The Annals of Applied Probability

The G/GI/N queue in the Halfin–Whitt regime

Josh Reed

Full-text: Open access


In this paper, we study the G/GI/N queue in the Halfin–Whitt regime. Our first result is to obtain a deterministic fluid limit for the properly centered and scaled number of customers in the system which may be used to provide a first-order approximation to the queue length process. Our second result is to obtain a second-order stochastic approximation to the number of customers in the system in the Halfin–Whitt regime. This is accomplished by first centering the queue length process by its deterministic fluid limit and then normalizing by an appropriate factor. We then proceed to obtain an alternative but equivalent characterization of our limiting approximation which involves the renewal function associated with the service time distribution. This alternative characterization reduces to the diffusion process obtained by Halfin and Whitt [Oper. Res. 29 (1981) 567–588] in the case of exponentially distributed service times.

Article information

Ann. Appl. Probab. Volume 19, Number 6 (2009), 2211-2269.

First available in Project Euclid: 25 November 2009

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60F17: Functional limit theorems; invariance principles 60K25: Queueing theory [See also 68M20, 90B22] 90B22: Queues and service [See also 60K25, 68M20]
Secondary: 60G15: Gaussian processes 60G44: Martingales with continuous parameter 60K15: Markov renewal processes, semi-Markov processes

Queueing theory diffusion approximation Gaussian process martingale weak convergence


Reed, Josh. The G / GI / N queue in the Halfin–Whitt regime. Ann. Appl. Probab. 19 (2009), no. 6, 2211--2269. doi:10.1214/09-AAP609.

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] Brémaud, P. (1981). Point Processes and Queues, Martingale Dynamics. Springer, New York.
  • [4] Evans, L. C. and Gariepy, R. F. (1992). Measure Theory and Fine Properties of Functions. Studies in Advanced Mathematics. CRC Press, Boca Raton, FL.
  • [5] Gamarnik, D. and Momcilovic, P. (2007). Steady-state analysis of a multi-server queue in the Halfin–Whitt regime. Preprint.
  • [6] Halfin, S. and Whitt, W. (1981). Heavy-traffic limits for queues with many exponential servers. Oper. Res. 29 567–588.
  • [7] Harrison, J. M. and Reiman, M. I. (1981). Reflected Brownian motion on an orthant. Ann. Probab. 9 302–308.
  • [8] Harrison, J. M. and Williams, R. J. (1987). Multidimensional reflected Brownian motions having exponential stationary distributions. Ann. Probab. 15 115–137.
  • [9] Jacod, J. and Shiryaev, A. N. (2003). Limit Theorems for Stochastic Processes, 2nd ed. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 288. Springer, Berlin.
  • [10] Jelenković, P., Mandelbaum, A. and Momčilović, P. (2004). Heavy traffic limits for queues with many deterministic servers. Queueing Syst. 47 53–69.
  • [11] Karlin, S. and Taylor, H. M. (1975). A First Course in Stochastic Processes, 2nd ed. Academic Press, New York.
  • [12] Kaspi, H. and Ramanan, K. (2006). Fluid limits of many-server queue. Preprint.
  • [13] Kiefer, J. and Wolfowitz, J. (1955). On the theory of queues with many servers. Trans. Amer. Math. Soc. 78 1–18.
  • [14] Krichagina, E. V. and Puhalskii, A. A. (1997). A heavy-traffic analysis of a closed queueing system with a GI/∞ service center. Queueing Syst. 25 235–280.
  • [15] Liptser, R. S. and Shiryaev, A. (1989). Theory of Martingales. Kluwer, Dordrecht.
  • [16] Mandelbaum, A. and Momcilovic, P. (2005). Queues with many servers: The virtual waiting-time process in the QED regime. Math. Oper. Res. To appear.
  • [17] 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.
  • [18] Reed, J. E. The G/GI/N queue in the Halfin–Whitt regime II: Idle time system equations. To appear.
  • [19] Ross, S. M. (1996). Stochastic Processes, 2nd ed. Wiley, New York.
  • [20] Whitt, W. (2002). Stochastic-Process Limits. Springer, New York.
  • [21] Whitt, W. (2005). Heavy-traffic limits for the G/H2*/n/m queue. Math. Oper. Res. 30 1–27.