2. ο Review of RESTART
ο Importance π° π of a state
ο Importance π° π and importance function π½ π
ο Two-queue tandem Jackson network
ο π°π,π for π > π as a function of π° π,π
ο Evaluation of π° π,π
ο Conclusions and future work
Contents
*
/ ixAP
*
/ ixAP
3. Review of RESTART (I)
M : No. of thresholdsCi β Ci +1: Importance regions
ο : Importance function
Ti : thresholds
ο = T1
C0 β C1
C1 β C2
C2 β C3
Rare Event
A
C3
ο = T3
ο = T2
C3
C2
C1
C0
4. Review of RESTART (II)
ο(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
οο½
ο½
i
j
ji Rr
1
5. ο Estimator of P:
π =
π π΄
π π π
o N : number of reference evens in the main trial
o ππ΄
0
βΆ number of rare events in the main trial
o π΅ π¨ : number of rare events in all the trials
ο Rare event probability π: probability of the system being in a
state of the rare set A when a reference event occurs
ο Example of reference event: a packet arrival at a queue
ο Rare event: reference event at which the system is in a state of the rare set A
Review of RESTART (III)
7. ο An importance function is appropriate if
Ξ¦ π₯ = Ξ¦ π¦ β πΌ π₯ β πΌ π¦ ; Ξ¦ π₯ > Ξ¦ π¦ β πΌ π₯ > πΌ π¦
ο An importance function is optimal if π½ π = π° π ; in general, if π½ π
is an increasing function of π° π
Evaluating π° π leads to the optimal importance function
Importance π° π and Importance function π½ π
ο The importance π° π is intrinsic of the system while the
importance function π½ π is defined by the user
ο The variance of π° ππ
at the states ππ with π½ ππ = π»π must be small
for an efficient application of RESTART
8. Types of events
ο Event AR : Customer arrival Transition π, π β π + 1, π
ο Event ES1: End of service in queue 1 Transition π, π β π β 1, π + 1
ο Event ES2: End of service in queue 2 Transition π, π β π, π β 1
Two-queue tandem Jackson network(I)
2ο 1ο π2
π1
π
π π
Loads: π1 = π π1, π2 = π π2 System state: π, π Rare set: π β₯ πΏ
πΌπ,π : importance of state π, π Reference event: arrival at the 2nd queue
10. ο Equation relating πΌπ, π to the importance of neighbor states:
πΌπ,π = π π΄π π, π πΌπ+1, π + π πΈπ1 π, π ππ
π΄
β π + πΌπβ1,π+1 + π πΈπ2 π, π πΌπ, πβ1 π = π2
πΏ
ο Previous paper said that this system could not be solved. The
problem is really that additional equation are required
π°π,π for π > π as a function of π° π,π
ο Solving for π°π+π,π:
πΌπ+1,π =
1
π π΄π (π, π)
πΌπ,π β π πΈπ1 π, π ππ
π΄
β π + πΌπβ1,π+1 β π πΈπ2 π, π πΌπ,πβ1
ο If π° π,π for βπ is known, π°π,π for βπ, π may be derived
Additional equations for deriving π° π,π are required
11. Evaluation of π° π,π (I)
πΌ0,π β πΌ0,πβ1 =
π=0
β
π π π β π πβ1 π
ο ππ π is the probability that the π + π th event ES1 occurred
after the state π, π finds at least L customers at the 2nd queue
π π π =
βπβ₯π+1 βπ/π+πβπβ₯πΏ
ππ π, π, π
π1
π + π1 + π2
ο πΊπ π, π, π 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 π, π
12. Evaluation of π° π,π (II)
π π π β π πβ1(π) =
βπβ₯π+1 π/π+πβπ=πΏ
ππ π, π, π
π1
π + π1 + π2
ο πΌπ π, π, π is as πΊπ π, π, π but only including the sequences in
which the 2nd queue never becomes empty
ο Example: πΌ π π, π, π is the sum of these probabilities:
0,1
π΄π
1,1
π΄π
2,1
πΈπ1
1,2
πΈπ1
(0,3)
πΈπ2
(0,2) ππ =
π2 π1
2 π2
π+π2
2 π+π1+π2
3
0,1
π΄π
1,1
π΄π
2,1
πΈπ1
1,2
πΈπ2
(1,1)
πΈπ1
(0,2) ππ =
π2 π1
2 π2
π+π2 π+π1+π2
4
0,1
π΄π
1,1
πΈπ1
0,2
π΄π
1,2
πΈπ1
0,3
πΈπ2
0,2 ππ =
π2 π1
2 π2
π+π2
3 π+π1+π2
2
0,1
π΄π
1,1
πΈπ1
0,2
π΄π
1,2
πΈπ2
1,1
πΈπ1
0,2 ππ =
π2 π1
2 π2
π+π2
2 π+π1+π2
3
0,1
π΄π
1,1
πΈπ1
0,2
πΈπ2
0,1
π΄π
(1,1)
πΈπ1
(0,2) ππ =
π2 π1
2 π2
π+π2
3 π+π1+π2
2
13. Evaluation of π° π,π (III)
ο 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 π π π + π π + π π when the first queue is empty
ο If π΅π π, π, π, π 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 π, π with the second queue never becoming
empty, and π½π π, π, π, π the sum of their probabilities:
ππ π, π, π =
π=0
β
ππ π, π, π, π
ππ π, π, π, π = ππ π, π, π, π
ππ π1
π+π
π2
π
π + π1 + π2
π+π+π+π
14. Evaluation of π° π,π (IV)
ο A recurrent formula to obtain π΅π π, π, π, π for sequences with k
events (i.e., π + π + π + π = π) as a function of π΅π π, π, π, π for
sequences with π β π events is provided in the paper
ο From this recurrent formula an expression of π΅π π, π, π, π is
derived. From π΅π π, π, π, π , we obtain π½π π, π, π, π , then πΌπ π, π, π ,
then ππ π β ππβπ π , then π° π,π β π° π,πβπ, then π° π,π β π° π,π and finally
π°π,π β π° π,π
ο Knowing π°π,π in relation to π° π,π is sufficient because we are only
interested in the relative importances. Nevertheless π° π,π may be
calculated using the normalization equation:
π=0
β
π1
π
π2
π
πΌπ,π = 0
15. Conclusions and future work (I)
ο Formulas of the importance in a two-queue tandem Jackson
network, which lead to the optimal importance function, have
been derived
ο Next step is to evaluate them for answering to these questions:
ο How are the equi-important lines π°π,π = constant? Are approximately
straight lines? Are they parallel? How they behave when they enter the
rare set?
ο How close to the optimal one are the importance functions proposed by
other authors? Can they be improved with closer approximations?
16. ο Using the concept of effective load, to extend the study to non-Markovian
networks
ο To derive approximations for more complex Jackson networks which allow
the choice of suitable importance functions
Conclusions and future work (II)
ο Plans for future work are:
ο To derive exact formulas of the importance for other simple queuing networks