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Waiting times for M/G/1 queues with service‐time or delay‐dependent server vacations

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  • P. H. Brill
  • C. M. Harris

Abstract

This article shows how to determine the stationary distribution of the virtual wait in M/G/1 queues with either one‐at‐a‐time or exhaustive server vacations, depending on either service times or accrued workload. For the first type of dependence, each vacation time is a function of the immediately preceding service time or of whether the server finds the system empty after returning from vacation. In this way, it is possible to model situations such as long service times followed by short vacations, and vice versa. For the second type of dependence, the vacation time assigned to an arrival to follow its service is a function of the level of virtual wait reached. By this device, we can model situations in which vacations may be shortened whenever virtual delays have gotten excessive. The method of analysis employs level‐crossing theory, and examples are given for various cases of service and vacation‐time distributions. A closing discussion relates the new model class to standard M/G/1 queues where the service time is a sum of variables having complex dependencies. © 1992 John Wiley & Sons, Inc.

Suggested Citation

  • P. H. Brill & C. M. Harris, 1992. "Waiting times for M/G/1 queues with service‐time or delay‐dependent server vacations," Naval Research Logistics (NRL), John Wiley & Sons, vol. 39(6), pages 775-787, October.
    • Handle: RePEc:wly:navres:v:39:y:1992:i:6:p:775-787
    DOI: 10.1002/1520-6750(199210)39:63.0.CO;2-K
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    References listed on IDEAS

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    1. Michel Scholl & Leonard Kleinrock, 1983. "On the M / G /1 Queue with Rest Periods and Certain Service-Independent Queueing Disciplines," Operations Research, INFORMS, vol. 31(4), pages 705-719, August.
    2. S. W. Fuhrmann & Robert B. Cooper, 1985. "Stochastic Decompositions in the M / G /1 Queue with Generalized Vacations," Operations Research, INFORMS, vol. 33(5), pages 1117-1129, October.
    3. P. H. Brill & M. J. M. Posner, 1977. "Level Crossings in Point Processes Applied to Queues: Single-Server Case," Operations Research, INFORMS, vol. 25(4), pages 662-674, August.
    4. P. H. Brill & M. J. M. Posner, 1981. "The System Point Method in Exponential Queues: A Level Crossing Approach," Mathematics of Operations Research, INFORMS, vol. 6(1), pages 31-49, February.
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