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An Invariance Relation and a Unified Method to Derive Stationary Queue-Length Distributions

Author

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  • Nam K. Kim

    (Chonnam National University, 300 Yongbong-dong, Buk-gu, Gwangju 500-757, Korea)

  • Kyung C. Chae

    (Department of Industrial Engineering, Korea Advanced Institute of Science and Technology, Daejon 305-701, Korea)

  • Mohan L. Chaudhry

    (Department of Mathematics and Computer Science, Royal Military College of Canada, P.O. Box 17000, STN Forces, Kingston, Ontario, Canada K7K 7B4)

Abstract

For a broad class of discrete- and continuous-time queueing systems, we show that the stationary number of customers in system (queue plus servers) is the sum of two independent random variables, one of which is the stationary number of customers in queue and the other is the number of customers that arrive during the time a customer spends in service. We call this relation an invariance relation in the sense that it does not change for a variety of single-sever queues (with batch arrivals and batch services) and some multiserver queues (with batch arrivals and deterministic service times) that satisfy a certain set of assumptions. Making use of this relation, we also present a simple method of deriving the probability generating functions (PGFs) of the stationary numbers in queue and in system, as well as some of their properties. This is illustrated by several examples, which show that new simple derivations of old results as well as new results can be obtained in a unified manner. Furthermore, we show that the invariance relation and the method we are presenting are easily generalized to analyze queues with batch Markovian arrival process (BMAP) arrivals. Most of the results are presented under the discrete-time setting. The corresponding continuous-time results, however, are covered as well because deriving the results for continuous-time queues runs exactly parallel to that for their discrete-time counterparts.

Suggested Citation

  • Nam K. Kim & Kyung C. Chae & Mohan L. Chaudhry, 2004. "An Invariance Relation and a Unified Method to Derive Stationary Queue-Length Distributions," Operations Research, INFORMS, vol. 52(5), pages 756-764, October.
    • Handle: RePEc:inm:oropre:v:52:y:2004:i:5:p:756-764
    DOI: 10.1287/opre.1040.0116
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    References listed on IDEAS

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    1. Dimitris Bertsimas & Daisuke Nakazato, 1995. "The Distributional Little's Law and Its Applications," Operations Research, INFORMS, vol. 43(2), pages 298-310, April.
    2. Warren B. Powell & Pierre Humblet, 1986. "The Bulk Service Queue with a General Control Strategy: Theoretical Analysis and a New Computational Procedure," Operations Research, INFORMS, vol. 34(2), pages 267-275, April.
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    Cited by:

    1. Yi, Xeung W. & Kim, Nam K. & Yoon, Bong K. & Chae, Kyung C., 2007. "Analysis of the queue-length distribution for the discrete-time batch-service Geo/Ga,Y/1/K queue," European Journal of Operational Research, Elsevier, vol. 181(2), pages 787-792, September.
    2. Dieter Claeys & Stijn De Vuyst, 2019. "Discrete-time modified number- and time-limited vacation queues," Queueing Systems: Theory and Applications, Springer, vol. 91(3), pages 297-318, April.
    3. Gopinath Panda & Veena Goswami, 2023. "Analysis of a Discrete-time Queue with Modified Batch Service Policy and Batch-size-dependent Service," Methodology and Computing in Applied Probability, Springer, vol. 25(1), pages 1-18, March.

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