5. System Usage
Host
1
Host
2
Host
3
Host
4
Host
5
)
(
1 t
N )
(
2 t
N
)
(
3 t
N
Route : static path through network, supporting Ni(t)
flows with Li(N(t)) allocated bandwidth.
Flows / Users : transfer documents of different sizes,
evenly split allocated bandwidth along route.
Dynamic. Not directed.
))
(
(
1 t
N
L ))
(
(
2 t
N
L
))
(
(
3 t
N
L
6. Simplification
1
C
2
C 3
C 4
C
5
C
6
C 7
C
)
(
1 t
N )
(
2 t
N
)
(
3 t
N
))
(
(
1 t
N
L ))
(
(
2 t
N
L
))
(
(
3 t
N
L
Extraneous elements have been removed.
7. Abstraction
1
C
2
C 3
C 4
C
5
C
6
C 7
C
)
(
1 t
N )
(
2 t
N
)
(
3 t
N
))
(
(
1 t
N
L ))
(
(
2 t
N
L
))
(
(
3 t
N
L
Routes are just subsets of links / resources.
Represented by [Aji] : whether resource j is used by route i.
Capacity constraint:
L
0
)
(
:
))
(
(
t
N
i
j
i
ji
i
C
t
A N
10. Allocation Efficiency
10
10 10 10
10
10 10
5
3
L 5
1
L 2
2
L
• An allocation L is feasible if capacity constraint
satisfied.
• A feasible allocation L is efficient if we don’t have
L for any other feasible .
• Defined at a point in time, regardless of usage.
11. Stability
• Stable Markov chain positive recurrent.
• Returns to each state with probability 1 in finite
mean time.
• Necessary, but not sufficient condition:
• How tight this is gives us an idea of utilization.
• Does not uniquely specify allocation.
j
C
A j
i
i
ji all
for
r
12. Maximize Overall Throughput
• That is, max
• No unique allocation.
• Could get unexpected results.
L
0
)
(
: t
N
i
i
i
10 10 10
1
L 2
L 3
L
4
L
13. Max-Min Fairness
• Increase allocation for each user, unless doing
so requires a corresponding decrease for a
user of equal or lower bandwidth to satisfy the
capacity constraints.
• Uniquely determined.
• Greedy algorithm. Not distributed.
12 12
1
L 2
L
3
L
14. Proportional Fairness
• L is proportionally fair if for any other
feasible allocation L* we have:
• Same as maximizing:
• Interpret as utility function.
• Distributed algorithms known.
0
)
(
*
L
L
L
i
i
i
i
i t
N
L
i
i
i t
N log
)
(
16. TCP Bias
RTT
timeout
• Congestion window based on
additive increase /
multiplicative decrease
mechanism.
• Increase for each ACK
received, once every
Round Trip Time.
• Timeouts based on RTT.
• Bias against long RTT.
17. Properties of a-Fair Allocations
• The optimal L exists and is unique.
• It’s positive: L > 0.
• Scale invariance: L(rN) = L(N), for r > 0.
• Continuity: L is continuous in N.
• System is stable when
Assume Ni(t) > 0.
Let L(N(t)) be a solution to the a-fair optimization.
j
C
A j
i
i
ji all
for
r
19. Fluid Models
))
(
(
)
(
)
0
(
)
( t
T
S
t
E
N
t
N i
i
i
i
i
Ni(0) : initial condition
Ei(t) : new arrival process
Ti(t) : cumulative bandwidth allocated
Si(t) : service process
Decompose into non-decreasing processes:
L
t
i
i s
s
t
T
0
d
))
(
(
)
( N
Consider a sequence indexed by r > 0:
r
rt
N
t
N
r
r )
(
)
(
21. Fluid Limit : Math
t
r
rt
E
t
r
rt
E
t
t
E
t
t
E
r
r
)
(
)
(
)
(
)
(
r
t
r
t
r
E
as
)
(
t
t
E
r
)
(
Look at slope:
with probability 1.
By strong law of large numbers for renewal processes:
Thus
22. Fluid Model Solution
A fluid model solution is an absolutely continuous
function
so that at each regular point t and each route i
and for each resource j
L
0
)
(
if
0
0
)
(
if
)
(
)
(
t
N
t
N
t
t
N
i
i
i
i
i
i
dt
d
N
)
(t
N
j
t
N
i
i
ji
t
N
i
i
ji C
A
t
A
i
i
L
0
)
(
:
0
)
(
:
)
( r
N
23. Fluid Analysis is Easier
A complex function f is absolutely continuous on I=[a,b]
if for every e > 0 there is a d > 0 such that
for any n and any disjoint collection of segments
(a1,b1),…,(an,bn) in I whose lengths satisfy
n
i
i
i
1
d
a
b
e
a
b
n
i
i
i f
f
1
)
(
)
(
If f is AC on I, the f is differentiable a.e. on I, and
x
a
dt
t
f
a
f
x
f )
(
'
)
(
)
(
Definition
Theorem
25. For Stability
• If fluid system empties in finite time, then system is
stable.
T
t
t
all
for
0
)
(
N
• Show that 0
)
(
t
dt
d
N
• In general, what happens as t when some of the
resources are saturated?
j
i
i
ji C
A
r
• We approach the invariant manifold, aka the set of
invariant states
0
where
all
for
)
(
:
L
i
i
i N
i
M r
a N
N
26. Towards a Formal Framework
• Interested in stochastic processes with samples
paths in D[0, , the space of right continuous real
functions having left limits.
)
(
)
(
lim t
x
s
x
t
s
)
(
)
(
lim
t
x
s
x
t
s
• Well behaved. At most countably many points of
discontinuity.
t
)
(t
x
27. Why we need a better metric.
…
…
t
t
)
(
1 t
x
)
(
2 t
x
What goes wrong in Lp ? L ?
28. Skorohod Topology
Let L be the set of strictly increasing Lipschitz
continuous functions mapping [0,) onto [0,),
such that
)
(
log
)
( '
t
)
)
(
(
),
(
sup
)
,
,
,
(
~
0
u
t
y
u
t
x
r
u
y
x
d
t
1
r
r
Put (standard bounded metric)
L
u
u
y
x
d
e
y
x
d u
d
)
,
,
,
(
~
)
(
inf
)
,
(
0
For functions mapping to any Polish (complete,
separable, metric) space.
29. Prohorov Metric
Let (S,d) be a metric space,
B(S) the s-algebra of Borel subsets of S,
P(S) family of Borel probability measures on S.
Define
S
F
F
Q
F
P
Q
P
closed
all
for
)
(
)
(
:
0
inf
)
,
( e
e
r e
)
,
(
inf
: e
e
y
x
d
S
x
F
F
y
The resulting metric space is Polish.
31. Outline of Proof
• Apply functional law of large numbers to load
processes.
• Derive dynamic equations for state and bounds.
• State contained in compact set with probability 1
in limit.
• State oscillations die down with probability 1 in
limit.
• Sequence is C-tight.
• Weak limit points are fluid solutions with
probability 1.
