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The Power-Series Algorithm Applied to the Shortest-Queue Model

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  • J. P. C. Blanc

    (Tilburg University, Tilburg, The Netherlands)

Abstract

An iterative numerical technique for the evaluation of queue length distribution is applied to multiserver systems with queues in parallel in which customers join (one of) the shortest queues upon arrival. The technique is based on power-series expansions of the state probabilities as functions of the load of the system. The convergence of the series is accelerated by applying a modified form of the epsilon algorithm. The shortest-queue model lends itself particularly well to a numerical analysis by means of the power-series algorithm due to a specific property of this model. Numerical values for the mean and the standard deviation of the total number of customers and the waiting times in stationary symmetrical systems are obtained for practically all values of the load for systems with up to ten queues and for a load not exceeding 75% for systems with up to 30 queues. Data are also presented for systems with four queues and unequal service rates.

Suggested Citation

  • J. P. C. Blanc, 1992. "The Power-Series Algorithm Applied to the Shortest-Queue Model," Operations Research, INFORMS, vol. 40(1), pages 157-167, February.
    • Handle: RePEc:inm:oropre:v:40:y:1992:i:1:p:157-167
    DOI: 10.1287/opre.40.1.157
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    Cited by:

    1. Partha Chakroborty & Rahul Gill & Pranamesh Chakraborty, 2016. "Analysing queueing at toll plazas using a coupled, multiple-queue, queueing system model: application to toll plaza design," Transportation Planning and Technology, Taylor & Francis Journals, vol. 39(7), pages 675-692, October.
    2. W. B. van den Hout & J. P. C. Blanc, 1995. "The Power-Series Algorithm for Markovian Queueing Networks," Springer Books, in: William J. Stewart (ed.), Computations with Markov Chains, chapter 19, pages 321-338, Springer.
    3. van den Hout, W.B. & Blanc, J.P.C., 1994. "The Power-Series Algorithm for a Wide Class of Markov Processes," Discussion Paper 1994-87, Tilburg University, Center for Economic Research.
    4. Danielle Tibi, 2019. "Martingales and buffer overflow for the symmetric shortest queue model," Queueing Systems: Theory and Applications, Springer, vol. 93(1), pages 153-190, October.
    5. P. Patrick Wang, 2000. "Workload distribution of discrete‐time parallel queues with two servers," Naval Research Logistics (NRL), John Wiley & Sons, vol. 47(5), pages 440-454, August.
    6. Blanc, J.P.C., 2009. "Bad luck when joining the shortest queue," European Journal of Operational Research, Elsevier, vol. 195(1), pages 167-173, May.
    7. Herwig Bruneel & Arnaud Devos, 2024. "Explicit Solutions for Coupled Parallel Queues," Mathematics, MDPI, vol. 12(15), pages 1-31, July.
    8. Herwig Bruneel & Arnaud Devos, 2023. "Asymptotic behavior of a system of two coupled queues when the content of one queue is very high," Queueing Systems: Theory and Applications, Springer, vol. 105(3), pages 189-232, December.
    9. Plinio S. Dester & Christine Fricker & Danielle Tibi, 2017. "Stationary analysis of the shortest queue problem," Queueing Systems: Theory and Applications, Springer, vol. 87(3), pages 211-243, December.
    10. M. Saxena & I. Dimitriou & S. Kapodistria, 2020. "Analysis of the shortest relay queue policy in a cooperative random access network with collisions," Queueing Systems: Theory and Applications, Springer, vol. 94(1), pages 39-75, February.
    11. Dieter Fiems & Balakrishna Prabhu & Koen Turck, 2013. "Analytic approximations of queues with lightly- and heavily-correlated autoregressive service times," Annals of Operations Research, Springer, vol. 202(1), pages 103-119, January.
    12. Blanc, J.P.C., 1990. "Performance evaluation of polling systems by means of the power-series algorithm," Other publications TiSEM a5f5fb56-c17c-4c46-8d5e-b, Tilburg University, School of Economics and Management.

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