# Perfect sampling of Jackson queueing networks

- 226 Downloads
- 1 Citations

## Abstract

We consider open Jackson networks with losses with mixed finite and infinite queues and analyze the efficiency of sampling from their exact stationary distribution. We show that perfect sampling is possible, although the underlying Markov chain may have an infinite state space. The main idea is to use a Jackson network with infinite buffers (that has a product form stationary distribution) to bound the number of initial conditions to be considered in the coupling from the past scheme. We also provide bounds on the sampling time of this new perfect sampling algorithm for acyclic or hyper-stable networks. These bounds show that the new algorithm is considerably more efficient than existing perfect samplers even in the case where all queues are finite. We illustrate this efficiency through numerical experiments. We also extend our approach to variable service times and non-monotone networks such as queueing networks with negative customers.

## Keywords

Perfect simulation Markov chain Jackson networks Bounding process## Mathematics Subject Classification

60H35 68U20 60K25 65C99## Notes

### Acknowledgments

This work was partially supported by the ANR Numerical Models Program (MARMOTE Project).

## References

- 1.Anselmi, J., Gaujal, B.: On the efficiency of perfect simulation in monotone queueing networks. In: IFIP Performance: 29th International Symposium on Computer Performance, Modeling, Measurements and Evaluation, Amsterdam. ACM Performance Evaluation Review (2011)Google Scholar
- 2.Anselmi, J., Gaujal, B.: Efficiency of simulation in monotone hyper-stable queueing networks. Queuing Syst. Theory Appl.
**76**(1), 51-72 (2013)Google Scholar - 3.Bolch, G., Greiner, S., de Meer, H., Trivedi, K.: Queueing Networks and Markov Chains. Wiley-Interscience, New York (2005)Google Scholar
- 4.Busic, A., Gaujal, B., Vincent, J.-M.: Perfect simulation and non-monotone Markovian systems. In 3rd International Conference Valuetools’08, Athens, Greece. ICST (2008)Google Scholar
- 5.Busic, A., Gaujal, B., Perronnin, F.: Perfect Sampling of Networks with Finite and Infinite Capacity Queues. In Al-Begain, K., Fiems, D., Vincent, J.-M. (eds.) 19th International Conference on Analytical and Stochastic Modelling Techniques and Applications (ASMTA) 2012, vol. 7314 of Lecture Notes in Computer Science, pp. 136–149, Grenoble, France. Springer, Heidelberg (2012a)Google Scholar
- 6.Busic, A., Gaujal, B., Pin, F.: Perfect sampling of Markov chains with piecewise homogeneous events. Perform. Eval.
**69**(6), 247–266 (2012b)CrossRefGoogle Scholar - 7.Chen, H., Yao, D.D.: Fundamentals of Queueing Networks. Springer, New York (2001)CrossRefGoogle Scholar
- 8.Dopper, J., Gaujal, B., Vincent, J.-M.: Bounds for the coupling time in queueing networks perfect simulation. In: Celebration of the 100th anniversary of Markov, Charleston, pp. 117–136 (2006)Google Scholar
- 9.Gaujal, B., Perronnin, F., Bertin, R.: Perfect simulation of a class of stochastic hybrid systems with an application to peer to peer systems. J. Discret. Event Dyn. Syst.
**18**(2), 211–240 (2008). Special Issue on Hybrid SystemsCrossRefGoogle Scholar - 10.Gelenbe, E.: Product-form queueing networks with negative and positive customers. J. Appl. Probab.
**28**(3), 656–663 (1991)CrossRefGoogle Scholar - 11.Goodman, J.B., Massey, W.A.: The non-ergodic Jackson network. J. Appl. Probab.
**21**(4), 860–869 (1984)CrossRefGoogle Scholar - 12.Jackson, J.R.: Job shop-like queueing systems. Manag. Sci.
**10**, 131 (1963)CrossRefGoogle Scholar - 13.Kelly, F.: Reversibility and Stochastic Networks. Wiley, Chichester (1979)Google Scholar
- 14.Kendall, W.S.: Notes on Perfect Simulation. Department of Statistics, University of Warwick, Warwick (2005)CrossRefGoogle Scholar
- 15.Kendall, W.S., Møller, J.: Perfect simulation using dominating processes on ordered spaces, with application to locally stable point processes. Adv. Appl. Probab.
**32**(3), 844–865 (2000)CrossRefGoogle Scholar - 16.Pin, F., Busic, A., Gaujal, B.: Acceleration of perfect sampling for skipping events. In Valuetools, Paris (2011)Google Scholar
- 17.Propp, J.G., Wilson, D.B.: Exact sampling with coupled Markov chains and applications to statistical mechanics. Rand. Struct. Alg.
**9**(1–2), 223–252 (1996)CrossRefGoogle Scholar - 18.Vincent, J.-M.: Perfect simulation of monotone systems for rare event probability estimation. In: WSC ’05: Proceedings of the 37th Conference on Winter Simulation Conference, pp. 528–537 (2005)Google Scholar
- 19.Walker, A.J.: An efficient method for generating discrete random variables with general distributions. ACM Trans. Math. Softw.
**3**, 253–256 (1977)CrossRefGoogle Scholar