state variables using (2.17), we can iteratively find the values of x1(k), . . . , x4(k) for k = 1, 2, . . . to produce the following optimal timetable, which can be read in the way one would read any standard real-life train timetable: 59
- 65. Departure Times S1 -0 5 12 17 24 29 37 43 51 57 S2 - 0 7 12 19 24 31 37 45 51 59 S3 - 2 7 14 19 26 31 39 45 53 59 S4 2 4 9 16 21 28 33 41 47 55 61 Table 2.3: Optimal Timetable of the Train Network. Recall that synchronisation results in departures from stations coinciding, so the times given here refer to departures from each station in all directions (for example, at time 5 one train departs on the inner-city loop at S1, a second departs for S3 and a third departs for S3). Note that we have shifted the variable k by 1 and 2 for the third and fourth rows respectively so that the departure times line up nicely. Specifically, the columns in Table 2.3 represent the vectors (x1(k), x2(k), x3(k + 1), x4(k + 2)) for k = −1, 0, 1, . . . , 9. This results in a sequence of departure times that is increasing (when read down and then across). This is purely for presentational reasons - we opt to make the departure timetable easy to read for someone travelling through the network starting at station S1. Notice that as per Section 2.3.4, the solution settles into a temporary periodic regime for k ≤ 5, with asymptotic growth rate λ1 = 6 and transient time 2. λ1 here corresponds to the maximal average weight of all the circuits of the event graph given in Figure 2.3 (where we think of the holding times as weights and the number of tokens in the initial marking as lengths; see Section 2.3.4). For k > 5, the solution enters a second periodic regime with asymptotic growth rate λ2 = 7. λ2 once again corresponds to the maximal average weight of all the circuits of the original event graph but with the holding times updated to reflect the adjusted travel times. There is an additional transient period of 2 time units (k = 6 to 8) during which the solution is temporarily aperiodic. In both cases the period is equal to 2, which therefore must be the cyclicity of the matrix A (k) for which the whole system is described by the evolution equation x(k + 1) = A (k) ⊗ x(k) (i.e. if we had not reduced the state space). We know that this matrix must be irreducible (again see the discussion in Section 2.3.4), but it is not immediate that the support of its critical graph (i.e. the set of edges (i, j) ∈ Ec(A (k)) for which [A (k)]ji = ε) is invariant under changes in k. In fact, in this case we can show that it is, which explains why the cyclicity does not change for the second periodic regime. 2.5 The Non-autonomous Case This section focuses on FIFO timed event graphs that are non-autonomous; that is, event graphs that include source transitions. Whilst we require one or two additional concepts, a lot of the theory is analogous to the autonomous case and we therefore omit proofs whenever appropriate. Firstly, we define a new class of transitions known as input transitions. Definition 2.21. An input transition consists of a source transition and a non-decreasing 60
- 66. sequence of realnumbers, called the input sequence. We denote the set of input transitions by I. The input sequence associated with transition qj ∈ I is denoted uj(k), k ≥ 1, and gives the time that qj fires for the k-th time (due to some external trigger). The input sequences form part of the specification of the event graph, i.e. they are known. As one might expect, in order for them to make sense, the input sequences must satisfy some condition: Definition 2.22. The input sequence uj(k), k ∈ Z, is weakly compatible if uj(1) ≥ 0. Regarding the initial condition, the concepts of weak compatibility and compatibility remain the same, but with the additional requirement that all the input sequences are weakly compatible. In what follows we will assume that we are dealing with a non-autonomous FIFO timed event graph with a weakly compatible initial condition. Similarly to before, let M := max µi | i = 1, . . . , |P| . Define the |Q| × |I| matrices B(k, k), . . . , B(k, k − M) by Bjl(k, k − m) := {i∈πq(j) | πp(i)=l,µi=m} αi(k) and the |I|-dimensional vector u(k) = (u1(k), . . . , u|I|(k)) , k ∈ N. We can then state the following theorem, which is the non-autonomous extension to Theorem 2.12 (and is proved in a similar way). This gives us what is known as the basic non-autonomous equation: Theorem 2.23. For a non-autonomous FIFO timed event graph, the state vector x(k) satisfies the evolution equation x(k) = M m=0 A(k, k − m) ⊗ x(k − m) ⊕ M m=0 B(k, k − m) ⊗ u(k − m) ⊕ v(k) for all k ∈ N, where xj(k) = uj(k) := ε for all k ≤ 0, and vj(k) is defined as in (2.4) for 1 ≤ k ≤ M (and is equal to ε otherwise). Once again, in the case of a compatible initial condition, by taking the continuation of x(k) to non-positive values of k as defined in (2.8) we can refine this result to give the following theorem, which is the non-autonomous analogy of Theorem 2.14: Theorem 2.24. For a non-autonomous FIFO timed event graph with a compatible initial con- dition, the state vector x(k) satisfies the evolution equation x(k) = M m=0 A(k, k − m) ⊗ x(k − m) ⊕ M m=0 B(k, k − m) ⊗ u(k − m) 61
