Abstract
We develop a strongly efficient rare-event simulation algorithm for computing the tail of the steady-state waiting time in a single server queue with regularly varying service times. Our algorithm is based on a state-dependent importance sampling strategy that is constructed so as to be straightforward to implement. The construction of the algorithm and its asymptotic optimality rely on a Lyapunov-type inequality that is used to bound the second moment of the estimator. The solution to the Lyapunov inequality is constructed using fluid heuristics. Our approach takes advantage of the regenerative ratio formula for the steady-state distribution—and does not use the first passage time representation that is particular to the delay in the G/G/1 queue. Hence, the strategy has the potential to be applied in more general queueing models.
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References
Anantharam, V.: How large delays build up in a GI/G/1 queue. Queueing Syst. Theory Appl. 5, 345–368 (1989)
Asmussen, S.: Applied Probability and Queues. Springer, New York (2003)
Asmussen, S., Binswanger, K.: Simulation of ruin probabilities for subexponential claims. Astin Bull. 27, 297–318 (1997)
Asmussen, S., Kroese, D.: Improved algorithms for rare event simulation with heavy tails. Adv. Appl. Probab. 38, 545–558 (2006)
Asmussen, S., Binswanger, K., Hojgaard, B.: Rare event simulation for heavy-tailed distributions. Bernoulli. 2, 303–322 (2000)
Blanchet, J., Glynn, P.: Efficient rare event simulation for the maximum of heavy-tailed random walks. Ann. Appl. Probab. (2007, to appear). See http://www.imstat.org/aap/future_papers.html
Blanchet, J., Li, C.: Efficient rare event simulation for geometric sums. In: Proc. of RESIM, Bamberg (2006)
Blanchet, J., Glynn, P., Liu, J.C.: Efficient rare event simulation for multiserver queues. Preprint (2007)
Bucklew, J.: Introduction to Rare-event Simulation. Springer, New York (2004)
Dupuis, P., Wang, H.: Importance sampling, large deviations, and differential games. Stoch. Stoch. Rep. 76, 481–508 (2004)
Dupuis, P., Sezer, A., Wang, H.: Importance sampling for tandem networks. Preprint (2005)
Dupuis, P., Leder, K., Wang, H.: Importance sampling for sums of random variables with regularly varying tails. TOMACS 17 (2006)
Foss, S., Konstantopoulos, T., Zachary, S.: Discrete and continuous time modulated random walks with heavy-tailed increments. Preprint (2007)
Embrechts, P., Klüppelberg, C., Mikosch, T.: Modelling Extremal Events for Insurance and Finance. Springer, New York (1997)
Goyal, A., Shahabuddin, P., Heidelberger, P., Nicola, V.F., Glynn, P.W.: A unified framework for simulating Markovian models of highly reliable systems. IEEE Trans. Comput. 41, 36–51 (1992)
Gut, A.: Stopped Random Walks: Limits Theorems and Applications. Springer, New York (1988)
Juneja, S., Shahabuddin, P.: Simulating heavy-tailed processes using delayed hazard rate twisting. ACM TOMACS 12, 94–118 (2002)
Juneja, S., Shahabuddin, P.: Rare event simulation techniques: an introduction and recent advances. In: Henderson, S., Nelson, B. (eds.) Handbook on Simulation, pp. 291–350. Elsevier, Amsterdam (2006)
Zachary, S.: A note on Veraverbeke’s theorem. Queueing Syst. Theory Appl. 46, 9–14 (2004)
Zwart, A.: Queueing Systems with Heavy Tails. Ph.D. Dissertation (2001)
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Blanchet, J., Glynn, P. & Liu, J.C. Fluid heuristics, Lyapunov bounds and efficient importance sampling for a heavy-tailed G/G/1 queue. Queueing Syst 57, 99–113 (2007). https://doi.org/10.1007/s11134-007-9047-4
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DOI: https://doi.org/10.1007/s11134-007-9047-4
Keywords
- State-dependent importance sampling
- Rare-event simulation
- Heavy-tails
- Fluid heuristics
- Lyapunov bounds
- Single-server queue
- Change-of-measure