## SPDE limits of many-server queues.(English)Zbl 1271.60098

Authors’ abstract: This paper studies a queueing system in which customers with independent and identically distributed service times arrive at a queue with many servers and enter service in the order of arrival. The state of the system is represented by a process that describes the total number of customers in the system, and a measure-valued process that keeps track of the ages of customers in service, leading to a Markovian description of the dynamics. Under suitable assumptions, a functional central limit theorem is established for the sequence of (centered and scaled) state processes as the number of servers goes to infinity. The limit process describing the total number in system is shown to be an Itō diffusion with a constant diffusion coefficient that is insensitive to the service distribution beyond its mean. In addition, the limit of the sequence of (centered and scaled) age processes is shown to be a diffusion taking values in a Hilbert space and is characterized as the unique solution of a stochastic partial differential equation that is coupled with the Itō diffusion describing the limiting number in system. Furthermore, the limit processes are shown to be semimartingales and to possess a strong Markov property.

### MSC:

 60K25 Queueing theory (aspects of probability theory) 60F17 Functional limit theorems; invariance principles 60H15 Stochastic partial differential equations (aspects of stochastic analysis) 90B22 Queues and service in operations research 68M20 Performance evaluation, queueing, and scheduling in the context of computer systems
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 [1] Asmussen, S. (2003). Applied Probability and Queues , 2nd ed. Springer, New York. · Zbl 1029.60001 [2] Berezanskii, Y. M. (1986). Selfadjoint Operators in Spaces of Functions of Infinitely Many Variables. Translations of Mathematical Monographs 63 . Amer. Math. Soc., Providence, RI. · Zbl 0596.47019 [3] Billingsley, P. (1968). Convergence of Probability Measures . Wiley, New York. · Zbl 0172.21201 [4] Brown, L., Gans, N., Mandelbaum, A., Sakov, A., Shen, H., Zeltyn, S. and Zhao, L. (2005). Statistical analysis of a telephone call center: A queueing science perspective. JASA 100 36-50. · Zbl 1117.62303 [5] Burton, B. (2005). Volterra Integral and Differential Equations . Elsevier, Amsterdam. · Zbl 1075.45001 [6] Centsov, N. N. (1956). Wiener random fields depending on several parameters. Dokl. Akad. Nauk. , SSSR 106 607-609. [7] Decreusefond, L. and Moyal, P. (2008). A functional central limit theorem for the $$M/GI/\infty$$ queue. Ann. Appl. Probab. 18 2156-2178. · Zbl 1154.60347 [8] Erlang, A. K. (1948). On the rational determination of the number of circuits. In The Life and Works of A. K. Erlang (E. Brockmeyer, H. L. Halstrom and A. Jensen, eds.). The Copenhagen Telephone Company, Copenhagen. [9] Ethier, S. N. and Kurtz, T. G. (1986). Markov Processes : Characterization and Convergence . Wiley, New York. · Zbl 0592.60049 [10] Evans, L. C. (1988). Partial Differential Equations. Graduate Studies in Mathematics 19 . Amer. Math. Soc., Providence, RI. [11] Friedman, E. (2006). Stochastic Differential Equations and Applications . Dover, New York. · Zbl 1113.60003 [12] Gamarnik, D. and Momčilović, P. (2008). Steady-state analysis of a multiserver queue in the Halfin-Whitt regime. Adv. in Appl. Probab. 40 548-577. · Zbl 1148.60070 [13] Glynn, P. and Whitt, W. (1991). A new view of the heavy-traffic limit theorem for infinite-server queues. Adv. in Appl. Probab. 2 188-209. · Zbl 0716.60105 [14] Halfin, S. and Whitt, W. (1981). Heavy-traffic limit theorems for queues with many servers. Oper. Res. 29 567-588. · Zbl 0455.60079 [15] Iglehart, D. L. (1965). Limit diffusion approximations for the many server queue and the repairman problem. J. Appl. Probab. 2 429-441. · Zbl 0216.47204 [16] Jacod, J. and Shiryaev, A. N. (1987). Limit Theorems for Stochastic Processes . Springer, Berlin. · Zbl 0635.60021 [17] Jagerman, D. (1974). Some properties of Erlang loss function. Bell System Techn. J. 53 525-551. · Zbl 0276.60092 [18] Jelenkovic, P., Mandelbaum, A. and Momčilović, P. (2004). Heavy traffic limits for queues with many deterministic servers. Queueing Syst. 47 53-69. · Zbl 1048.60069 [19] Kang, W. N. and Ramanan, K. (2010). Fluid limits of many-server queues with reneging. Ann. Appl. Probab. 20 2204-2260. · Zbl 1208.60094 [20] Kang, W. N. and Ramanan, K. (2012). Asymptotic approximations for stationary distributions of many-server queues with abandonment. Ann. Appl. Probab. 22 477-521. · Zbl 1245.60087 [21] Kaspi, H. and Ramanan, K. (2011). Law of large numbers limits for many-server queues. Ann. Appl. Probab. 21 33-114. · Zbl 1208.60095 [22] Krichagina, E. and Puhalskii, A. (1997). A heavy traffic analysis of a closed queueing system with a $$\mathrm{GI}/\mathrm{G}/\infty$$ service center. Queueing Syst. 25 235-280. · Zbl 0892.60090 [23] Mitoma, I. (1983). On the sample continuity of $$\mathcal{S}'$$-processes. J. Math. Soc. Japan 35 629-636. · Zbl 0507.60029 [24] Mitoma, I. (1983). Tightness of probabilities on $$\mathcal{C}([0,1],S')$$ and $$\mathcal{D}([0,1],S)$$. Ann. Probab. 11 989-999. · Zbl 0527.60004 [25] Puhalskii, A. A. and Reed, J. (2010). On many servers queues in heavy traffic. Ann. Appl. Probab. 20 129-195. · Zbl 1201.60088 [26] Puhalskii, A. A. and Reiman, M. (2000). The multiclass $$\mathrm{GI}/\mathrm{PH}/\mathrm{N}$$ queue in the Halfin-Whitt regime. Adv. in Appl. Probab. 32 564-595. · Zbl 0962.60089 [27] Reed, J. (2009). The $$\mathrm{G}/\mathrm{GI}/\mathrm{N}$$ queue in the Halfin-Whitt regime. Ann. Appl. Probab. 19 2211-2269. · Zbl 1181.60137 [28] Revuz, D. and Yor, M. (1999). Continuous Martingales and Brownian Motion , 3rd ed. Springer, Berlin. · Zbl 0917.60006 [29] Rogers, D. and Williams, D. (1994). Diffusion , Markov Processes and Martingales , Volume 2: Foundations , 2nd ed. Wiley, New York. · Zbl 0826.60002 [30] Sharpe, M. (1988). General Theory of Markov Processes . Academic Press, San Diego. · Zbl 0649.60079 [31] Walsh, J. B. (1986). An introduction to stochastic partial differential equations. In École d’Été Probabilités de Saint-Flour XIV. Lecture Notes in Math. 1180 265-439. Springer, Berlin. · Zbl 0608.60060 [32] Whitt, W. (2005). Heavy-traffic limits for the $$\mathrm{G}/\mathrm{H2}/\mathrm{n}/\mathrm{m}$$ queue. Queueing Syst. 30 1-27. · Zbl 1082.90019
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