- 67. for all k∈ N, where the continuation of u(k) to non-positive values of k is defined by wi(k) = αi(k) ⊗ uj(k − µi) for all pi ∈ σ(qj) with µi ≥ 1, and for all 1 ≤ k ≤ µi. We can say that a non-autonomous event graph is live if its associated autonomous event graph (i.e. the one associated with the equation x(k) = M m=0 A(k, k − m) ⊗ x(k − m)) is live. If this is the case, as before, for k ∈ Z, m = 1, . . . , M, we let A(k, k − m) := A∗ (k, k) ⊗ A(k, k − m), and B(k, k − m) := A∗ (k, k) ⊗ B(k, k − m). Then the following theorem is the non-autonomous analogue to Theorem 2.18: Theorem 2.25. For a live non-autonomous FIFO timed event graph with a weakly compatible initial condition, the evolution equation of the state vector x(k) can be written as x(k) = M m=1 A(k, k − m) ⊗ x(k − m) ⊕ M m=1 B(k, k − m) ⊗ u(k − m) ⊕ v(k) where xj(k) := ε ∀k ≤ 0. If the initial condition is compatible then these equations can also be similarly simplified as we did in Section 2.3.3. Finally, after reducing the state space like we did before, if we define the (M × |I|)-dimensional vector u(k) := u(k + 1) u(k) ... u(k + 2 − M) and the (|I| × M) × (|Q | × M) matrix B(k) := B(k + 1, k + 1) B(k + 1, k) . . . B(k + 1, k + 2 − M) E E . . . E ... ... ... ... E E . . . E then we can state the standard form of the basic non-autonomous equation, which is analogous to Corollary 2.20: Corollary 2.26. The extended state space vector x(k) satisfies the (M × |Q |)-dimensional 62
- 68. first-order recurrence relation x(k+ 1) = A(k) ⊗ x(k) ⊕ B(k) ⊗ u(k) ⊕ v(k) (2.19) for k ∈ N, with the standard simplification if the initial condition is compatible. To end this chapter, let U(k) be the diagonal (|I|×|I|)-dimensional matrix with entries Ujj(k) := uj(k) − uj(k − 1). These are known as the inter-input times. In the case of a compatible initial condition, if x(k) := (u(k), x(k)) and A(k) is the matrix defined by A(k) := U(k + 1) E B(k) A(k) , then it is clear that (2.19) can be rewritten as x(k) = A(k) ⊗ x(k) (2.20) for k ∈ N. This transformed evolution equation corresponds to an equivalent autonomous event graph in which each source transition j is viewed as a recycled transition (i.e. supplemented with a self-loop), where the holding times of the recycling place are given by the sequence Ujj(k) . Note, however, that in contrast to the remark in Section 2.3.4, the matrix A(k) now fails to be irreducible. We can therefore conclude that any system modelled by an equation of the form (2.20) is autonomous if and only if A(k) is irreducible, and non-autonomous otherwise. 63
- 69. Chapter 3 Stochastic EventSystems Over Max-plus 3.1 Introduction & Stochastic Background In this chapter we consider stochastic FIFO event graphs; that is, FIFO event graphs in which the holding times and lag times are random variables rather than predetermined constants. We are interested in the conditions under which these graphs enter a stationary regime. Most of the material in this chapter is taken from [1], [2] and [12]. We begin by laying some foundations of Ergodic theory and describing the statistical assumptions we will use throughout. The manner in which we introduce the concepts below may be surprising - specifically, what we state as definitions are actually theorems which follows from the classical definition of ergodicity. Whilst the reader may be more familiar with the latter approach, we find that in the context of max-plus stochastic event systems it is intuitively helpful to proceed in this way. Definition 3.1. Let (Ω, F, P) be a probability space. The mapping θ : Ω → Ω is a shift operator on (Ω, F, P) if it is bijective and measurable, and if it is such that P is left invariant by θ, namely E[f] = E[f ◦ θ] for all measurable and integrable functions f : Ω → R (where E represents the expectation w.r.t. P). Definitions 3.2. Let θ be a shift operator on (Ω, F, P). (i) A sequence of R-valued random variables a(k, ω) k∈Z defined on (Ω, F, P) is θ-stationary if a(k, ω) d = a(0; θk(ω)) ∀k ≥ 0, where θk is the composition of θ by itself k times. (ii) θ is ergodic if the almost sure (a.s.) limit lim k→∞ 1 k k l=1 f ◦ θ l (ω) = E[f] holds for all measurable and integrable functions f : Ω → R. 64
- 70. A simple exampleof an ergodic sequence of random variables is a sequence of independent identically distributed (i.i.d.) random variables, and this will often be the obvious choice for the sequence of holding times in the modelling of real-life systems such as rail networks. Notation. Recall from Definition 1.3 that for a, b ∈ Rmax, we say a ≤ b if a ⊕ b = b. In this chapter we introduce the notation a ∧ b to mean the greatest lower bound of a and b, i.e. a ≤ b ⇐⇒ a ⊕ b = b ⇐⇒ a ∧ b = a. (3.1) The ∧ operation is associative, commutative and idempotent but it does not have an identity element, and does not always distribute over ⊗. For our purposes, we can think of a∧b to mean min {a, b}. We must take extra care to ensure that some fundamental concepts of probability are applicable in the max-plus setting. For example, if X is a random variable taking values in Rmax defined on a probability space (Ω, F, P) then it may take the value ε with positive probability, which confuses the idea of integrability. We therefore call X ∈ Rmax integrable if X ⊕ e and X ∧ e are both integrable, and if E[X ⊕ e] is finite. The expected value of X is then given by E[X] = E[X ⊕e] ⊗ E[X ∧e]. Moreover, a random matrix A ∈ Rn×m max (i.e. a matrix whose entries are random variables taking values in Rmax) is called integrable if the entries aij are integrable ∀i, j, and its expected value is given by the matrix E[A] with entries E[A] ij = E[aij]. 3.2 Statistical Assumptions In this chapter we consider the max-plus evolution equation x(k + 1) = A(k) ⊗ x(k), k ∈ Z (3.2) with initial condition x(0) = x0. We will often stress the dependence of x(k) on the initial condition by writing x(k; x0). We assume that A(k) k∈Z is a sequence of random matrices defined on a common probability space (Ω, F, P) endowed with an ergodic shift operator θ, and that x0 is also a random variable defined on this space. If we let A := A(0) (3.3) then θ-stationarity gives us that A(k) d = A ◦ θ k ∀k ∈ Z. We assume in addition that each entry of A is either a.s. equal to ε or non-negative and integrable, and that each diagonal of A is non-negative. Throughout the rest of this chapter, what we have described above will be known as the SENI framework (stationary, ergodic, non-negative and integrable). Note that the assumptions of the SENI framework mean that the sequence of matrices A(k) has fixed support (i.e. the topologies of the communication graphs G(A(k)) are non-random and do not vary with k). A formal definition is given below. 65
