Hostname: page-component-586b7cd67f-t8hqh Total loading time: 0 Render date: 2024-11-25T20:08:13.151Z Has data issue: false hasContentIssue false
  • English
  • Français

THE N-NETWORK MODEL WITH UPGRADES

Published online by Cambridge University Press:  18 March 2010

Douglas G. Down  and
Mark E. Lewis
Douglas G. Down
Affiliation:
Department of Computing and Software, McMaster University, Hamilton, ON L8S 4L7, CanadaE-mail: [email protected]
Mark E. Lewis
Affiliation:
School of Operations Research and Information Engineering, Cornell University, 226 Rhodes Hall, Ithaca, NY 14853 E-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

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.

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 24 , Issue 2 , April 2010 , pp. 171 - 200
Copyright
Copyright © Cambridge University Press 2010

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Ahn, H., Duenyas, I. & Zhang, R. (2004). Optimal control of a flexible server. Advances In Applied Probability 36(1): 139–170.CrossRefGoogle Scholar
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: 665–668.CrossRefGoogle Scholar
3.Armony, M. & Maglaras, C. (2004). Contact centers with a call-back option and real-time delay information. Operations Research 52(4): 527–545.CrossRefGoogle Scholar
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): 271–292.CrossRefGoogle Scholar
5.Armony, M., Shimkin, N. & Whitt, W. (2007). The impact of delay announcements in many-server queues with abandonment. Preprint.Google Scholar
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): 608–649.CrossRefGoogle Scholar
7.Brémaud, P. (1981). Point processes and queues. Martingale dynamics. New York: Springer-Verlag.CrossRefGoogle Scholar
8.Buyukkoc, C., Varaiya, P. & Walrand, J. (1985). The cμ-rule revisited. Advances in Applied Probability 17(1): 237–238.CrossRefGoogle Scholar
9.Dai, J. (1999). Stability of fluid and stochastic processing networks. Princeton, NJ: Centre for Mathematical Physics and Stochastics.Google Scholar
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: 1889–1904.CrossRefGoogle Scholar
11.Dai, J. & Tezcan, T. (2008). Optimal control of parallel server systems with many servers in heavy traffic. Queueing Systems 59: 95–134.CrossRefGoogle Scholar
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.Google Scholar
13.Gans, N., Koole, G. & Mandelbaum, A. (2003). Telephone call centers: Tutorial, review and research prospects. Manufacturing and Service Operations Management 5(2): 79–141.CrossRefGoogle Scholar
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): 237–253.CrossRefGoogle Scholar
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): 822–848.CrossRefGoogle Scholar
16.Lippman, S. (1975). Applying a new device in the optimization of exponential queueing system. Operations Research 23(4): 687–710.CrossRefGoogle Scholar
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): 836–855.CrossRefGoogle Scholar
18.Sennott, L.I. (1999). Stochastic dynamic programming and the control of queueing systems. Wiley Series in Probability and Statistics. New York: Wiley.Google Scholar
19.Stanford, D.A. & Grassman, W.K. (1993). The bilingual server system: A queuing model featuring fully and partially qualified servers. INFOR 31(4): 261–278.Google Scholar
20.Ward, A.R. & Glynn, P.W. (2003). A diffusion approximation for a Markovian queue with reneging. Queueing Systems 43(1): 103–128.CrossRefGoogle Scholar
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): 371–400.CrossRefGoogle Scholar