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Algorithms for Queueing Systems with Reneging and Priorities Modeled as Quasi-Birth-Death Processes

Author

Listed:
  • Amir Rastpour

    (Faculty of Business and Information Technology, Ontario Tech University, Oshawa, Ontario L1H 7K5, Canada)

  • Armann Ingolfsson

    (Alberta School of Business, University of Alberta, Edmonton, Alberta T6G 2R6, Canada)

  • Burhaneddin Sandıkçı

    (Department of Industrial Engineering, Istanbul Technical University, 34367 Istanbul, Turkey)

Abstract

We consider a Markovian multiserver queueing system with two customer classes, preemptive priorities, and reneging. We formulate this system as an infinite level-dependent quasi-birth-death process (LDQBD). We introduce an algorithm that endogenously truncates the level and calculates lower and upper bounds on stationary probabilities of this LDQBD such that the gap between the bounds can be any desired amount. Our algorithm can be applied to any LDQBD for which the rate matrices become elementwise nonincreasing above some level. This appears to be the first algorithm that provides bounds on stationary probabilities for an infinite-level LDQBD. To obtain these bounds, the algorithm first obtains lower and upper bounds on the rate matrices of the LDQBD using a novel method, which can be applied to any LDQBD. We then extend this algorithm to approximate performance measures of the system of interest and calculate exact lower and upper bounds for those that can be expressed as probabilities, such as the probability that an incoming low-priority customer will wait. We generate a wide range of instances with up to 100 servers and compare the solution times and accuracy of our algorithm with two existing algorithms. These numerical experiments indicate that our algorithm is faster than the other two algorithms for a given accuracy requirement. We investigate the impact of changing service rates on the proportion of low-priority customers served and their wait time, and we demonstrate how ignoring one of these measures can possibly mislead decision makers. Summary of Contribution: We contribute to operations research by modeling a practically important queueing system and developing an algorithm to accurately compute performance measures for that system. We also contribute to computer science by providing error and complexity analysis for the algorithm to solve a broad class of two-dimensional Markov chains with infinite state space.

Suggested Citation

  • Amir Rastpour & Armann Ingolfsson & Burhaneddin Sandıkçı, 2022. "Algorithms for Queueing Systems with Reneging and Priorities Modeled as Quasi-Birth-Death Processes," INFORMS Journal on Computing, INFORMS, vol. 34(3), pages 1693-1710, May.
    • Handle: RePEc:inm:orijoc:v:34:y:2022:i:3:p:1693-1710
    DOI: 10.1287/ijoc.2021.1141
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    References listed on IDEAS

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    1. Armann Ingolfsson & Ling Tang, 2012. "Efficient and Reliable Computation of Birth-Death Process Performance Measures," INFORMS Journal on Computing, INFORMS, vol. 24(1), pages 29-41, February.
    2. Douglas R. Miller, 1981. "Computation of Steady-State Probabilities for M / M /1 Priority Queues," Operations Research, INFORMS, vol. 29(5), pages 945-958, October.
    3. O Jouini & A Roubos, 2014. "On multiple priority multi-server queues with impatience," Journal of the Operational Research Society, Palgrave Macmillan;The OR Society, vol. 65(5), pages 616-632, May.
    4. Mohammad Delasay & Armann Ingolfsson & Bora Kolfal, 2016. "Modeling Load and Overwork Effects in Queueing Systems with Adaptive Service Rates," Operations Research, INFORMS, vol. 64(4), pages 867-885, August.
    5. Hendrik Baumann & Werner Sandmann, 2016. "Structured Modeling and Analysis of Stochastic Epidemics with Immigration and Demographic Effects," PLOS ONE, Public Library of Science, vol. 11(3), pages 1-19, March.
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    Cited by:

    1. Opher Baron & Dmitry Krass & Arik Senderovich & Eliran Sherzer, 2024. "Supervised ML for Solving the GI / GI /1 Queue," INFORMS Journal on Computing, INFORMS, vol. 36(3), pages 766-786, May.
    2. Di Crescenzo, A. & Gómez-Corral, A. & Taipe, D., 2025. "A computational approach to extreme values and related hitting probabilities in level-dependent quasi-birth–death processes," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 228(C), pages 211-224.

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