Annals of Applied Probability

Law of large numbers limits for many-server queues

Haya Kaspi and Kavita Ramanan

Full-text: Open access


This work considers a many-server queueing system in which customers with independent and identically distributed service times, chosen from a general distribution, enter service in the order of arrival. The dynamics of the system are represented in terms of a process that describes the total number of customers in the system, as well as a measure-valued process that keeps track of the ages of customers in service. Under mild assumptions on the service time distribution, as the number of servers goes to infinity, a law of large numbers (or fluid) limit is established for this pair of processes. The limit is characterized as the unique solution to a coupled pair of integral equations which admits a fairly explicit representation. As a corollary, the fluid limits of several other functionals of interest, such as the waiting time, are also obtained. Furthermore, when the arrival process is time-homogeneous, the measure-valued component of the fluid limit is shown to converge to its equilibrium. Along the way, some results of independent interest are obtained, including a continuous mapping result and a maximality property of the fluid limit. A motivation for studying these systems is that they arise as models of computer data systems and call centers.

Article information

Ann. Appl. Probab., Volume 21, Number 1 (2011), 33-114.

First available in Project Euclid: 17 December 2010

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: 60H99: None of the above, but in this section 35D99: None of the above, but in this section

Multi-server queues GI∕G∕N queue fluid limits mean-field limits strong law of large numbers measure-valued processes call centers


Kaspi, Haya; Ramanan, Kavita. Law of large numbers limits for many-server queues. Ann. Appl. Probab. 21 (2011), no. 1, 33--114. doi:10.1214/09-AAP662.

Export citation


  • [1] Asmussen, S. (2003). Applied Probability and Queues, 2nd ed. Applications of Mathematics (New York) 51. Springer, New York.
  • [2] 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. J. Amer. Statist. Assoc. 100 36–50.
  • [3] Decreusefond, L. and Moyal, P. (2008). Fluid limit of a heavily loaded EDF queue with impatient customers. Markov Process. Related Fields 14 131–158.
  • [4] Dupuis, P. and Ellis, R. S. (1997). A Weak Convergence Approach to the Theory of Large Deviations. Wiley, New York.
  • [5] Ethier, S. N. and Kurtz, T. G. (1986). Markov Processes: Characterization and Convergence. Wiley, New York.
  • [6] Gromoll, H. C., Puha, A. L. and Williams, R. J. (2002). The fluid limit of a heavily loaded processor sharing queue. Ann. Appl. Probab. 12 797–859.
  • [7] Gromoll, H. C., Robert, P. and Zwart, B. (2008). Fluid limits for processor-sharing queues with impatience. Math. Oper. Res. 33 375–402.
  • [8] Halfin, S. and Whitt, W. (1981). Heavy-traffic limits for queues with many exponential servers. Oper. Res. 29 567–588.
  • [9] Harrison, J. M. (1985). Brownian Motion and Stochastic Flow Systems. Wiley, New York.
  • [10] Jacod, J. and Shiryaev, A. N. (1987). Limit Theorems for Stochastic Processes. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 288. Springer, Berlin.
  • [11] Jakubowski, A. (1986). On the Skorokhod topology. Ann. Inst. H. Poincaré Probab. Statist. 22 263–285.
  • [12] Kang, W. and Ramanan, K. (2010). Fluid limits of many-server queues with reneging. Ann. Appl. Probab. 20 2204–2260.
  • [13] Kang, W. and Ramanan, K. (2011). Asymptotic approximations for the stationary distributions of many-server queues. Ann. Appl. Probab. To appear.
  • [14] Kaspi, H. and Ramanan, K. (2010). SPDE limits of many server queues. Preprint.
  • [15] Mandelbaum, A., Massey, W. A. and Reiman, M. I. (1998). Strong approximations for Markovian service networks. Queueing Syst. 30 149–201.
  • [16] Parthasarathy, K. R. (1967). Probability Measures on Metric Spaces. Probability and Mathematical Statistics 3. Academic Press, New York.
  • [17] Pang, G., Talreja, R. and Whitt, W. (2007). Martingale proofs of many-server heavy-traffic limits for Markovian queues. Probab. Surv. 4 193–267.
  • [18] Ramanan, K. and Reiman, M. I. (2003). Fluid and heavy traffic diffusion limits for a generalized processor sharing model. Ann. Appl. Probab. 13 100–139.
  • [19] Reed, J. (2009). The GGIN queue in the Halfin–Whitt regime. Ann. Appl. Probab. 19 2211–2269.
  • [20] Rudin, W. (1974). Real and Complex Analysis, 2nd ed. McGraw-Hill, New York.
  • [21] Shilov, G. E. (1968). Generalized Functions and Partial Differential Equations. Gordon and Breach, New York.
  • [22] Whitt, W. (2006). Fluid models for multiserver queues with abandonments. Oper. Res. 54 37–54.