Hostname: page-component-76fb5796d-wq484 Total loading time: 0 Render date: 2024-04-25T10:24:14.462Z Has data issue: false hasContentIssue false
  • English
  • Français

On the Optimality of Trunk Reservation in Overflow Processes

Published online by Cambridge University Press:  27 July 2009

Viên Nguyen
Viên Nguyen
Affiliation:
Graduate School of Business Stanford UniversityStanford, California94305
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper discusses an optimal dynamic policy for a queueing system with M servers, no waiting room, and two types of customers. Customer types differ with respect to the reward that is paid on commencement of service, but service times are exponentially distributed with the same mean for both types of customers. The arrival stream of one customer type is generated by a Poisson process, and the other customer type arrives according to the overflow process of an M/M/m/m queue. The objective is to determine a policy for admitting customers to maximize the expected long-run average reward.

By posing the problem in the framework of Markov decision processes and exploiting properties of submodular functions, the optimal policy is shown to be a “generalized trunk reservation policy”; in other words, the optimal policy accepts higher-paying customers whenever possible and accepts lower-paying customers only if fewer than c1 servers are busy, where i is the number of busy servers in the overflow queue. Computational issues are also discussed. More specifically, approximations of the overflow process by an interrupted Poisson process and a Poisson process are investigated.

Type
Articles
Information
Probability in the Engineering and Informational Sciences , Volume 5 , Issue 3 , July 1991 , pp. 369 - 390
Copyright
Copyright © Cambridge University Press 1991

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

REFERENCES

Derman, C. (1970). Finite state Markovian decision. New York: Academic Press.Google Scholar
Foschini, G. & Gopinath, B. (1981). Optimum allocation of servers to two types of competing customers. IEEE Transactions on Communication 29: 1051–1055.CrossRefGoogle Scholar
Howard, R. (1960). Dynamic programming and Markov processes. New York: Wiley.Google Scholar
Kuczura, , Anatol., (1973). The interrupted Poisson process as an overflow process. Bell System Technical Journal 52: 437–448.CrossRefGoogle Scholar
Lippman, S. (1975). Applying a new device in the optimization of exponential queueing systems. Operations Research 23: 687–710.CrossRefGoogle Scholar
Odoni, A. (1969). On finding the maximal gain for Markov decision processes. Operations Research 17: 857–860.CrossRefGoogle Scholar
Ott, T. & Krishnan, K. (1985). State dependent routing of telephone traffic and the use of separable routing schemes. Proceedings of the 11th International Teletraffic Congress, Kyoto, Japan.Google Scholar
Ross, S. (1983). Introduction to stochastic dynamical programming. New York: Academic Press.Google Scholar
Schweitzer, P. (1971). Iterative solution of the functional equation of undiscounted Markov renewal programming. Journal of Mathematical Analysis and Applications 34: 495–501.CrossRefGoogle Scholar
Serfozo, R. (1981). Optimal control of random walks, birth and death processes, and queues. Advances in Applied Probability 13: 61–83.CrossRefGoogle Scholar
Stidham, S. Jr, (1985). Optimal control of admission to a queueing system. IEEE Transactions on Automated Control AC-30: 705–713.CrossRefGoogle Scholar
Tijms, H. (1986). Stochastic modelling and analysis: A computational approach. New York: Wiley.Google Scholar
Tijms, H. & Eikenboom, A. (1986). A simple technique in Markovian control with applications to resource allocation in communication networks. Operations Research Letters 5: 25–32.CrossRefGoogle Scholar
Van Nunen, J. & Puterman, M. (1983). Computing optimal control limits for G1/M/S queueing systems with controlled arrivals. Management Science 29: 725–734.CrossRefGoogle Scholar
White, D.J. (1963). Dynamic programming, Markov chains, and the method of successive approximations. Journal of Mathematical Analysis and Applications 6: 373–376.CrossRefGoogle Scholar