Annals of Applied Probability

Fluid limits of many-server queues with reneging

Weining Kang and Kavita Ramanan

Full-text: Open access


This work considers a many-server queueing system in which impatient customers with i.i.d., generally distributed service times and i.i.d., generally distributed patience times enter service in the order of arrival and abandon the queue if the time before possible entry into service exceeds the patience time. The dynamics of the system is represented in terms of a pair of measure-valued processes, one that keeps track of the waiting times of the customers in queue and the other that keeps track of the amounts of time each customer being served has been in service. Under mild assumptions, essentially only requiring that the service and reneging distributions have densities, as both the arrival rate and the number of servers go to infinity, a law of large numbers (or fluid) limit is established for this pair of processes. The limit is shown to be the unique solution of a coupled pair of deterministic integral equations that admits an explicit representation. In addition, a fluid limit for the virtual waiting time process is also established. This paper extends previous work by Kaspi and Ramanan, which analyzed the model in the absence of reneging. A strong motivation for understanding performance in the presence of reneging arises from models of call centers.

Article information

Ann. Appl. Probab., Volume 20, Number 6 (2010), 2204-2260.

First available in Project Euclid: 19 October 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

Many-server queues GI/G/N queue fluid limits reneging abandonment strong law of large numbers measure-valued processes call centers


Kang, Weining; Ramanan, Kavita. Fluid limits of many-server queues with reneging. Ann. Appl. Probab. 20 (2010), no. 6, 2204--2260. doi:10.1214/10-AAP683.

Export citation


  • [1] Baccelli, F. and Hebuterne, G. (1981). On queues with impatient customers. In Performance ’81 (E. Gelenbe, ed.) 159–179. North-Holland, Amsterdam.
  • [2] Bassamboo, A., Harrison, J. M. and Zeevi, A. (2005). Dynamic routing and admission control in high-volume service systems: Asymptotic analysis via multi-scale fluid limits. Queueing Syst. 51 249–285.
  • [3] Boxma, O. J. and de Waal, P. R. (1994). Multiserver queues with impatient customers. In Proceedings of ITC 14 743–756. Elsevier, Amsterdam.
  • [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. J. Amer. Statist. Assoc. 100 36–50.
  • [5] Dupuis, P. and Ellis, R. S. (1997). A Weak Convergence Approach to the Theory of Large Deviations. Wiley, New York.
  • [6] Ethier, S. N. and Kurtz, T. G. (1986). Markov Processes: Characterization and Convergence. Wiley, New York.
  • [7] Gans, N., Koole, G. and Mandelbaum, A. (2003). Telephone call centers: Tutorial, review and research prospects. Manufacturing Service Oper. Management 5 79–141.
  • [8] Garnett, O., Mandelbaum, A. and Reiman, M. I. (2002). Designing a call center with impatient customers. Manufacturing Service Oper. Management 4 208–227.
  • [9] Harrison, J. M. (2005). A method for staffing large call centers based on stochastic fluid models. Manufacturing Service Oper. Management 7 20–36.
  • [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] Jacobsen, M. (2006). Point Process Theory and Applications: Markov Point and Piecewise Deterministic Processes. Birkhäuser, Boston, MA.
  • [12] Kallenberg, O. (1975). Random Measures. Akademie Verlag, Berlin.
  • [13] Kang, W. N. and Ramanan, K. (2010). Asymptotic approximations for the stationary distributions of many-server queues. Preprint. Available at
  • [14] Kaspi, H. and Ramanan, K. (2010). Law of large numbers limits for many-server queues. Ann. Appl. Probab. To appear.
  • [15] Kaspi, H. and Ramanan, K. (2010). SPDE limits for many-server queues. Preprint.
  • [16] Mandelbaum, A., Massey, W. A. and Reiman, M. I. (1998). Strong approximations for Markovian service networks. Queueing Syst. 30 149–201.
  • [17] Mandelbaum, A. and Momcilovic, P. (2010). Queues with many servers and impatient customers. Preprint.
  • [18] Zeltyn, S. and Mandelbaum, A. (2005). Call centers with impatient customers: Many-server asymptotics of the M/M/n+G queue. Queueing Syst. 51 361–402.
  • [19] Mandelbaum, A. and Zeltyn, S. (2009). Staffing many-server queues with impatient customers: Constraint satisfaction in call centers. Oper. Res. 57 1189–1205.
  • [20] Parthasarathy, K. R. (1967). Probability Measures on Metric Spaces. Probability and Mathematical Statistics 3. Academic Press, New York.
  • [21] 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.
  • [22] Whitt, W. (2006). Fluid models for multiserver queues with abandonments. Oper. Res. 54 37–54.
  • [23] Whitt, W. (2002). Stochastic-process Limits: An Introduction to Stochastic-Process Limits and Their Application to Queues. Springer, New York.
  • [24] Zhang, J. (2010). Fluid models of many-server queues with abandonment. Preprint.