1

RESTART Simulation Simulation of Non-of Non-Markovian Queueing NetworksMarkovian Queueing Networks

Manuel Villén-Altamirano, José Villén-Altamirano, Enrique Vázquez-Gallo

Universidad Politécnica de Madrid

2

CONTENTSCONTENTS

Description of RESTART and previous results

Effective Importance Functions

Simulation results

Conclusions

3

Description of RESTART (I)(t)

t (time)

L

T3

T2

T1

B 3

B 2

B 1

B 3

B1D1 D1

D2D2

D3D 3

D 1D1

D 2

D3

A

=

=

1

P Pr Pr

Pr Pr

:Number of trials at

i i

i i

A

C

i

i jj

L

T

R B

r R

: No. of simulated events ( retrials not included ):No. of events ( retrials included )

NN AANr

NP

M

A

ˆ

C1 C2 C3 … CM A

P A = P C1 P C2 / C1 … P A/CM

4

Description of RESTART (II)

i

: No. of simulated events ( retrials not included ):No. of events in retrials from sets C

NN AAi

1

ˆM

i i

AiNP

r N

1

1,..., then: /M

i i ii

A C i M P A P C P A C

LT3

T2

T1

Q1

Q2

2

21 2

1

Pr =Pr Q

Pr =Pr

ln

ln

i i

P A L

C T

Q Q

5

Gain Obtained with RESTART

2

0

1 1Gain

ln 1V R Tf f f f P P

Factors f 1 reflect inefficiency due to:

f T - not optimal thresholds f 0 - algorithm overhead

f R - not optimal Ri f V - variance at Bi

6

Factor fR

Optimal values of ri

11

011

Rii

fPP

ri

Rounding Algorithm R1 = r1 rounded to an integer number, R2 = r2/ R1 rounded

to an integer number, . . . , Ri = ri / R1 · . . . ·Ri-1 rounded to an integer number.

7

Factor fT

2

minmin

2

min

ln

1 1min

PP

PfTiiPP Min

The thresholds must be set as close as possible

Pmin fT1 1

0.5 1.04

0.1 1.5

0.01 4.6

0.001 20.9

8

Factor fO

This factor usually takes low values with exponential times. However the rescheduling of Hyperexponential or Erlang times is more time consuming.

iyMaxf 0

Affects to computational time, not to number of events

ye = overhead per event: evaluate , compare with Ti , …

yri = overhead per retrial: restore state at Bi , re-schedule, ...

y0 = ye yi = ye yri

9

Factor fV (I): Rescheduling It is convenient to reschedule at Bi, for each retrial, the scheduled arrival times and the

scheduled end of service times. Otherwise, there would be high correlation between retrials. If these times are exponentially distributed, the rescheduling is straightforward, due to the

memory-less property of this distribution. For other distributions we use the following procedures: we obtain a random value of the

whole e.g., service time of a customer. If the end of service time is greater than the value of the clock at the current time (Bi), the residual lifetime is obtained as the difference between the two amounts. Otherwise a new random value is obtained and so on.

If after 50 attempts the new end of service time is lower than the value of the clock at the current time (Bi), it is not rescheduled.

Start of service

Scheduled end of service

Bi

10

CONTENTSCONTENTS

Description of RESTART and previous results

Effective Importance Functions

Simulation results

Conclusions

11

Factor fV (II)

iv sMaxf * */ /

* 2 * 2/ /

( ) ( )' 1

( ) ( )i iA X A Xi

i i iA A i A i

V P V Pas KK P P

Xi : system state at Bi

: importance of state Xi

: expected value of

i : factor reflecting the autocovariance of

P XA i

*

P XA i

*

P XA i

*

P iA*

1iXAP

2iXAP

3iXAP

A

iC1iX

2iX

3iX

iC

iTF

12

Importance Function for Jackson Networks (I)

Importance function for three-queue Jackson tandem network: if 1 > 2 > 3

Villén-Altamirano, J. 2010. Importance function for RESTART simulation of general Jackson networks. European Journal of Operation Research 203 (1), 156 – 165.

1 2 3 2 1 3 1 2 3 ,or ,If if Q Q Q

12 3 1 1 2 3

3

ln,

lnIf Q Q Q

21 3 2 3 1 2 1 2 3

3

ln ,or ,If

lnif Q Q Q

1 21 2 3

3 3

ln ln

ln lnQ Q Q

13

Importance Function for

General Jackson Networks (II)

*

1 1 2 21 1

ln ln

ln ln

H Ktg tgi tg tgj

i i j j tgi jtg tg

Q Q Q

1

K

tg j jtgj tg

tgtg tg

p

2 1 1 21*

,K

tg j i i ij j jtg tg tgtgj

tgitg

Min p p p

1 21 1

1

11

1 ;

K K

l lj jtg j jtg tgl i j j

i K

i ij jtgj

p p p

p p

1 21

2 1

H

i il ltg l ltg tgi l j l j

jj jtg

p p p

p

Pr tgP Q L

2 2tg j jtg l ltg tg tgtgl j

tgjtg

p p p

14

Importance Functions for Non-Jackson Networks (I)

Would be fit for other networks the importance function derived for Jackson networks? Or at least, would be easy to modify it?

The importance function is a linear combination of the queue length of the nodes. The coefficients are function of the load of the nodes. In general: the lower the load of a node, the higher the value of the coefficient.

Importance Functions for Non-Jackson Networks (II)

For Jackson networks the value of the load (), can be calculated from the formulas:

P(X>=n) = P(X>=2n / X>=n) = ^n (1)

For non-Jackson networks the value of that will be used in previous formulas of the importance function (derived for Jackson networks) is calculated with Equation (1).

We will call it “effective load” and it does not match the actual load. The probability P(X>=n) is evaluated by crude simulation for a low value of n.

16

CONTENTSCONTENTS

Description and previous results

Effective Importance Function

Simulation results

Conclusions

Models Simulated (I) Example 1: Example 1: 2-node network with strong feedback2-node network with strong feedback

Three sets of loads: Three sets of loads: 1 2 32 /3; 1/ 2; 1/3

1 2 31/3; 1/ 2; 1/3

1 2 31/5; 1/ 4; 1/3

0,2

0,8

0,2

0,8

=2

=2 1 2

1 22 / 3; 1/ 3

1 2 3

Example 2: Example 2: Three-queue tandem networkThree-queue tandem network ..

18

Example 3: Example 3: Network with 7 nodes . Arrival rate γi = 1; i = 1, …,7

Transition Probability Matrix Transition Probability Matrix

Models Simulated (II)

1 2 3 4 5 6 t Ext.

1 0.1 0.1 0.1 0.1 0.2 0.2 0 0.2

2 0.1 0.1 0.1 0.1 0.2 0.2 0 0.2

3 0.1 0.1 0.1 0.1 0.2 0.2 0 0.2

4 0.1 0.1 0.1 0.1 0.2 0.2 0 0.2

5 0.1 0.1 0.1 0.1 0 0.1 0.3 0.2

6 0.1 0.1 0.1 0.1 0.1 0 0.3 0.2

t 0.1 0.1 0.1 0.1 0.1 0.1 0.2 0.2

Three sets of loads: Three sets of loads: 1 20.5; 0.41; 0.33i j tg

1 20.32; 0.41; 0.33i j tg

1 20.28; 0.30; 0.33i j tg

Models Simulated (III)

Example 4: A large network with 15 nodes: 4 of the nodes are at “distance” 3, and so their queue

lengths are not included in the importance function. A customer leaving a node can go to 8 nodes with probability

0.1 (to each one) or can leave the network with probability 0.2. The load of the target node is similar to the loads of other 2

nodes, and lower than the loads of the other 12 nodes.

This paper deals with networks with: Interarrival times: Exponential, Erlang or Hyperexponential

Service times: Exponential or Erlang CV Pearson: Erlang (3, β) = 0.58 ; Hyperexponential: 1.42

20

Simulation Results (I)Example 1: Example 1: 2-node network with strong feedback.2-node network with strong feedback.

Rare event probability: Relative error = 0.1Interarrival times: Exponential, Service times: ExponentialInterarrival times: Hyper-Exponential, Service times: ErlangImportance function:

1510tgP Q L

0.22;etg

Model

L P 1 aEventsmillions

Timeminute

s

Gain(events)

fVGain(time)

f0

Exp-Exp

31

1.4x10-

15

0.67

0.36

3.3 2.1 8.2x1010 6.11,2x101

0 6.9

Hyp-Erl

23

2.2x10-

15

0.75

0.19

1.8 1.6 6.1x1010 6.1 4.3x109 14

0.33tg

1 tgaQ Q

Robustness: Acceptable results for coefficients a between 0 and 0.21.