- 71. Definition 3.3. Thesequence of matrices A(k) ∈ Rn×n max has fixed support if the probability that [A(k)]ij equals ε is either 0 or 1 and does not depend on k. The remainder of this section is devoted to showing that both autonomous and non-autonomous FIFO stochastic event graphs are described by an evolution equation which falls into the SENI framework. The Autonomous Case By the remark that follows Corollary 2.20, for an autonomous FIFO timed event graph with a compatible initial condition, the extended state space vector x(k) satisfies the evolution equation x(k) = A(k) ⊗ x(k) which is of the form (3.2). Suppose in addition that the holding times αi(k) and the lag times wi(k), i = 1, . . . , |P|, are random variables defined on a common probability space endowed with an ergodic shift operator θ, and that the sequence αi(k) k∈Z is θ-stationary. Then it is easily verified that the matrices A(k) satisfy the θ-stationarity property and that each entry of A := A(0) is either a.s. equal to ε or non-negative and integrable. Furthermore, by the FIFO assumption we have that xj(k + 1) ≥ xj(k) ∀k, and therefore Ajj(k) ≥ e for all j. Therefore, under the appropriate statistical assumptions, any autonomous FIFO stochastic event graph with compatible initial condition falls into the SENI framework. The Non-autonomous Case By the remark that follows Corollary 2.26, the evolution of a non- autonomous FIFO timed event graph with a compatible initial condition is described by the equation x(k) = A(k) ⊗ x(k) which is also of the form (3.2). If the holding times αi(k) and the inter-input times Ujj(k) are sequences of non-negative and integrable random variables satisfying θ-stationarity, then it is again straightforward to check that the matrices A(k) are also θ-stationary. The additional conditions mentioned above also follow in the same way as the autonomous case. Therefore the SENI framework also covers the non-autonomous case, provided we make additional assumptions on the inter-input times. 3.3 Asymptotic Firing Rates In Section 2.3.4 we showed that in the case of constant holding times, any strongly connected timed event graph will eventually reach a periodic regime in which each transition fires every λ units of time (where λ is a constant determined by the event graph in question). The aim of this section is to develop a corresponding theory for stochastic event graphs, i.e. to examine whether the firing rates settle into a steady state distribution after a long period of time. We will consider the case when the matrix A defined in (3.3) (herein assumed to be of dimension n × n) is both a.s. irreducible and a.s. reducible, which is a well-defined distinction owing to 66
- 72. the assumption offixed support within the SENI framework. This corresponds to stochastic event graphs which are a.s. strongly connected and a.s. not strongly connected respectively. We begin with some new notation. Notation. For A ∈ Rn×m max , let |A|⊕ := n i=1 m j=1 aij and |A|∧ := n i=1 m j=1 aij. Thus |A|⊕ gives the maximal entry of A and |A|∧ gives the minimal entry of A. Lemma 3.4. For all pairs of matrices A, B ∈ Rn×m max such that the product A ⊗ B is well defined, we have |A ⊗ B|⊕ ≤ |A|⊕ ⊗ |B|⊕ |A ⊗ B|⊕ ≥ |A|∧ ⊗ |B|⊕ |A ⊗ B|⊕ ≥ |A|⊕ ⊗ |B|∧ and |A ⊗ B|∧ ≥ |A|∧ ⊗ |B|∧ |A ⊗ B|∧ ≤ |A|∧ ⊗ |B|⊕ |A ⊗ B|∧ ≤ |A|⊕ ⊗ |B|∧ Proof. Since Aik ≤ |A|⊕ for all i, k, i,j k Aik ⊗ Bkj ≤ |A|⊕ ⊗ j,k Bkj = |A|⊕ ⊗ |B|⊕ The proofs of the other formulae follow similarly. We now recall an important theorem from Ergodic theory known as Kingman’s Theorem on subadditive ergodic processes. This will be used in the material below. We do not offer a proof here; readers are referred to [14] for the full working. Theorem 3.5 (Kingman’s subadditive ergodic theorem). Let ξm,n, m > n ∈ Z, be an integrable random process defined on the probability space (Ω, F, P) such that ξm,m+k d = ξ0,k ◦ θm ∀m ∈ Z, k > 0 (stationarity) 67
- 73. and ξm,n ≤ ξm,k+ ξk,n a.s. ∀ m < k < n (subadditivity). Assume in addition that there exists a positive constant c such that E[ξ0,k] ≥ −ck for all k > 0. Then there exists a constant γ such that the following two equations hold: lim k→∞ ξ0,k k = γ a.s., lim k→∞ E[ξ0,k] k = γ = inf k E[ξ0,k] k . We now able to begin our task of characterising the asymptotic behaviour of the firing rates in a stochastic event graph. Theorem 3.6. Within the SENI framework, there exists a non-negative constant a ∈ Rmax (known as the maximal Lyapunov exponent) such that, for all finite initial conditions x0, the limit lim k→∞ |x(k; x0)|⊗1/k ⊕ = a (3.4) holds almost surely. If the initial condition is integrable, in addition we have lim k→∞ E |x(k; x0)|⊗1/k ⊕ = lim k→∞ E |x(k; x0)|⊕ ⊗1/k = a. (3.5) Proof. Using the integrability assumptions of the SENI framework, we obtain by induction that |x(k; e)|⊕ is integrable for all k ≥ 0. We therefore have e ≤ E |x(k; e)|⊕ < ∞ ∀k ≥ 0. Now let ξm,m+k = |x(k; e)|⊕ ◦ θm , m ∈ Z, k ≥ 0. Since |x(k, e)|⊕ = |A ◦ θk−1 ⊗ · · · ⊗ A ⊗ e|⊕ = |A ◦ θk−1 ⊗ · · · ⊗ A|⊕ , we obtain from Lemma 3.4 that for any k ≥ 1 and for all 0 ≤ p ≤ k: |A ◦ θk−1 ⊗ · · · ⊗ A ◦ θp ⊗ A ◦ θp−1 ⊗ · · · ⊗ A|⊕ ◦ θm ≤ |A ◦ θk−1 ⊗ · · · ⊗ A ◦ θp |⊕ ◦ θm ⊗ |A ◦ θp−1 ⊗ · · · ⊗ A|⊕ ◦ θm ; that is, ξm,m+k ≤ ξm,m+p +ξm+p,m+k. Thus ξm,m+k is a non-negative and integrable subadditive process. Using Theorem 3.5, we obtain lim k→∞ (ξ0,k)⊗1/k = lim k→∞ E (ξ0,k)⊗1/k = a a.s., for some constant a < ∞, which concludes the proof for |x(k; e)|⊕ . Now, using that x(k) = 68
