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RESTART Simulation of Non-Markovian Queueing Networks. Manuel Villén-Altamirano, José Villén-Altamirano, Enrique Vázquez-Gallo Universidad Politécnica de Madrid. CONTENTS. Description of RESTART and previous results Effective Importance Functions Simulation results Conclusions.  (t).

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slide1

RESTART Simulation of Non-Markovian Queueing Networks

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

Universidad Politécnica de Madrid

slide2

CONTENTS

  • Description of RESTART and previous results
  • Effective Importance Functions
  • Simulation results
  • Conclusions
slide3

(t)

L

A

B

B

3

3

T

3

D

D

D

3

3

3

B

2

T

2

D

D

D

2

2

2

B

1

T

1

D

D

D

B1

D

1

1

1

1

t (time)

C1 C2 C3 … CM A

P A = P C1  P C2 / C1  …  P A/CM

Description of RESTART (I)

slide4

Q2

T

3

L

T

2

T

1

Q1

Description of RESTART (II)

slide5

Gain Obtained with RESTART

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

slide6

Factor fR

  • Optimal values of ri

Rounding Algorithm

  • R1 = r1rounded to an integer number, R2 = r2/ R1 rounded to an integer number, . . . , Ri = ri / R1 · . . . ·Ri-1 rounded to an integer number.
slide7

Factor fT

  • The thresholds must be set as close as possible
slide8

Affects to computational time, not to number of events

Factor fO

  • ye = overhead per event: evaluate  , compare with Ti , …
  • yri = overhead per retrial: restore state at Bi , re-schedule, ...
  • y0 = ye yi = ye yri
  • This factor usually takes low values with exponential times. However the rescheduling of Hyperexponential or Erlang times is more time consuming.
factor f v i rescheduling
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

Scheduledend of service

Bi

slide10

CONTENTS

  • Description of RESTART and previous results
  • Effective Importance Functions
  • Simulation results
  • Conclusions
slide11

A

Factor fV (II)

  • Xi : system state at Bi
  • : importance of state Xi
  • : expected value of
  • i : factor reflecting the autocovariance of
importance function for jackson networks i
Importance Functionfor 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.
importance functions for non jackson networks i
Importance Functions forNon-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
Importance Functions forNon-Jackson Networks (II)
  • For Jackson networks the value of the load (), can be calculatedfrom 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.

slide16

CONTENTS

  • Description and previous results
  • Effective Importance Function
  • Simulation results
  • Conclusions
models simulated i

0,2

0,2

 =2

2

1

0,8

 =2

0,8

3

2

1

Models Simulated (I)
  • Example 1: 2-node networkwithstrongfeedback

Example 2: Three-queue tandem network.

Three sets of loads:

slide18

Models Simulated (II)

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

Transition Probability Matrix

Three sets of loads:

models simulated iii
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

slide20

Simulation Results (I)

  • Example 1: 2-node networkwithstrongfeedback.
  • Rare event probability:
  • Relative error = 0.1
  • Interarrival times: Exponential, Service times: Exponential
  • Interarrival times: Hyper-Exponential, Service times: Erlang
  • Importance function:
  • Robustness: Acceptable results for coefficients a between 0 and 0.21.
simulation results ii
Simulation Results (II)
  • Example 2: Three-queue tandem network
    • Rare event probability:
    • Actual loads of the target network: Effective (H-E):
    • Importance function:
  • Robustness: Acceptable results for coefficients a and b up to 10% lower or greater than optimal
simulation results iii
Simulation Results (III)
  • Example 3: Network with 7 nodos:
    • Rare set probability:
    • Importance function:
  • Robustness: Acceptable results are obtained for coefficients aandb 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.
slide23

Simulation Results (IV)

  • Example 4: Large network with 15 nodes
  • Rare event probability:
  • Relative error = 0.1
  • Coefficients 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)

  • 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.

23

slide24

Simulation Results (V)

Analogousresults(betterthantheHyp-Erl case butworsethantheExp-Exp case)havebeenobtainedforthese 4 topologieswiththefollowingdistributions:

Exponential-Erlang, Hyper-Exponential-Exponential

Erlang-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
slide25

Conclusions

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