The Annals of Applied Probability

Scheduling control for queueing systems with many servers: Asymptotic optimality in heavy traffic

Rami Atar

Full-text: Open access


A multiclass queueing system is considered, with heterogeneous service stations, each consisting of many servers with identical capabilities. An optimal control problem is formulated, where the control corresponds to scheduling and routing, and the cost is a cumulative discounted functional of the system’s state. We examine two versions of the problem: “nonpreemptive,” where service is uninterruptible, and “preemptive,” where service to a customer can be interrupted and then resumed, possibly at a different station. We study the problem in the asymptotic heavy traffic regime proposed by Halfin and Whitt, in which the arrival rates and the number of servers at each station grow without bound. The two versions of the problem are not, in general, asymptotically equivalent in this regime, with the preemptive version showing an asymptotic behavior that is, in a sense, much simpler. Under appropriate assumptions on the structure of the system we show: (i) The value function for the preemptive problem converges to V, the value of a related diffusion control problem. (ii) The two versions of the problem are asymptotically equivalent, and in particular nonpreemptive policies can be constructed that asymptotically achieve the value V. The construction of these policies is based on a Hamilton–Jacobi–Bellman equation associated with V.

Article information

Ann. Appl. Probab., Volume 15, Number 4 (2005), 2606-2650.

First available in Project Euclid: 7 December 2005

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60K25: Queueing theory [See also 68M20, 90B22] 68M20: Performance evaluation; queueing; scheduling [See also 60K25, 90Bxx] 90B22: Queues and service [See also 60K25, 68M20] 90B36: Scheduling theory, stochastic [See also 68M20] 60F05: Central limit and other weak theorems 49L20: Dynamic programming method

Multiclass queueing systems heavy traffic scheduling and routing asymptotically optimal controls Hamilton–Jacobi–Bellman equation


Atar, Rami. Scheduling control for queueing systems with many servers: Asymptotic optimality in heavy traffic. Ann. Appl. Probab. 15 (2005), no. 4, 2606--2650. doi:10.1214/105051605000000601.

Export citation


  • Ata, B. and Kumar, S. (2005). Heavy traffic analysis of open processing networks with complete resource pooling: Asymptotic optimality of discrete review policies. Ann. Appl. Probab. 15 331–391.
  • Atar, R. (2005). A diffusion model of scheduling control in queueing systems with many servers. Ann. Appl. Probab. 15 820–852.
  • Atar, R., Mandelbaum, A. and Reiman, M. (2004). Scheduling a multi-class queue with many exponential servers: Asymptotic optimality in heavy-traffic. Ann. Appl. Probab. 14 1084–1134.
  • Bell, S. L. and Williams, R. J. (2001). Dynamic scheduling of a system with two parallel servers in heavy traffic with resource pooling: Asymptotic optimality of a threshold policy. Ann. Appl. Probab. 11 608–649.
  • Billingsley, P. (1999). Convergence of Probability Measures, 2nd ed. Wiley, New York.
  • Ethier, S. N. and Kurtz, T. G. (1986). Markov Processes. Characterization and Convergence. Wiley, New York.
  • Gans, N., Koole, G. and Mandelbaum, A. (2003). Telephone call centers: Tutorial, review and research prospects. Manufacturing and Service Operations Management 5 79–141.
  • Halfin, S. and Whitt, W. (1981). Heavy-traffic limits for queues with many exponential servers. Oper. Res. 29 567–588.
  • Harrison, J. M. (2000). Brownian models of open processing networks: Canonical representation of workload. Ann. Appl. Probab. 10 75–103.
  • Harrison, J. M. and López, M. J. (1999). Heavy traffic resource pooling in parallel-server systems. Queueing Systems Theory Appl. 33 339–368.
  • Karatzas, I. and Shreve, S. E. (1991). Brownian Motion and Stochastic Calculus, 2nd ed. Springer, New York.
  • Mandelbaum, A. and Reiman, M. I. Private communication.
  • Mandelbaum, A. and Stolyar, A. L. (2004). Scheduling flexible servers with convex delay costs: Heavy traffic optimality of the generalized $c\mu$ rule.\goodbreak Oper. Res. 52 836–855.
  • Williams, R. J. (2000). On dynamic scheduling of a parallel server system with complete resource pooling. In Analysis of Communication Networks: Call Centres, Traffic and Performance. Fields Inst. Commun. 28 49–71. Amer. Math. Soc., Providence, RI.