- 74. A(k − 1)⊗ · · · ⊗ A(0) ⊗ x0, by Lemma 3.4 we have that for all finite initial conditions x0 and ∀k ≥ 0: |x(k; e)|⊕ ⊗ |x0|∧ ≤ |x(k; x0)|⊕ ≤ |x(k; e)|⊕ ⊗ |x0|⊕ . Therefore |x(k; e)|⊗1/k ⊕ ⊗ |x0|⊗1/k ∧ ≤ |x(k; x0)|⊗1/k ⊕ ≤ |x(k; e)|⊗1/k ⊕ ⊗ |x0|⊗1/k ⊕ (3.6) for all k ≥ 0. We then immediately obtain property (3.4) by letting k tend to ∞. Similarly, if x0 is integrable then we can take expectations in (3.6) and use the fact that limk→∞ E[(ξ0,k)⊗1/k] = a to obtain (3.5). Note that we have only shown the existence of the constant a here, whereas in the deterministic case we were able to specify its value. Computing exactly the maximal Lyapunov exponent of products of matrices over the max-plus semiring is a long-standing problem, and only for a few special cases are explicit formulae known [12]. However, it can be shown (see [2]) that if the communication graph G(A) contains at least one circuit with two vertices i0, j0 in this circuit such that E[Aj0i0 (k)] > e, then the maximal Lyapunov exponent a is strictly positive. This corresponds to having a circuit with at least one place with positive mean holding time in the stochastic event graph. Theorem 3.6 tells us that |x(k)|⊕ grows like a⊗k. Using the same techniques we could prove an analogous result for the growth rate of |x(k)|∧ which is also a constant b, called the minimal Lyapunov exponent, and with b ≤ a. In the following section we examine the growth rate of the individual state variables xj(k) within the SENI framework. In the irreducible case we show that all state variables have the same asymptotic growth rate equal to the maximal Lyapunov exponent defined above, and a similar result holds in the reducible case. This matches our intuition that the speed with which a system operates is determined by its slowest component. 3.3.1 The Strongly Connected Case We assume that A is a.s. irreducible. Recall that within the SENI framework we also assume that the diagonal entries of A are a.s. non-negative. It is then easy to see that the matrices G(k) := A(k + n − 1) ⊗ A(k + n − 2) ⊗ · · · ⊗ A(k), k ∈ Z are such that Gij(k) ≥ e for all i, j = 1, . . . , n. This allows us to state the following corollary. Corollary 3.7. Within the SENI framework, if the matrix A is irreducible, then for all finite initial conditions x0 and for all j = 1, . . . , n, we have that lim k→∞ xj(k; x0) ⊗1/k = a a.s., 69
- 75. where a isthe maximal Lyapunov exponent of Theorem 3.6. If x0 is integrable, we also have lim k→∞ E xj(k; x0) ⊗1/k = a. Proof. From the remark above we obtain that xj(k; x0) ≥ xi(k − n; x0) for all i, j = 1, . . . , n and k > n. This gives that |x(k − n; x0)|⊕ ≤ xj(k; x0) ≤ |x(k; x0)|⊕ ∀j = 1, . . . , n and the result follows by letting k tend to ∞ and applying Theorem 3.6. In the integrable case, the proof of the convergence of the expectations in then immediate. Corollary 3.7 tells us that in the case of a strongly connected stochastic event graph, all tran- sitions have the same asymptotic firing rate. In this case the maximal Lyapunov exponent a is also called the cycle time of the stochastic event graph, and its inverse a−1 is often referred to as the throughput. By viewing the deterministic case of Chapter 2 as a special case, if A(k) = A for all k ∈ N then the maximal Lyapunov exponent is simply the unique eigenvalue λ of A. 3.3.2 The General Case If G(A) is not strongly connected then it can be decomposed into a number N(A) > 1 of maximal strongly connected subgraphs (m.s.c.s.’s), in the same way as we did in Section 1.4 for the critical graph Gc(A). Note that due to the assumptions of the SENI framework, the number N(A) of m.s.c.s.’s and their topologies are non-random. Let Gr(A) = (Vr(A), Er(A)) denote the r-th m.s.c.s. of G(A) and let jr := min j ∈ Vr(A) be the smallest numbered vertex in the r-th m.s.c.s. We call {j1, . . . , jN(A)} a set of representative vertices of the m.s.c.s’s of G(A). We now introduce some new notation and a simple definition. Notation. For i ∈ V(A), let [i] denote the subset of vertices of the m.s.c.s. containing vertex i. We then let x(i)(k) denote the subvector of x(k) associated with the vertex set [i]; that is, the vector x(k) restricted to the entries corresponding to vertices in the m.s.c.s. containing vertex i. Similarly, A(i)(j)(k) denotes the block extracted from A(k) by keeping rows associated with the vertex set [i] and the columns associated with the vertex set [j]. Consistent with the notation used in Section 2.2, we let π+(i) be the set of ascendants of vertex i; that is, the set of all vertices j such that a path exists from j to i (which by convention does not include i itself). In addition, we let π∗(i) := {i} ∪ π+(i). We then let x(<i)(k) denote the subvector of x(k) associated with the vertex set j∈π+(i)[j] (so x(<i)(k) denotes the restriction of x(k) to the entries corresponding to vertices in the m.s.c.s’s of all the ascendants of vertex i, not including the m.s.c.s. of i itself), and x(≤i)(k) denote the subvector of x(k) associated with the vertex set j∈π∗(i)[j]. Finally, the matrices A(≤i)(≤j), A(<i)(<j) etc. are defined in the same way. Definition 3.8. The reduced graph of G(A) is the graph G(A) = (V(A), E(A)), with vertex set V(A) := {j1, . . . , jN(A)} (one vertex per m.s.c.s.) and where (j1, j2) ∈ E(A) iff (k, l) ∈ E(A) 70
