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Analysis of a MAP/PH/1 Queue with Discretionary Priority Based on Service Stages

Author

Listed:
  • Ning Zhao

    (Faculty of Science, Kunming University of Science and Technology, Kunming, Yunnan, P. R. China)

  • Zhaotong Lian

    (Faculty of Business Administration, University of Macau, Macau, P. R. China)

  • Kan Wu

    (School of Mechanical and Aerospace Engineering, Nanyang Technological University, Singapore)

Abstract

In this paper, we study a MAP/PH/1 queue with two classes of customers and discretionary priority. There are two stages of service for the low-priority customer. The server adopts the preemptive priority discipline at the first stage and adopts the nonpreemptive priority discipline at the second stage. Such a queuing system can be modeled into a quasi-birth-and-death (QBD) process. But there is no general solution for this QBD process since the generator matrix has a block structure with an infinite number of blocks and each block has infinite dimensions. We present an approach to derive the bound for the high-priority queue length. It guarantees that the probabilities of ignored states are within a given error bound, so that the system can be modeled into a QBD process where the block elements of the generator matrix have finite dimensions. The sojourn time distributions of both high and low priority customers are obtained. Some managerial insights are given after comparing the discretionary priority rule with the preemptive and nonpreemptive disciplines numerically.

Suggested Citation

  • Ning Zhao & Zhaotong Lian & Kan Wu, 2015. "Analysis of a MAP/PH/1 Queue with Discretionary Priority Based on Service Stages," Asia-Pacific Journal of Operational Research (APJOR), World Scientific Publishing Co. Pte. Ltd., vol. 32(06), pages 1-22, December.
    • Handle: RePEc:wsi:apjorx:v:32:y:2015:i:06:n:s0217595915500426
    DOI: 10.1142/S0217595915500426
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    References listed on IDEAS

    as
    1. Kim, Kilhwan & Chae, Kyung C., 2010. "Discrete-time queues with discretionary priorities," European Journal of Operational Research, Elsevier, vol. 200(2), pages 473-485, January.
    2. B. Kumar & A. Vijayakumar & D. Arivudainambi, 2002. "An M/G/1 Retrial Queueing System with Two-Phase Service and Preemptive Resume," Annals of Operations Research, Springer, vol. 113(1), pages 61-79, July.
    3. Morris A. Cohen & Paul R. Kleindorfer & Hau L. Lee, 1988. "Service Constrained (s, S) Inventory Systems with Priority Demand Classes and Lost Sales," Management Science, INFORMS, vol. 34(4), pages 482-499, April.
    4. Martin Paterok & Markus Ettl, 1994. "Sojourn Time and Waiting Time Distributions for M/GI/1 Queues with Preemption-Distance Priorities," Operations Research, INFORMS, vol. 42(6), pages 1146-1161, December.
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