Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-23T11:01:45.287Z Has data issue: false hasContentIssue false
  • English
  • Français

The Waiting-Time Distribution for the GI/G/1 Queue under the D-Policy

Published online by Cambridge University Press:  27 July 2009

Jingwen Li  and
Shun-Chen Niu
Jingwen Li
Affiliation:
Department of Decision SciencesNational University of Singapore, 10 Kent Ridge Crescent Singapore 0511
Shun-Chen Niu
Affiliation:
School of ManagementUniversity of Texas at Dallaas P. O. Box 830688 Richardson, Texas, 75083-0688
Get access Rights & Permissions [Opens in a new window]

Abstract

We study a generalization of the GI/G/l queue in which the server is turned off at the end of each busy period and is reactivated only when the sum of the service times of all waiting customers exceeds a given threshold of size D. Using the concept of a “randomly selected” arriving customer, we obtain as our main result a relation that expresses the waiting-time distribution of customers in this model in terms of characteristics associated with a corresponding standard GI/G/1 queue, obtained by setting D = 0. If either the arrival process is Poisson or the service times are exponentially distributed, then this representation of the waiting-time distribution can be specialized to yield explicit, transform-free formulas; we also derive, in both of these cases, the expected customer waiting times. Our results are potentially useful, for example, for studying optimization models in which the threshold D can be controlled.

Type
Articles
Information
Probability in the Engineering and Informational Sciences , Volume 6 , Issue 3 , July 1992 , pp. 287 - 308
Copyright
Copyright © Cambridge University Press 1992

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

Avi-Itzhak, B., Maxwell, W.L., & Miller, L.W. (1965). Queueing with alternating priorities. Operations Research 13: 306–318.CrossRefGoogle Scholar
Balachandran, K.R. (1971). Queue length dependent priority queues. Management Science 17: 463–471.Google Scholar
Balachandran, K.R. (1973). Control policies for a single server system. Management Science 19: 1013–1018.Google Scholar
Balachandran, K.R. & Tijms, H. (1975). On the D-policy for the M/G/1 queue. Management Science 21: 1073–1076.Google Scholar
Boxma, O.J. (1976). Note on a control problem of Balachandran and Tijms. Management Science 22: 916–917.CrossRefGoogle Scholar
Cooper, R.B. (1981). Introduction to queueing theory, 2nd ed.Amsterdam: Elsevier North- Holland. (First edition, 1972, Macmillan. Republished, 1990, by CEEPress, The George Washington University, Washington, DC.)Google Scholar
Doshi, B.T. (1985). A note on stochastic decomposition in a GI/G/1 queue with vacations or set-up times. Journal of Applied Probability 22: 419–428.Google Scholar
Doshi, B.T. (1990). Single-server queues with vacations. In Takagi, H. (ed.), Stochastic analysis of computer and communication systems. Amsterdam: Elsevier North-Holland.Google Scholar
Heyman, D.P. (1968). Optimal operating policies for M/G/1 queueing systems. Operations Research 16: 362–382.Google Scholar
Heyman, D.P. (1977). The T-policy for the M/G/1 queue. Management Science 23: 775–778.CrossRefGoogle Scholar
Li, J. (1989). Sample-average analyses of some generalizations of the M/G/1 queue. Doctoral dissertation, University of Texas at Dallas, Richardson.Google Scholar
Neuts, M.F. (1986). Generalizations of the Pollaczek-Khinchin integral equations in the theory of queues. Advances in Applied Probability 18: 952–990.CrossRefGoogle Scholar
Neuts, M.F. (1989). Structured stochastic matrices of M/C/1 type and their applications. New York: Marcel Dekker.Google Scholar
Niu, S-C. (1988). Representing workloads in GI/G/1 queues through the preemptive-resume LIFO queue discipline. Queueing Systems 3: 157–178.CrossRefGoogle Scholar
Niu, S-C. & Cooper, R.B. (1989). Duality and other results forM/G/l and GI/M/1 queues, via a new ballot theorem. Mathematics of Operations Research 14: 281–293.Google Scholar
Niu, S-C. & Cooper, R.B. (1989). Transform-free analysis of M/G/1/K and related queues. Mathematics of Operations Research (to appear, 1993).Google Scholar
Ross, S.M. (1983). Stochastic processes. New York: Wiley.Google Scholar
Rubin, I. & Zhang, Z. (1987). Switch on policies for communications and queueing systems. In Proceedings of the Third International Conference on Data Communication. Amsterdam: Elsevier North-Holland, pp. 329–339.Google Scholar
Shanthikumar, J.G. (1981). Optimal control of an M/G/1 priority queue via N-control. American Journal of Mathematical Management Science 1: 191–212.Google Scholar
Welch, P.D. (1964). On a generalized M/G/1 queueing process in which the first customer of each busy period receives exceptional service. Operations Research 12: 736–752.Google Scholar
Yadin, M. & Naor, P. (1963). Queueing systems with a removable service station. Operational Research Quarterly 14: 393–405.Google Scholar