- 76. for some k∈ Vj1 (A) and l ∈ Vj2 (A). In other words, a vertex j1 of the reduced graph G(A) corresponds to a collection of vertices in the original communication graph G(A) (vertices in the m.s.c.s. containing j1), and the edge (j1, j2) exists in G(A) if there is an edge in G(A) from some vertex in the m.s.c.s. containing j1 to some vertex in the m.s.c.s. containing j2. We will assume without loss of generality that the vertices of the reduced graph are labelled 1, 2, . . . , N(A), and that this numbering is such that (r, s) ∈ E(A) implies r < s. In particular, the source vertices in the reduced graph (corresponding to the m.s.c.s’s in G(A) which have no predecessors) are labelled 1, . . . , N0, where N0 < N(A). Note that from now on, the notation π, π+ and π∗ will be used to represent the usual sets of predecessor vertices in the reduced graph. For all 1 ≤ r, s ≤ N(A) we will make use of the restrictions x(r), x(≤r), A(r)(s), A(≤r)(≤s) etc. in the same way as defined above, along with the additional notation A(r) := A(r)(r), A(≤r) := A(≤r)(≤r) and A(<r) := A(<r)(<r). The maximal Lyapunov exponents of Theorem 3.6 associated with these matrices will be denoted a(r), a(≤r) and a(<r) respectively. In general, x(r)(k) does not coincide with the solution of the evolution equation y(k + 1) = A(r)(k) ⊗ y(k), k ≥ 0, with initial condition y(0) := x(r)(0). However, the sequence {x(≤r)(k)} is the solution of the evolution equation x(≤r)(k + 1) = A(≤r)(k) ⊗ x(≤r)(k), k ≥ 0, with initial condition x(≤r)(0) (and the same is also true of {x(<r)(k)}). We are now in a position where we are able to begin to characterise the asymptotic firing rates in a general stochastic event graph. Similarly to Theorem 3.6, this concerns the growth rate of the quantities |x(r)(k)|⊕ (for each r ∈ {1, . . . , N(A)}), but we will quickly be able to deduce some implications for the growth rate of the individual state variables xj(k) in the material that follows. Lemma 3.9. Within the SENI framework, for all finite initial conditions x0 and for each r ∈ {1, . . . , N(A)}, we have that lim k→∞ |x(r)(k; x0)|⊗1/k ⊕ = a(≤r) a.s. If x0 is integrable, we also have lim k→∞ E |x(r)(k; x0)|⊗1/k ⊕ = a(≤r). 71
- 77. Proof. From thedefinitions we have that |x(r)(k; x0)|⊕ ≤ |x(≤r)(k; x0)|⊕ . This gives that lim inf k→∞ |x(r)(k; x0)|⊗1/k ⊕ ≤ a(≤r). (3.7) Now let i ∈ s∈π∗(r) Vs(A). We know that for all j ∈ Vr(A) there exists a path of length less than n from i to j in G(A). Using this, together with the fact that the diagonal entries of A are a.s. non-negative, we obtain xj(k + 1; x0) ≥ {i∈Vs(A) : s∈π∗(r)} xi(k − n; x0) ∀j ∈ Vr(A), provided k ≥ n. Therefore |x(r)(k + 1; x0)|⊕ ≥ |x(≤r)(k − n; x0)|⊕ for k ≥ n, and so lim sup k→∞ |x(r)(k; x0)|⊗1/k ⊕ ≥ a(≤r) a.s. (3.8) The result then follows from combining (3.7) and (3.8). Corollary 3.10. Within the SENI framework, for all finite initial conditions x0 and for each r ∈ {1, . . . , N(A)}, we have that lim k→∞ xj(k; x0) ⊗1/k = a(≤r) a.s., ∀j ∈ Vr(A). If x0 is integrable, we also have lim k→∞ E xj(k; x0) ⊗1/k = a(≤r) ∀j ∈ Vr(A). Proof. The result is obtained immediately by following the same lines as in Corollary 3.7. Notice that since the reduced graph G(A) necessarily contains no circuits, the vector x(r)(k) satisfies the equation x(r)(k + 1) = A(r)(k) ⊗ x(r)(k) ⊕ s(r, k + 1) (3.9) where s(r, k + 1) := A(r)(<r)(k) ⊗ x(<r)(k). (3.10) We use this observation to prove the following property, which gives us a simple formula relating the maximal Lyapunov exponents a(s) of the individual m.s.c.s.’s with the constants a(≤r) which characterise the growth rate of the variables xj(k) for j ∈ Vr(A). Theorem 3.11. For any r ∈ {1, . . . , N(A)}, we have that a(≤r) is obtained from the constants a(s), 1 ≤ s ≤ r, by the relation a(≤r) = s∈π∗(r) a(s). (3.11) 72
- 78. Proof. Note thatby the enumeration convention discussed after Definition 3.8, we have that if s ∈ π∗(r) then necessarily s ≤ r. To begin, we firstly prove that lim k→∞ |s(r, k)|⊗1/k ⊕ = a(<r) a.s., (3.12) for all N0 < r ≤ N(A). From (3.10) we obtain |s(r, k + 1)|⊕ ≤ |A(k)|⊕ ⊗ s∈π+(r) |x(s)(k)|⊕ , so that |s(r, k + 1)|⊗1/k ⊕ ≤ |A(k)|⊗1/k ⊕ ⊗ |x(<r)(k)|⊗1/k ⊕ . (3.13) Since we are working within the SENI framework, the integrability assumption on A implies that lim k→∞ |A(k)|⊗1/k ⊕ = e a.s. Letting k go to ∞ in (3.13) then implies lim inf k→∞ |s(r, k)|⊗1/k ⊕ ≤ a(<r) a.s. (3.14) Now, by using the same type of arguments as in Lemma 3.9, from (3.10) we also obtain |s(r, k + 1)|⊕ ≥ |x(<r)(k − n)|⊕ a.s., which in turn implies that lim sup k→∞ |s(r, k)|⊗1/k ⊕ ≥ a(<r) a.s. This, combined with (3.14), completes the proof of (3.12). From (3.10) again we can see that |x(r)(k + 1)|⊕ ≥ |s(r, k + 1)|⊕ , so |x(r)(k + 1)|⊗1/k ⊕ ≥ |s(r, k + 1)|⊗1/k ⊕ and therefore a(≤r) ≥ a(<r). By Corollary 3.10, for all j ∈ Vr(A) we have that (x(r))j (k) ∼ ak (≤r), whereas |s(r, k)|⊕ ∼ ak (<n), so that if a(≤r) > a(<r), then there exists a finite integer-valued random variable K such that A(r)(k) ⊗ x(r)(k) ≥ s(r, k) ∀k ≥ K. We therefore have that for all k ≥ K, (3.9) reads x(r)(k + 1) = A(r)(k) ⊗ x(r)(k). 73
- 79. Let y(k; x0)denote the solution of the equation y(k + 1) = A(r)(k) ⊗ y(k), k ≥ 0, with initial condition y(0) := (x0)(r). On the event {K = h}, we have x(r)(k) = A(r)(k) ⊗ · · · ⊗ A(r)(h) ⊗ x(r)(h) = A(r)(k − h) ⊗ · · · ⊗ A(r)(0) ⊗ x(r)(h) ◦ θ−h ◦ θh = y(k − h; x(r)(h) ◦ θ−h ) ◦ θh , for all k ≥ h. Thus on the event {K = h} lim k→∞ |x(r)(k)|⊗1/k ⊕ = lim k→∞ |y(k − h; x(r)(h) ◦ θ−h )|⊗1/k ⊕ ◦ θh = a(r) a.s., where we use the a.s. convergence result of Theorem 3.6 applied to the matrix A(r)(k). Since K is necessarily finite, ∞ h=0{K = h} = Ω, so for all j ∈ Vr(A) we have lim k→∞ xj(k) ⊗1/k = a(r) a.s. Therefore a(≤r) ≥ a(<r) and a(≤r) > a(<r) implies that a(≤r) = a(r); that is, a(≤r) = a(<r) ⊕ a(r). The proof of (3.11) then follows from this relation by an immediate induction on r. Overall, in this section we have shown that all the transitions in a strongly connected component of a stochastic event graph have the same asymptotic firing rate. Moreover, this quantity is equal to the maximum of the maximal Lyapunov exponents a(s) (namely, the asymptotic firing rates of the strongly connected components in isolation), where s varies over the ascendants of the m.s.c.s. in question. 3.4 Queuing Systems and Timed Event Graphs 3.4.1 Introduction Stochastic timed event graphs are a useful tool used to study several types of queuing sys- tems. Generally speaking, places represent queues of items and transition firings represent the completed service of items at each upstream queue. If items arrive into the system from the outside world, they do so through an input transition and the queuing system is referred to as open. Otherwise, the system is closed. Similarly, a queue may also allow items to exit the system completely, which is modelled by an output transition. Thus closed queuing systems can be modelled by autonomous stochastic event graphs, and open queuing systems by their non-autonomous counterparts. It should be clear from Chapter 2 that a discrete event system is linear in the max-plus sense if and only if it can be modelled by a timed event graph. Unfortunately, the interactions between queues in a general queuing system may be governed by a variety of phenomena, not all of 74
- 80. which preserve linearityin the max-plus sense, and therefore not all queuing networks can be represented by stochastic event graphs. For example, it is clear that we cannot allow for different classes of items within the system. The aim of this section is to identify the class of queuing system for which we can derive necessary and sufficient conditions for max-plus linearity. We begin by exploring some of the possible interactions between queues. • Fork: A departure from one queue may generate simultaneous arrivals at more than one downstream queue. This can be interpreted as an item being split up into several (sub) items. • Join: If service can only commence if one item from each of the upstream queues has arrived, we call this a join queue. Service of an item at a join queue consumes one item from each of the upstream queues. Clearly, the join operation is tantamount to synchronising arrival streams. • Blocking: Upon service completion, an item finds no place at the next queue (i.e. the queue has a finite buffer). Note that due to a fork mechanism, an item may have to wait for buffer places at several downstream queues. We assume that any arrival processes can never be blocked. • Variable Origins: We say that a queue admits variable origins if an arrival to the queue may originate from different upstream queues over time. Note that a join mechanism does not imply variable origins since the upstream queues are fixed. • Variable Destinations: After completing service at a queue, an item may be split up according to a fork mechanism. If the set of queues receiving a (sub) item upon departure varies over time then this phenomenon is called variable destinations. We say that a queuing system admits no routing if all queues admit neither variable origins nor variable destinations. • Internal Overtaking: In general, the order in which items leave a queue is different from the order in which they enter (for example, if there is more than one server). Internal overtake-freeness can be forced by a so-called resequencing mechanism, whereby an item whose service is completed remains on its service place until the service of all items that entered the queue before this particular item is finished. In addition to the way in which queues can interact with each other, the way in which items are processed at the queues can vary. This is known as the queuing discipline. The most common example is the first come, first served (FCFS) queuing discipline. If the buffer at one queue simultaneously blocks several other upstream queues, then the order in which this blocking is resolved is determined via a blocking discipline, like, for example, first blocked, first unblocked (FBFU). If an item is blocked, we assume that the item is blocked at the end of service and remains on its service place until a space at the next queue becomes available. This is referred to as blocking after service. We can summarise the assumptions we use throughout this section as a definition: 75
- 81. Definition 3.12. Aqueuing system satisfies condition (A) if it only has one class of items, no state-dependent service times, all queues are FCFS with blocking after service, and blocking is resolved according to FBFU. Note that a queuing system satisfying condition (A) is not necessarily max-plus linear (and so cannot necessarily be represented by a stochastic event graph). However, as was our aim, it turns out that (A) does specify the class of queuing system for which we can derive necessary and sufficient conditions for max-plus linearity. In other words, reducing our analysis to queuing systems of this type imposes no restriction in the context of stochastic event graphs. The main reason for requiring the assumption (A) is that in a max-plus linear model we have no information about the physical state of the system (in terms of queue lengths), and this means that any dependence of the service times on the physical state cannot be incorporated. The intuition should be reasonably clear and we choose to omit the details here; a full discussion can be found in [12]. It is possible to derive a general recursive formula (in max-plus) describing the departure times from queues in a queuing system satisfying (A). Once again, the full working can be found in [12]. Having done this, it remains to identify conditions under which this formula can be reduced to the form of the basic non-autonomous equation of Theorem 2.23 (or its autonomous counterpart), which, as we have remarked above, will mean that the queuing system is max-plus linear, and can be represented by a stochastic event graph. It boils down to whether the general recursive formula is of finite order. A simple set of sufficient conditions for max-plus linearity are summarised in the following result: Theorem 3.13. A queuing system satisfying (A) is max-plus linear if it admits no internal overtaking, and if all resequencing queues have a finite buffer. The proof is not difficult but of considerable length, and can be found in [12]. 3.4.2 Example: The G/G/1 Queue The event graph in Figure 3.1 below represents the FIFO G/G/1 queue with an infinite buffer. The G/G part states that both the interarrival times and holding times form general stationary sequences. The 1 indicates that there is a single server (we assume that there is an infinite buffer in front of the server). The input u1 into transition q1 features the external input stream of items, p1 is the infinite buffer which stores the items to be served, q2 represents the single server and the holding times in p2 represent the service times. 76
- 82. q1 p1 q2 p2 Figure3.1: Stochastic event graph representation of the G/G/1 queue with an infinite buffer. Items enter the system via q1 and are stored in p1. Departures occur whenever q2 fires since this will remove one token from the system. Note that p2 will never contain more than one token; i.e. there can never be more than one person in service at any one time. Notice that transition q2 has two upstream places but only one downstream place, so whenever q2 fires (i.e. whenever the service of an item is completed) the total number of tokens in the system reduces by one. Thus departures from the queue are modelled without the use of a sink transition. By construction there will always be one token in place p2 (corresponding to the item in service), whereas p1 could hold any number of tokens. The holding times in p1 are taken to be e, so in particular they are non-random. Recalling the notation of Chapter 2, notice that M = 1 and that both |Q | and |I| are equal to 1. Hence the matrices A(k, k − 1) and B(k, k − 1) in the basic non-autonomous equation of Theorem 2.23 are one-dimensional: A(k, k − 1) = α(k) , B(k, k − 1) = (e). Let A(k) = α(k+1). If the initial condition is weakly compatible (which will happen if u(1) ≥ e, and the lag times w1, w2 satisfy w1 ≤ e, w2 ≤ α(1) and w1 ⊕ w2 ≥ e), then it is also compatible (since each transition is followed by at most one place with non-zero initial marking), so the basic non-autonomous equation gives us that the firing times satisfy x(k + 1) = A(k) ⊗ x(k) ⊕ u(k), k ≥ 0, provided we now take x(0) = w2 ⊗ α(1)⊗−1 and u(0) = w1. 3.4.3 Stability Analysis of Waiting Times In this final section we present a celebrated result in the area of stochastic max-plus theory. Recall that if in a G/G/1 queue the expected interarrival time is larger than the expected service time, then the sequence of waiting times converges, independent of the initial condition, to a unique stationary regime (see [16]). Our aim is to generalise this result to the waiting times in open max-plus linear queuing systems. Consider an open queuing system (i.e. one in which items can arrive from the outside world) with one input transition and J queues, satisfying the conditions of Theorem 3.13. It is therefore max-plus linear, and so the vector of departure times from each queue, denoted x(k), satisfies 77
- 83. the basic non-autonomousequation x(k) = M m=0 A(k, k − m) ⊗ x(k − m) ⊕ M m=0 B(k, k − m) ⊗ u(k − m) (3.15) with x(0) = e (a compatible initial condition), and where u(k) denotes the time of the k-th arrival to the system. Note that similarly to the example of the previous section, the matrices B(k, k − m) and the vectors u(k) are all scalars. Following the reasoning laid out in Section 2.5, (3.15) can be transformed into the following first-order recurrence relation: x(k + 1) = A(k) ⊗ x(k) ⊕ B(k) ⊗ u(k + 1) . (3.16) We let Wj(k) := xj(k) − u(k), so Wj(k) denotes the time the k-th item arriving to the system spends in the system until completion of service at queue j. The vector of k-th sojourn times, denoted by W(k) = (W1(k), . . . , WJ (k)) , follows the recurrence relation W(k + 1) = A(k) ⊗ C(U(k + 1)) ⊗ W(k) ⊕ B(k), k ≥ 0 (3.17) where C(h) denotes a diagonal matrix with −h on the diagonal and ε elsewhere, and U(k) := u(k) − u(k − 1) denotes the k-th interarrival time (which we called ‘inter-input time’ in the discussion at the end of Section 2.5). Once again we use the assumptions of the SENI framework (so A(k) is a.s. irreducible ∀k, has non-negative elements on the diagonal and has fixed support), with a minor adjustment given that we are working with the inhomogeneous evolution equation (3.16). We assume that the sequences A(k), B(k) are jointly stationary and ergodic, and independent of the arrival process u(k). Finally, the interarrival times U(k) form another stationary and ergodic sequence of positive random variables with mean ν ∈ (0, ∞) (for example, the arrival process could be Poisson). We are now able to begin our task of characterising the stability of waiting times in open max- plus linear queuing systems. The main result is that if the maximal Lyapunov exponent of the sequence of matrices A(k) in (3.16) is greater than the expected interarrival time, then the sequence of waiting times W(k) converges with strong coupling to a unique stationary regime. The proof comes in three steps. For ease of reading we will occasionally revert to the use of notation from conventional algebra, but it should be noted that none of the operators we use is invalid or ill-defined within the max-plus semiring. Step 1 (The Loyne’s Scheme) Let M(k) denote the vector of sojourn times at time 0 provided that the sequence of waiting time vectors was started at time −k in B(−(k + 1)). For k > 0 we set u(−k) = − k−1 i=0 U(−i) 78
