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Simple Bounds for Finite Single-Server Exponential Tandem Queues

Author

Listed:
  • Nico M. Van Dijk

    (Vrije Universiteit, Amsterdam, The Netherlands)

  • Bernard F. Lamond

    (University of Arizona, Tucson, Arizona)

Abstract

Tandem queue configurations naturally arise in multistage stochastic systems such as assembly lines in manufacturing or multiphase transmissions in telecommunications. As finite capacity or storage constraints (buffers) are usually involved, the celebrated closed product form expression is generally not applicable. In this paper, a new bounding methodology for nonproduct form systems is applied to finite single-server exponential tandem queues. The methodology is based on modifying the original system into product form systems that provide bounds for some performance measure of interest. The product form modifications given for this finite tandem queue propose a computationally attractive and intuitively obvious lower and upper bound for the call congestion and throughput. Numerical results indicate that the bounds are reasonable indicators of the order of magnitude. This can be useful for quick engineering purposes as will be illustrated by an optimal design example. A formal proof of the bounds is given. This proof extends standard techniques for comparing stochastic systems and is of interest in itself.

Suggested Citation

  • Nico M. Van Dijk & Bernard F. Lamond, 1988. "Simple Bounds for Finite Single-Server Exponential Tandem Queues," Operations Research, INFORMS, vol. 36(3), pages 470-477, June.
    • Handle: RePEc:inm:oropre:v:36:y:1988:i:3:p:470-477
    DOI: 10.1287/opre.36.3.470
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    Citations

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    Cited by:

    1. Kozlowski, Dawid & Worthington, Dave, 2015. "Use of queue modelling in the analysis of elective patient treatment governed by a maximum waiting time policy," European Journal of Operational Research, Elsevier, vol. 244(1), pages 331-338.
    2. Waal, P.R. & Dijk, N.M. van, 1988. "Monotonicity of performance measures in a processor sharing queue," Serie Research Memoranda 0051, VU University Amsterdam, Faculty of Economics, Business Administration and Econometrics.
    3. Ivo Adan & Gerard Hooghiemstra, 1998. "The M/M/c with critical jobs," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 47(3), pages 341-353, October.
    4. Dijk, N.M. van, 1989. "Simple performance estimates and error bounds for slotted ALOHA loss systems," Serie Research Memoranda 0013, VU University Amsterdam, Faculty of Economics, Business Administration and Econometrics.
    5. Dijk, N.M. van, 1989. "Sensitivity error bounds for non-exponental stochastic networks," Serie Research Memoranda 0093, VU University Amsterdam, Faculty of Economics, Business Administration and Econometrics.
    6. Dijk, N.M. van, 1989. "The importance of bias-terms for error bounds and comparison results," Serie Research Memoranda 0036, VU University Amsterdam, Faculty of Economics, Business Administration and Econometrics.
    7. Dijk, N.M. van & Korezlioglu, H., 1991. "On product form approximations for communication networks with losses : error bounds," Serie Research Memoranda 0011, VU University Amsterdam, Faculty of Economics, Business Administration and Econometrics.
    8. Nakade, Koichi, 2000. "New bounds for expected cycle times in tandem queues with blocking," European Journal of Operational Research, Elsevier, vol. 125(1), pages 84-92, August.

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