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A unified approach for large queue asymptotics in a heterogeneous multiserver queue

  • Masakiyo Miyazawa (a1)

We are interested in a large queue in a GI/G/k queue with heterogeneous servers. For this, we consider tail asymptotics and weak limit approximations for the stationary distribution of its queue length process in continuous time under a stability condition. Here, two weak limit approximations are considered. One is when the variances of the interarrival and/or service times are bounded, and the other is when they become large. Both require a heavy-traffic condition. Tail asymptotics and heavy-traffic approximations have been separately studied in the literature. We develop a unified approach based on a martingale produced by a good test function for a Markov process to answer both problems.

Corresponding author
* Postal address: Department of Information Sciences, Tokyo University of Science, 2641 Yamazaki, Noda, Chiba 278-8510, Japan. Email address:
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[1] Alsmeyer, G. (1996).The ladder variables of a Markov random walk.Tech. Rep., University of Münster.
[2] Alsmeyer, G. (1997).The Markov renewal theorem and related results.Markov Process. Relat. Fields 3,103127.
[3] Asmussen, S. (2003).Applied Probability and Queues(Appl. Math. 51),2nd edn.Springer,New York.
[4] Braverman, A.,Dai, J. and Miyazawa, M. (2015).Heavy traffic approximation for the stationary distribution of a generalized Jackson network: the BAR approach.Submitted.
[5] Chen, H. and Ye, H.-Q. (2011).Methods in diffusion approximation for multi-server system: sandwich, uniform attraction and state-space collapse.In Queueing Networks(Internat. Ser. Operat. Res. Manag. Sci. 154),Springer,New York,pp.489530.
[6] Dai, J. G. (1995).On positive Harris recurrence of multiclass queueing networks: a unified approach via fluid limit models.Ann. Appl. Prob. 5,4977.
[7] Davis, M. H. A. (1976).The representation of martingales of jump processes.SIAM J. Control Optimization 14,623638.
[8] Davis, M. H. A. (1984).Piecewise-deterministic Markov processes: a general class of nondiffusion stochastic models.J. R. Statist. Soc. B 46,353388.
[9] Davis, M. H. A. (1993).Markov Models and Optimization(Monogr. Statist. Appl. Prob. 49).Chapman and Hall,London.
[10] Dembo, A. and Zeitouni, O. (1998).Large Deviations Techniques and Applications(Appl. Math. 38),2nd edn.Springer,New York.
[11] Ethier, S. N. and Kurtz, T. G. (1986).Markov Processes: Characterization and Convergence.John Wiley,New York.
[12] Foss, S. and Korshunov, D. (2012).On large delays in multi-server queues with heavy tails.Math. Operat. Res. 37,201218.
[13] Glynn, P. and Whitt, W. (1994).Logarithmic asymptotics for steady-state tail probabilities in a single-server queue.J. Appl. Prob. 31,131156.
[14] Halfin, S. and Whitt, W. (1981).Heavy-traffic limits for queues with many exponential servers.Operat. Res. 29,567588.
[15] Jacod, J. and Shiryaev, A. N. (2003).Limit Theorems for Stochastic Processes,2nd edn.Springer,Berlin.
[16] Kingman, J. F. C. (1962).On queues in heavy traffic.J. R. Statist. Soc. B 24,383392.
[17] Kobayashi, M. and Miyazawa, M. (2012).Revisiting the tail asymptotics of the double QBD process: refinement and complete solutions for the coordinate and diagonal directions.In Matrix-Analytic Methods in Stochastic Models,eds G. Latouche et al.,Springer,New York,pp.147181.
[18] Kobayashi, M. and Miyazawa, M. (2014).Tail asymptotics of the stationary distribution of a two dimensional reflecting random walk with unbounded upward jumps.Adv. Appl. Prob. 46,365399.
[19] Köllerström, J. (1974).Heavy traffic theory for queues with several servers. I.J. Appl. Prob. 11,544552.
[20] Miyazawa, M. (2011).Light tail asymptotics in multidimensional reflecting processes for queueing networks.TOP 19,233299.
[21] Miyazawa, M. (2015).Diffusion approximation for stationary analysis of queues and their networks: a review.J. Operat. Res. Soc. Japan 58,104148.
[22] Miyazawa, M. (2015).A superharmonic vector for a nonnegative matrix with QBD block structure and its application to a Markov-modulated two-dimensional reflecting process.Queueing Systems 81,148.
[23] Miyazawa, M. (2016).A unified approach for large queue asymptotics in a heterogeneous multiserver queue.Preprint. Available at
[24] Miyazawa, M. and Zwart, B. (2012).Wiener-Hopf factorizations for a multidimensional Markov additive process and their applications to reflected processes.Stochastic Systems 2,67114.
[25] Neuts, M. and Takahashi, Y. (1981).Asymptotic behavior of the stationary distributions in the GI/PH/c queue with heterogeneous servers.Z. Wahrscheinlichkeitsth. 57,441452.
[26] Palmowski, Z. and Rolski, T. (2002).A technique of the exponential change of measure for Markov processes.Bernoulli 8,767785.
[27] Reed, J. (2009).The G/GI/N queue in the Halfin‒Whitt regime.Ann. Appl. Prob. 19,22112269.
[28] Reiman, M. I. (1984).Open queueing networks in heavy traffic.Math. Operat. Res. 9,441458.
[29] Sadowsky, J. (1995).The probability of large queue lengths and waiting times in a heterogeneous multiserver queue. II. Positive recurrence and logarithmic limits.Adv. Appl. Prob. 27,567583.
[30] Sadowsky, J. and Szpankowski, W. (1995).The probability of large queue lengths and waiting times in a heterogeneous multiserver queue. I. Tight limits.Adv. Appl. Prob. 27,532566.
[31] Sakuma, Y. (2011).Asymptotic behavior for MArP/PH/2 queue with join the shortest queue discipline.J. Operat. Res. Soc. Japan 54,4664.
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Advances in Applied Probability
  • ISSN: 0001-8678
  • EISSN: 1475-6064
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