The document discusses the importance of states and importance functions in tandem Jackson networks, focusing on evaluating rare event probabilities and the application of a restart simulation method. It outlines formulas for determining the importance of states in a two-queue system and emphasizes deriving optimal importance functions. Future work includes exploring the behavior of equi-important lines and extending this analysis to more complex queuing networks.

1. Authors: Manuel Villén-Altamirano, Arcadio Reyes, Eduardo Casilari
Universidad de Málaga
Presenter: José Villén-Altamirano
Universidad Politécnica de Madrid

2.  Review of RESTART
 Importance 𝑰 𝒙 of a state
 Importance 𝑰 𝒙 and importance function 𝚽 𝒙
 Two-queue tandem Jackson network
 𝑰𝒊,𝒋 for 𝒊 > 𝟎 as a function of 𝑰 𝟎,𝒋
 Evaluation of 𝑰 𝟎,𝒋
 Conclusions and future work
Contents
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3. Review of RESTART (I)
M : No. of thresholdsCi – Ci +1: Importance regions
 : Importance function
Ti : thresholds
 = T1
C0 – C1
C1 – C2
C2 – C3
Rare Event
A
C3
 = T3
 = T2
C3
C2
C1
C0

4. Review of RESTART (II)
(t)
t (time)
T3
T2
T1
B3
B2
B1
B3
B1D1 D1
D2D2
D3D 3
D1D1
D2
D3
A
RESTART: REpetitive Simulation Trials After Reaching Thresholds
Oversampling factor at Ci - Ci+1:
No. of trials at Bi : Ri


i
j
ji Rr
1

5.  Estimator of P:
𝑃 =
𝑁 𝐴
𝑁 𝑟 𝑀
o N : number of reference evens in the main trial
o 𝑁𝐴
0
∶ number of rare events in the main trial
o 𝑵 𝑨 : number of rare events in all the trials
 Rare event probability 𝑃: probability of the system being in a
state of the rare set A when a reference event occurs
 Example of reference event: a packet arrival at a queue
 Rare event: reference event at which the system is in a state of the rare set A
Review of RESTART (III)

6. Importance 𝑰 𝒙 of a state
 𝑰 𝒙 : limit as Nr → ∞ of the expected number of events A in
the Nr reference events following state 𝑥 , minus 𝑁𝑟 𝑃
 Equation relating 𝑰 𝒙 to the importance of neighbor states
𝐼 𝑥 =
∀𝑦∈Ω
𝜋 𝑥,𝑦 𝑛 𝑥,𝑦
𝐴
− 𝑛 𝑥,𝑦
𝑅
𝑃 + 𝐼 𝑦
𝜋 𝑥,𝑦 : transition probability from x to y
𝑛 𝑥,𝑦
𝐴
: no. of rare events in the transition
𝑛 𝑥,𝑦
𝑅 : no. of reference events in the transition

7.  An importance function is appropriate if
Φ 𝑥 = Φ 𝑦 ⇒ 𝐼 𝑥 ≈ 𝐼 𝑦 ; Φ 𝑥 > Φ 𝑦 ⇔ 𝐼 𝑥 > 𝐼 𝑦
 An importance function is optimal if 𝚽 𝒙 = 𝑰 𝒙 ; in general, if 𝚽 𝒙
is an increasing function of 𝑰 𝒙
Evaluating 𝑰 𝒙 leads to the optimal importance function
Importance 𝑰 𝒙 and Importance function 𝚽 𝒙
 The importance 𝑰 𝒙 is intrinsic of the system while the
importance function 𝚽 𝒙 is defined by the user
 The variance of 𝑰 𝒙𝒊
at the states 𝒙𝒊 with 𝚽 𝒙𝒊 = 𝑻𝒊 must be small
for an efficient application of RESTART

8. Types of events
 Event AR : Customer arrival Transition 𝑖, 𝑗 → 𝑖 + 1, 𝑗
 Event ES1: End of service in queue 1 Transition 𝑖, 𝑗 → 𝑖 − 1, 𝑗 + 1
 Event ES2: End of service in queue 2 Transition 𝑖, 𝑗 → 𝑖, 𝑗 − 1
Two-queue tandem Jackson network(I)
2 1 𝜇2
𝜇1
𝜆
𝑖 𝑗
Loads: 𝜌1 = 𝜆 𝜇1, 𝜌2 = 𝜆 𝜇2 System state: 𝑖, 𝑗 Rare set: 𝑗 ≥ 𝐿
𝐼𝑖,𝑗 : importance of state 𝑖, 𝑗 Reference event: arrival at the 2nd queue