21

Simulation Results (II)Example 2: Example 2: Three-queue tandem networkThree-queue tandem network

Rare event probability: Actual loads of the target network: Effective (H-E):Importance function:

1510tgP Q L 0.08;0.15;0.30t

Robustness: Acceptable results for coefficients a and b up to 10% lower or greater than optimal

Model

L P a bEventsmillions

Timeminute

sfV

Gain(time)

f0

Exp-Exp

31

1.7x10-

15 0.370.63

11.9 3.3 119.7x10

9 4.6

Exp-Exp

31

1.6x10-

15 0.630.63

3.3 1.0 6.51.9x10

10 4.3

Exp-Exp

31

1.6x10-

15 1 1 1.0 0.25 1.01.0x10

11 1.9

Hyp-Erl

14

7.0x10-

15 0.130.50

21.9 16.6 771.2x10

8 14

Hyp-Erl

19

1.3x10-

15 0.320.32

9.1 6.4 192.5x10

9 13

Hyp-Erl

33

1.9x10-

15 0.900.90

2.0 0.9 2.85,7x10

9 7.0

0.33;t

1 2 tgaQ bQ Q

22

Simulation Results (III)Example 3: NExample 3: Network with 7 nodos:etwork with 7 nodos:

Rare set probability: Importance function:

1510 ;tgP Q L 4 6

1 5i j t

i j

a Q b Q Q

Robustness: Acceptable results are obtained for coefficients a and b up to 20% lower or greater than optimal ones. Similar results are obtained (for the Hyp-Erl case) with the coefficients derived for Jackson networks.

Model L P a bEventsmillions

Timeminute

sfV

Gain(time)

F0

Exp-Exp 30 2.5x10-15 0.2

30.45 3.8 1.1 1.8

4.4x101

0 4.2

Exp-Exp 30 2.5x10-15 0.3

60.45 1.9 0.55 1.3

7.7x101

0 3.7

Exp-Exp 30 2.6x10-15 0.4

00.61 1.6 0.50 1.0

8.7x101

1 3.7

Hyp-Erl 31 2.9x10-15 0.23

0.49 3.0 1.7 2.1 2.3 1010 6.1

Hyp-Erl 32 3.0x10-15 0.35

0.43 2.4 1.21.4

3.7 1010 5.4

Hyp-Erl 33 5.3x10-15 0.41

0.66 2.1 1.21.1

2.5 1010 6.0

23

Simulation Results (IV)Example Example 44: : Large network with 15 nodesLarge network with 15 nodes

Rare event probability: Relative error = 0.1Coefficients of the importance function: 0,0,0,0,0.22,0.20,0.20,0.22,0.37,0.35,0.35,0.37,0.34,0.42,1 (Exp-Exp)

0,0,0,0,0.25,0.22,0.22,0.24,0.42,0.39,0.38,0.41,0.35,0.43,1 (Hyp-Erl)

1510tgP Q L

Model L PEventsMillions

TimeMinutes

fVGain(time)

f0

Exp-Exp

321.1x10-

15 10.5 3.11.9

1.4x101

1

2.7

Hyp-Erl

312.0x10-

15 20.3 15.94.1

1.5x101

0

6.6

Robustness: Acceptable results for coefficients up to 50% lower than those given by the formula. Similar results are obtained (for the Hyp-Erl case) with the set of coefficients derived for Jackson networks.

24

Simulation Results (V)

Analogous results Analogous results ((better than the Hyp-Erl case but worse than the Exp-better than the Hyp-Erl case but worse than the Exp-Exp caseExp case)) have been obtained for these 4 topologies with the following have been obtained for these 4 topologies with the following distributions:distributions:

Exponential-Erlang, Hyper-Exponential-ExponentialExponential-Erlang, Hyper-Exponential-Exponential

Erlang-Exponential and Erlang-ErlangErlang-Exponential and Erlang-Erlang

For simulating the networks of 7 and 15 nodes the importance function derived for Markovian networks has been used for these four cases.

For the three-queue tandem network and for the tho-node network with strong feedback the same formulas have been used, but with the “effective loads” calculated as: P(X>=2n / X>=n) = ^n

25

Conclusions

Probabilities of the order of 10-15 have been estimated, within short or moderate computational times, in 48 types of networks with different topologies and loads, and different interarrival and service times. Most of them are “difficult” networks for estimating rare event probabilities.

Formulas of the importance function derived for Jackson networks lead to very good results in Non-Jackson networks changing the actual loads of the nodes by the “effective loads”.

These type of formulas could be applied to many other non-Jackson networks for estimating rare event probabilities.

Worst results are obtained when the dependence of the target queue on the queue length of the other queues is very high. As a consequence, the efficiency of RESTART often improves with the complexity of the system.