- 84. By (3.17) weobtain M(1) = A(−1) ⊗ C(U(0)) ⊗ B(−2) ⊕ B(−1) and for M(2) we have to replace B(−2) by A(−2) ⊗ C(A(−1)) ⊗ B(−3) ⊕ B(−2) which yields M(2) = A(−1) ⊗ C(U(0)) ⊗ A(−2) ⊗ C(U(−1)) ⊗ B(−3) (3.18) ⊕ A(−1) ⊗ C(U(0)) ⊗ B(−2) ⊕ B(−1). (3.19) By induction, we obtain for M(k): M(k) = k j=0 j i=1 A(−i) ⊗ C(U(−i + 1)) ⊗ B(−(j + 1)), (3.20) where we set the product j i=1 A(−i) ⊗ C(U(−i + 1)) to E for j = 0. Notice that the sequence M(k) is monotonically increasing in k. For k ≥ 0: M(k) = k j=0 j i=1 A(−i) ⊗ C(U(−i + 1)) ⊗ B(−(j + 1)) ≤ k+1 i=1 A(−i) ⊗ C(U(−i + 1)) ⊗ B(−(k + 1)) ⊕ k j=0 j i=1 A(−i) ⊗ C(U(−i + 1)) ⊗ B(−(j + 1)) = k+1 j=0 j i=1 A(−i) ⊗ C(U(−i + 1)) ⊗ B(−(j + 1)) = M(k + 1). Note also that for any y ∈ Rmax, the matrix C(y) commutes with any matrix A ∈ RJ×J max : C(y) ⊗ A = A ⊗ C(y). Furthermore, for y, z ∈ Rmax, it holds that C(y) ⊗ C(z) = C(z) ⊗ C(y) = C(y ⊗ z) and therefore j i=1 C(U(−i + 1)) = C j i=1 U(−i + 1) = C(−u(−j)). 79
- 85. Now, if weset D(k) := k i=1 A(−i) ⊗ B(−(k + 1)), k ≥ 1 (3.21) and, for k = 1, set D(0) = B(−1), then (3.20) now reads M(k) = k j=0 C(−u(−j)) ⊗ D(j). (3.22) Step 2 (Pathwise Limit) We now show that the limit of M(k) as k tends to ∞ exists, and establish a condition for the limit to be a.s. finite. Because M(k) is monotonically increasing, the random variable M, defined by M := lim k→∞ M(k) = j≥0 C(−u(−j)) ⊗ D(j) (3.23) is either equal to ∞ or finite. To derive a sufficient condition for M to be a.s. finite, we first study three individual limits: (i) Since we are working under the assumptions of the SENI framework, by Theorem 3.6 a number a exists (the maximal Lyapunov exponent) such that for any x ∈ RJ max lim k→∞ k j=1 A(−j) ⊗ x ⊗1/k ⊕ = a a.s. (ii) By the strong law of large numbers (which is essentially a special case of Corollary 3.10), we have that lim k→∞ C(−u(−k)) ⊗1/k ⊕ = lim k→∞ u(−k)⊗1/k = − lim k→∞ 1 k 0 i=−k U(i) = −ν a.s. (iii) Ergodicity of B(k) implies that for each j, there exists a bj ∈ Rmax{ε} such that lim k→∞ 1 k k i=1 Bj(−i) = bj a.s. which implies that it holds with probability one that bj = lim k→∞ 1 k k i=1 Bj(−i) 80
- 86. = lim k→∞ 1 k Bj(−k) +lim k→∞ k − 1 k 1 k − 1 k−1 i=1 Bj(−i) = lim k→∞ 1 k Bj(−k) + bj and thus lim k→∞ 1 k Bj(−k) = lim k→∞ 1 k Bj(−(k + 1)) = 0 a.s. We conclude that lim k→∞ B(−k) ⊗1/k ⊕ = 0 a.s. Now, from Lemma 3.4 and using the definition of the matrix D(k) in (3.21), we have that C(−u(−k)) ⊗ D(k) ⊕ = C(−u(−k)) ⊗ k i=1 A(−i) ⊗ B(−(k + 1)) ⊕ ≤ C(−u(−k)) ⊕ ⊗ k i=1 A(−i) ⊗ e ⊕ ⊗ B(−(k + 1)) ⊕ . Using the limits (i)-(iii) discussed above, we obtain lim k→∞ C(−u(−k)) ⊗ D(k) ⊗1/k ⊕ ≤ a − ν a.s. and so ν > a implies lim k→∞ C(−u(−k)) ⊗ D(k) ⊕ = ε a.s. Hence, for k sufficiently large, the vector C(−u(−k)) ⊗ D(k) has only negative elements. Re- ferring to (3.22) and noting that M(k) ≥ 0 by definition, we see that M(k) is dominated by the maximum over finitely many vectors whose elements are all finite, and therefore ν > a implies that M is an a.s. finite random variable (similarly, ν < a implies M = ∞ a.s.). Step 3 (Stationarity and Uniqueness) Under our statistical assumptions, let θ denote an ergodic shift operator such that A(k) = A ◦ θk, B(k) = B ◦ θk and U(k) = U ◦ θk (for appropriately defined random variables, as we discussed in Section 3.2). We then have that (3.18) reads M(2) = A ◦ θ−1 ⊗ C(U) ⊗ M(1) ◦ θ−1 ⊕ B ◦ θ−1 , and by induction M(k + 1) = A ◦ θ−1 ⊗ C(U) ⊗ M(k) ◦ θ−1 ⊕ B ◦ θ−1 . (3.24) 81
- 87. Letting k tendto ∞ in (3.24) shows that M = A ◦ θ−1 ⊗ C(U) ⊗ M ⊕ B ◦ θ−1 ; that is, M is the stationary solution of (3.17), and it remains to show uniqueness. To do this, let M(k, w) denote the vector of sojourn times at time 0 provided that the sequence is started at −k and with initial vector w ∈ RJ max. This gives M(k, w) = k i=1 A(−i) ⊗ C(U(−i + 1)) ⊗ w ⊕ k−1 j=0 C(−u(−j)) ⊗ D(j). Assuming that w has at least one finite element, we have that |w|⊕ < ∞. Following the same argument as in step 2 above, we obtain that for ν > a: lim k→∞ k i=1 A(−i) ⊗ C(U(−i + 1)) ⊗ w ⊕ = ε a.s., and so lim k→∞ k i=1 A(−i) ⊗ C(U(−i + 1)) ⊗ w ⊕ k−1 j=0 C(−u(−j)) ⊗ D(j) = M a.s. Thus for any initial value w, M(k, w) has the same limit as M(k), and uniqueness has been established. If W(k, w) is the vector of k-th sojourn times initiated at w, then M(k, w) and W(k, w) are equal in distribution, and so M is the unique (weak) limit of W(k, w) , independent of the w chosen. Finally, we refer to [12] for the proof that W(k) converges with strong coupling to the stationary regime M. We can summarise this result in the following theorem: Theorem 3.14. Assume we are working under the (modified) assumptions of the SENI frame- work, and denote the maximal Lyapunov exponent of {A(k)} by a. If ν > a, then the sequence W(k) of sojourn times at each queue converges with strong coupling to a unique stationary regime M (defined in (3.23)). This result is essentially an example of a max-plus multiplicative ergodic theorem. Whilst we have worked in the context of queuing systems, the same theory applies for general stochastic event graphs. From what we have proved above, it is straightforward to show that other increments of the form xj(k + 1) − xi(k), for i, j ∈ {1, . . . , J}, also couple with a stationary and ergodic process [1]. The autonomous case turns out to be considerably more involved and is dealt with in [2]. 82
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