Hostname: page-component-586b7cd67f-t7fkt Total loading time: 0 Render date: 2024-11-22T21:55:21.561Z Has data issue: false hasContentIssue false
  • English
  • Français

Vacation queues with Markov schedules

Published online by Cambridge University Press:  01 July 2016

M. A. Wortman  and
Ralph L. Disney
M. A. Wortman*
Affiliation:
Texas A & M University
Ralph L. Disney*
Affiliation:
Texas A & M University
*
∗Postal address for both authors: Industrial Engineering, College of Engineering, Texas A & M University, College Station, TX 77843-3131, USA.
∗Postal address for both authors: Industrial Engineering, College of Engineering, Texas A & M University, College Station, TX 77843-3131, USA.
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper identifies a probability structure for queues that belong to the class of vacation systems operating according to Markov schedules, admitting a wide variety of server scheduling disciplines including most disciplines associated with those M/GI/1/L vacation systems reported in the literature. The conditions that define these schedules are identified, and it is shown that when these conditions are satisfied, queueing behavior is governed by an underlying Markov renewal/semi-regenerative structure. A simple example is examined (the M/GI/1 vacation system with limited batch service) to demonstrate the usefulness and generality of the underlying probability structure.

Keywords

MARKOV RENEWAL PROCESSESSEMI-REGENERATIVE PROCESSES
Type
Research Article
Information
Advances in Applied Probability , Volume 22 , Issue 3 , September 1990 , pp. 730 - 748
Copyright
Copyright © Applied Probability Trust 1990 

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.)

Footnotes

This material is based upon work supported in part by the National Science Foundation under Grant No. ECS 8501217.

References

Çinlar, E. (1969) Markov renewal theory. Adv. Appl. Prob. 1, 123–87.Google Scholar
Çlinar, E. (1975) Introduction to Stochastic Processes. Prentice-Hall, Englewood Cliffs, N.J. Google Scholar
Cooper, R. B. (1981) Introduction to Queueing Theory, 2nd edn. North-Holland, New York.Google Scholar
Disney, R. L. and Kiessler, P. C. (1987) Traffic Processes in Queueing Networks: A Markov Renewal Approach. The Johns Hopkins University Press, Baltimore, MD.Google Scholar
Doshi, B. T. (1986) Queueing systems with vacations—a survey. Queueing Systems 1, 29–66.Google Scholar
Fuhrmann, S. W. and Cooper, R. B. (1985) Stochastic decompositions in the M/G/1 queue with generalized vacations. Operat. Res. 33, 1117–29.CrossRefGoogle Scholar
Harris, C. M. and Marchal, W. G. (1988) State dependence in M/G/1 server-vacation models. Operat. Res. 36, 560–65.Google Scholar
Kielson, J. and Servi, L. D. (1988) A distributional form of Little's law. Operat. Res. Letters 7, 223–7.Google Scholar
Kleinrock, L. (1975). Queueing Systems, Vol. 1: Theory. Wiley, New York.Google Scholar
Kohlas, J. (1982) Stochastic Models in Operations Research. Cambridge University Press, New York.Google Scholar
Lucantoni, D. M., Meier-Hellstern, K. S. and Neuts, M. F. (1990) A single server queue with server vacations and a class of non-renewal arrival processes. Adv. Appl. Prob. 22, 676–705.Google Scholar
Shanthikumar, J. G. (1988) On decomposition in M/G/1 type queues with generalized server vacations. Operat. Res. 36, 566–69.CrossRefGoogle Scholar
Takagi, H. (1987) Queueing analysis of vacation models. IBM TRL Research Report.Google Scholar
Wolff, R. W. (1982) Poisson arrivals see time averages. Operat. Res. 30, 223–231.Google Scholar
Wortman, M. A. (1988) Vacation Queues with Markov Schedules. Ph.D. Dissertation, Virginia Polytechnic Institute and State University.Google Scholar