9. No. of refer. events No. of rare events
 Event AR : 0 0
 Event ES1: 1 𝑛𝑗
𝐴
=
0 𝑖𝑓 𝑗 < 𝐿
1 𝑖𝑓 𝑗 ≫ 𝐿
 Event ES2: 0 0
Probabilities of occurrence
 If 𝑖 > 0, 𝑗 > 0: 𝜋 𝐴𝑅 𝑖, 𝑗 = 𝜆
𝜆+𝜇1+𝜇2
𝜋 𝐸𝑆1 𝑖, 𝑗 = 𝜇1
𝜆+𝜇1+𝜇2
𝜋 𝐸𝑆2 𝑖, 𝑗 = 𝜇2
𝜆+𝜇1+𝜇2
 If 𝑖 > 0, 𝑗 = 0: 𝜋 𝐴𝑅 𝑖, 0 = 𝜆
𝜆+𝜇1
𝜋 𝐸𝑆1 𝑖, 0 = 𝜇1
𝜆+𝜇1
𝜋 𝐸𝑆2 𝑖, 0 = 0
 If 𝑖 = 0, 𝑗 > 0: 𝜋 𝐴𝑅 0, 𝑗 = 𝜆
𝜆+𝜇2
𝜋 𝐸𝑆1 0, 𝑗 = 0 𝜋 𝐸𝑆2 0, 𝑗 = 𝜇2
𝜆+𝜇2
 If 𝑖 = 0, 𝑗 = 0: 𝜋 𝐴𝑅 0,0 =1 𝜋 𝐸𝑆1 0,0 = 0 𝜋 𝐸𝑆2 0,0 = 0
Two-queue tandem Jackson network (II)

10.  Equation relating 𝐼𝑖, 𝑗 to the importance of neighbor states:
𝐼𝑖,𝑗 = 𝜋 𝐴𝑅 𝑖, 𝑗 𝐼𝑖+1, 𝑗 + 𝜋 𝐸𝑆1 𝑖, 𝑗 𝑛𝑗
𝐴
− 𝑃 + 𝐼𝑖−1,𝑗+1 + 𝜋 𝐸𝑆2 𝑖, 𝑗 𝐼𝑖, 𝑗−1 𝑃 = 𝜌2
𝐿
 Previous paper said that this system could not be solved. The
problem is really that additional equation are required
𝑰𝒊,𝒋 for 𝒊 > 𝟎 as a function of 𝑰 𝟎,𝒋
 Solving for 𝑰𝒊+𝟏,𝒋:
𝐼𝑖+1,𝑗 =
1
𝜋 𝐴𝑅(𝑖, 𝑗)
𝐼𝑖,𝑗 − 𝜋 𝐸𝑆1 𝑖, 𝑗 𝑛𝑗
𝐴
− 𝑃 + 𝐼𝑖−1,𝑗+1 − 𝜋 𝐸𝑆2 𝑖, 𝑗 𝐼𝑖,𝑗−1
 If 𝑰 𝟎,𝒋 for ∀𝒋 is known, 𝑰𝒊,𝒋 for ∀𝒊, 𝒋 may be derived
Additional equations for deriving 𝑰 𝟎,𝒋 are required

11. Evaluation of 𝑰 𝟎,𝒋 (I)
𝐼0,𝑗 − 𝐼0,𝑗−1 =
𝑚=0
∞
𝑞 𝑗 𝑚 − 𝑞 𝑗−1 𝑚
 𝒒𝒋 𝒎 is the probability that the 𝒎 + 𝟏 th event ES1 occurred
after the state 𝟎, 𝒋 finds at least L customers at the 2nd queue
𝑞 𝑗 𝑚 =
∀𝑙≥𝑚+1 ∀𝑛/𝑗+𝑚−𝑛≥𝐿
𝑆𝑗 𝑙, 𝑚, 𝑛
𝜇1
𝜆 + 𝜇1 + 𝜇2
 𝑺𝒋 𝒍, 𝒎, 𝒏 is the sum of the probabilities of all the possible
sequences of l events AR, m events ES1 and n events ES2 which
can occur after the state 𝟎, 𝒋

