Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-26T07:05:58.158Z Has data issue: false hasContentIssue false
  • English
  • Français

An M/G/l Vacation Model with Two Service Modes

Published online by Cambridge University Press:  27 July 2009

Shoichi Nishimura  and
Yong Jiang
Shoichi Nishimura
Affiliation:
Department of Applied Mathematics, Faculty of Science, Science University of Tokyo, 1-3, Kagurazawa, Shinjuku-ku Tokyo, 162, Japan
Yong Jiang
Affiliation:
Faculty of Economics, Tokyo Metropolitan University, 1-1, Minami-Osawa, Hachioji-shi Tokyo, 192–03, Japan
Get access Rights & Permissions [Opens in a new window]

Abstract

Consider an M/G/1 type queueing system with two service modes: regular speed and high speed. The service rule is characterized by two switch-over levels nR and nH, where nR and nH are given integers with 0 ≤ nH < nR. The server switches from regular speed mode to high speed mode when the number of customers present at a service completion epoch is equal to or larger than nR and switches from high speed mode to regular speed mode when the number of customers present decreases to nH. A key feature of the model is that the server takes a vacation for setup operations before a new service mode is available. This paper derives for the general model an expression for the generating function of the equilibrium queue-length distribution in terms of the switch-over levels. Two unknown parameters appear in the generating functions. Using a recursive method, we solve these unknown parameters and obtain a computationally tractable algorithm for the steady-state probabilities.

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 9 , Issue 3 , July 1995 , pp. 355 - 374
Copyright
Copyright © Cambridge University Press 1995

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.Altman, E. & Nain, P. (1993). Optimal control of the M/G/l queueing with repeated vacations of the server. IEEE Transactions on Automatic Control 38: 1766–1775.CrossRefGoogle Scholar
2.Doshi, B.T. (1986). Queueing system with vacation —A survey. Queueing System 1: 29–66.CrossRefGoogle Scholar
3.Federgruen, A. & Tijms, H.C. (1980). Computation of the stationary distribution of the queue size in an M/G/l queueing system with variable server rate. Journal of Applied Probability 17: 515–522.CrossRefGoogle Scholar
4.Fuhrmann, S.W. (1984). A note on the M/G/l queue with server vacations. Operations Research 32: 1368–1373.CrossRefGoogle Scholar
5.Fuhrmann, S.W. & Copper, R.B. (1985). Stochastic decompositions in the M/G/l queue with generalized vacations. Operations Research 33: 1117–1129.CrossRefGoogle Scholar
6.Kella, O. (1989). The threshold policies in the M/G/l with server vacations. Naval Research Logistics 36: 111–123.3.0.CO;2-3>CrossRefGoogle Scholar
7.Kella, O. (1990). Optimal control of the vacation scheme in an M/G/l queue. Operations Research 38: 724–728.CrossRefGoogle Scholar
8.Nishigaya, T., Mukumoto, K. & Fukuda, A. (1991). M/G/l system with set–up time for server replacement. Transactions of Institute of Electronics, Information and Communication Engineers J74–A–10: 1586–1593.Google Scholar
9.Takagi, H. (1991).Queueing analysis. Vol. 1: Vacation and priority systems, Pt. I. Amsterdam: Elsevier Science.Google Scholar
10.Teghem, J. Jr., (1986). Control of the service process in a queueing system. European Journal of Operational Research 23: 141–158.CrossRefGoogle Scholar
11.Yamada, K. & Nishimura, S. (1994). A queueing system with a setup time for switching of the service distribution. Journal of the Operations Research Society of Japan 37: 271–286.CrossRefGoogle Scholar