The Annals of Applied Probability

On many-server queues in heavy traffic

Anatolii A. Puhalskii and Josh E. Reed

Full-text: Open access


We establish a heavy-traffic limit theorem on convergence in distribution for the number of customers in a many-server queue when the number of servers tends to infinity. No critical loading condition is assumed. Generally, the limit process does not have trajectories in the Skorohod space. We give conditions for the convergence to hold in the topology of compact convergence. Some new results for an infinite server are also provided.

Article information

Ann. Appl. Probab., Volume 20, Number 1 (2010), 129-195.

First available in Project Euclid: 8 January 2010

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60K25: Queueing theory [See also 68M20, 90B22]
Secondary: 60F17: Functional limit theorems; invariance principles 60G15: Gaussian processes 60G44: Martingales with continuous parameter

Many-server queues heavy traffic weak convergence Skorohod space martingales Gaussian processes


Puhalskii, Anatolii A.; Reed, Josh E. On many-server queues in heavy traffic. Ann. Appl. Probab. 20 (2010), no. 1, 129--195. doi:10.1214/09-AAP604.

Export citation


  • [1] Billingsley, P. (1999). Convergence of Probability Measures, 2nd ed. Wiley, New York.
  • [2] Borovkov, A. A. (1967). Limit laws for queueing processes in multichannel systems. Sibirsk. Mat. Ž. 8 983–1004.
  • [3] Borovkov, A. A. (1980). Asimptoticheskie Metody v Teorii Massovogo Obsluzhivaniya. Nauka, Moscow.
  • [4] Brémaud, P. (1981). Point Processes and Queues. Springer, New York.
  • [5] Dellacherie, C. (1972). Capacités et Processus Stochastiques. Springer, Berlin.
  • [6] Doob, J. L. (1984). Classical Potential Theory and Its Probabilistic Counterpart. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 262. Springer, New York.
  • [7] Dudley, R. M. (1990). Nonlinear functionals of empirical measures and the bootstrap. In Probability in Banach Spaces, 7 (Oberwolfach, 1988). Progress in Probability 21 63–82. Birkhäuser, Boston, MA.
  • [8] Dudley, R. M. (1999). Uniform Central Limit Theorems. Cambridge Studies in Advanced Mathematics 63. Cambridge Univ. Press, Cambridge.
  • [9] Ethier, S. N. and Kurtz, T. G. (1986). Markov Processes. Wiley, New York.
  • [10] Halfin, S. and Whitt, W. (1981). Heavy-traffic limits for queues with many exponential servers. Oper. Res. 29 567–588.
  • [11] Ibragimov, I. A. and Rozanov, Y. A. (1978). Gaussian Random Processes. Applications of Mathematics 9. Springer, New York.
  • [12] Jacod, J. and Shiryaev, A. N. (1987). Limit Theorems for Stochastic Processes. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 288. Springer, Berlin.
  • [13] Jelenković, P., Mandelbaum, A. and Momčilović, P. (2004). Heavy traffic limits for queues with many deterministic servers. Queueing Syst. 47 53–69.
  • [14] Kaspi, H. and Ramanan, K. (2006). Fluid limits of many-server queues. Preprint.
  • [15] Krichagina, E. V. and Puhalskii, A. A. (1997). A heavy-traffic analysis of a closed queueing system with a GI/∞ service center. Queueing Systems Theory Appl. 25 235–280.
  • [16] Liptser, R. and Shiryayev, A. (1989). Theory of Martingales. Kluwer, Dordrecht.
  • [17] Louchard, G. (1988). Large finite population queueing systems. I. The infinite server model. Comm. Statist. Stochastic Models 4 473–505.
  • [18] Mandelbaum, A. and Momcilovic, P. (2008). Queues with many servers: The virtual waiting-time process in the QED regime. Math. Oper. Res. 33 561–586.
  • [19] Pang, G., Talreja, R. and Whitt, W. (2007). Martingale proofs of many-server heavy-traffic limits for Markovian queues. Probab. Surv. 4 193–267.
  • [20] Puhalskii, A. (2001). Large Deviations and Idempotent Probability. Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics 119. Chapman & Hall/CRC, Boca Raton, FL.
  • [21] Puhalskii, A. A. and Reiman, M. I. (2000). The multiclass GI/PH/N queue in the Halfin–Whitt regime. Adv. in Appl. Probab. 32 564–595.
  • [22] Reed, J. E. (2009). The G/GI/N queue in the Halfin–Whitt regime I: Infinite server queue system equations. Ann. Appl. Probab. 19 2211–2269.
  • [23] van der Vaart, A. W. and Wellner, J. A. (1996). Weak Convergence and Empirical Processes. Springer, New York.
  • [24] Whitt, W. (2002). Stochastic-Process Limits. Springer, New York.
  • [25] Whitt, W. (2005). Heavy-traffic limits for the G/H*2/n/m queue. Math. Oper. Res. 30 1–27.