12. Evaluation of 𝑰 𝟎,𝒋 (II)
𝑞 𝑗 𝑚 − 𝑞 𝑗−1(𝑚) =
∀𝑙≥𝑚+1 𝑛/𝑗+𝑚−𝑛=𝐿
𝑈𝑗 𝑙, 𝑚, 𝑛
𝜇1
𝜆 + 𝜇1 + 𝜇2
 𝑼𝒋 𝒍, 𝒎, 𝒏 is as 𝑺𝒋 𝒍, 𝒎, 𝒏 but only including the sequences in
which the 2nd queue never becomes empty
 Example: 𝑼 𝟏 𝟐, 𝟐, 𝟏 is the sum of these probabilities:
0,1
𝐴𝑅
1,1
𝐴𝑅
2,1
𝐸𝑆1
1,2
𝐸𝑆1
(0,3)
𝐸𝑆2
(0,2) 𝑃𝑟 =
𝜆2 𝜇1
2 𝜇2
𝜆+𝜇2
2 𝜆+𝜇1+𝜇2
3
0,1
𝐴𝑅
1,1
𝐴𝑅
2,1
𝐸𝑆1
1,2
𝐸𝑆2
(1,1)
𝐸𝑆1
(0,2) 𝑃𝑟 =
𝜆2 𝜇1
2 𝜇2
𝜆+𝜇2 𝜆+𝜇1+𝜇2
4
0,1
𝐴𝑅
1,1
𝐸𝑆1
0,2
𝐴𝑅
1,2
𝐸𝑆1
0,3
𝐸𝑆2
0,2 𝑃𝑟 =
𝜆2 𝜇1
2 𝜇2
𝜆+𝜇2
3 𝜆+𝜇1+𝜇2
2
0,1
𝐴𝑅
1,1
𝐸𝑆1
0,2
𝐴𝑅
1,2
𝐸𝑆2
1,1
𝐸𝑆1
0,2 𝑃𝑟 =
𝜆2 𝜇1
2 𝜇2
𝜆+𝜇2
2 𝜆+𝜇1+𝜇2
3
0,1
𝐴𝑅
1,1
𝐸𝑆1
0,2
𝐸𝑆2
0,1
𝐴𝑅
(1,1)
𝐸𝑆1
(0,2) 𝑃𝑟 =
𝜆2 𝜇1
2 𝜇2
𝜆+𝜇2
3 𝜆+𝜇1+𝜇2
2

13. Evaluation of 𝑰 𝟎,𝒋 (III)
 In order to make event occurrence probabilities independent
from the system state, we introduce a dummy event, dES1,
which does not change the system state and which occurs with
probability 𝝁 𝟏 𝝀 + 𝝁 𝟏 + 𝝁 𝟐 when the first queue is empty
 If 𝑵𝒋 𝒍, 𝒎, 𝒏, 𝒓 is the is number of distinct sequences of l events
AR, m events ES1, n events ES2 and r events dES1 which can
occur after the state 𝟎, 𝒋 with the second queue never becoming
empty, and 𝑽𝒋 𝒍, 𝒎, 𝒏, 𝒓 the sum of their probabilities:
𝑈𝑗 𝑙, 𝑚, 𝑛 =
𝑟=0
∞
𝑉𝑗 𝑙, 𝑚, 𝑛, 𝑟
𝑉𝑗 𝑙, 𝑚, 𝑛, 𝑟 = 𝑁𝑗 𝑙, 𝑚, 𝑛, 𝑟
𝜆𝑙 𝜇1
𝑚+𝑟
𝜇2
𝑛
𝜆 + 𝜇1 + 𝜇2
𝑙+𝑚+𝑟+𝑛

14. Evaluation of 𝑰 𝟎,𝒋 (IV)
 A recurrent formula to obtain 𝑵𝒋 𝒍, 𝒎, 𝒏, 𝒓 for sequences with k
events (i.e., 𝒍 + 𝒎 + 𝒏 + 𝒓 = 𝒌) as a function of 𝑵𝒋 𝒍, 𝒎, 𝒏, 𝒓 for
sequences with 𝒌 − 𝟏 events is provided in the paper
 From this recurrent formula an expression of 𝑵𝒋 𝒍, 𝒎, 𝒏, 𝒓 is
derived. From 𝑵𝒋 𝒍, 𝒎, 𝒏, 𝒓 , we obtain 𝑽𝒋 𝒍, 𝒎, 𝒏, 𝒓 , then 𝑼𝒋 𝒍, 𝒎, 𝒏 ,
then 𝒒𝒋 𝒎 − 𝒒𝒋−𝟏 𝒎 , then 𝑰 𝟎,𝒋 − 𝑰 𝟎,𝒋−𝟏, then 𝑰 𝟎,𝒋 − 𝑰 𝟎,𝟎 and finally
𝑰𝒊,𝒋 − 𝑰 𝟎,𝟎
 Knowing 𝑰𝒊,𝒋 in relation to 𝑰 𝟎,𝟎 is sufficient because we are only
interested in the relative importances. Nevertheless 𝑰 𝟎,𝟎 may be
calculated using the normalization equation:
𝑖=0
∞
𝜌1
𝑖
𝜌2
𝑗
𝐼𝑖,𝑗 = 0

15. Conclusions and future work (I)
 Formulas of the importance in a two-queue tandem Jackson
network, which lead to the optimal importance function, have
been derived
 Next step is to evaluate them for answering to these questions:
 How are the equi-important lines 𝑰𝒊,𝒋 = constant? Are approximately
straight lines? Are they parallel? How they behave when they enter the
rare set?
 How close to the optimal one are the importance functions proposed by
other authors? Can they be improved with closer approximations?

16.  Using the concept of effective load, to extend the study to non-Markovian
networks
 To derive approximations for more complex Jackson networks which allow
the choice of suitable importance functions
Conclusions and future work (II)
 Plans for future work are:
 To derive exact formulas of the importance for other simple queuing networks