Skip to main content Accessibility help


  • Douglas G. Down (a1) and Mark E. Lewis (a2)


In this article we introduce a new method of mitigating the problem of long wait times for low-priority customers in a two-class queuing system. To this end, we allow class 1 customers to be upgraded to class 2 after they have been in queue for some time. We assume that there are ci servers at station i, i=1, 2. The servers at station 1 are flexible in the sense that they can work at either station, whereas the servers at station 2 are dedicated. Holding costs at rate hi are accrued per customer per unit time at station i, i=1, 2. This study yields several surprising results. First, we show that stability analysis requires a condition on the order of the service rates. This is unexpected since no such condition is required when the system does not have upgrades. This condition continues to play a role when control is considered. We provide structural results that include a c-μ rule when an inequality holds and a threshold policy when the inequality is reversed. A numerical study verifies that the optimal control policy significantly reduces holding costs over the policy that assigns the flexible server to station 1. At the same time, in most cases the optimal control policy reduces waiting times of both customer classes.



Hide All
1.Ahn, H., Duenyas, I. & Zhang, R. (2004). Optimal control of a flexible server. Advances In Applied Probability 36(1): 139170.
2.Aksin, Z., Armony, M. & Mehrotra, V. (2007). The modern call-center: A multi-disciplinary perspective on operations management research. Production and Operations Management 16: 665668.
3.Armony, M. & Maglaras, C. (2004). Contact centers with a call-back option and real-time delay information. Operations Research 52(4): 527545.
4.Armony, M. & Maglaras, C. (2004). On customer contact centers with a call-back option: Customer decisions, routing rules and system design. Operations Research 52(2): 271292.
5.Armony, M., Shimkin, N. & Whitt, W. (2007). The impact of delay announcements in many-server queues with abandonment. Preprint.
6.Bell, S. & Williams, R. (2001). Dynamic scheduling of a system with two parallel servers in heavy traffic with resource pooling: Asymptotic optimality of a threshold policy. Annals of Applied Probability 11(3): 608649.
7.Brémaud, P. (1981). Point processes and queues. Martingale dynamics. New York: Springer-Verlag.
8.Buyukkoc, C., Varaiya, P. & Walrand, J. (1985). The cμ-rule revisited. Advances in Applied Probability 17(1): 237238.
9.Dai, J. (1999). Stability of fluid and stochastic processing networks. Princeton, NJ: Centre for Mathematical Physics and Stochastics.
10.Dai, J. & Meyn, S. (1995). Stability and convergence of moments for multiclass queueing networks via fluid limit models. IEEE Transactions on Automatic Control 40: 18891904.
11.Dai, J. & Tezcan, T. (2008). Optimal control of parallel server systems with many servers in heavy traffic. Queueing Systems 59: 95134.
12.Dai, J. & Tezcan, T. (in press). Dynamic control of n-systems with many servers: Asymptotic optimality of a static priority policy in heavy traffic. Operations Research.
13.Gans, N., Koole, G. & Mandelbaum, A. (2003). Telephone call centers: Tutorial, review and research prospects. Manufacturing and Service Operations Management 5(2): 79141.
14.Gurvich, I. & Whitt, W. (2009). Scheduling flexible servers with convex delay costs in many-server service systems. Manufacturing and Service Operations Management 11(2): 237253.
15.Harrison, J. (1998). Heavy traffic analysis of a system with parallel servers: Asymptotic optimality of discrete-review policies. Annals of Applied Probability 8(3): 822848.
16.Lippman, S. (1975). Applying a new device in the optimization of exponential queueing system. Operations Research 23(4): 687710.
17.Mandelbaum, A. & Stolyar, A. (2004). Scheduling flexible servers with convex delay costs: Heavy-traffic optimality of the generalized cμ-rule. Operations Research 52(6): 836855.
18.Sennott, L.I. (1999). Stochastic dynamic programming and the control of queueing systems. Wiley Series in Probability and Statistics. New York: Wiley.
19.Stanford, D.A. & Grassman, W.K. (1993). The bilingual server system: A queuing model featuring fully and partially qualified servers. INFOR 31(4): 261278.
20.Ward, A.R. & Glynn, P.W. (2003). A diffusion approximation for a Markovian queue with reneging. Queueing Systems 43(1): 103128.
21.Ward, A.R. & Glynn, P.W. (2005). A diffusion approximation for a GI/GI/1 queue with balking or reneging. Queueing Systems 50(4): 371400.

Related content

Powered by UNSILO


  • Douglas G. Down (a1) and Mark E. Lewis (a2)


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed.