IDEAS home Printed from https://ideas.repec.org/a/wly/jnljam/v2021y2021i1n5550363.html

An Efficient Convolution Algorithm for the Non‐Markovian Two‐Node Cyclic Network

Author

Listed:
  • Muhammad El-Taha
  • Thomas Michaud

Abstract

Consider a closed cyclic queueing model that consists of two nodes and a total of M customers. Each node buffer can accommodate all M customers. Node 1 has N ≤ M servers, each having an exponential service time with rate λ. The second node consists of a single server with a general service time distribution function B(.). The well‐known machine repair model with spares, where a set of identical machines, N, is served by a single repair person, is a key application of this model. This model has several other applications in performance evaluation, manufacturing, computer networks, and in reliability studies as it can be easily used to compute system availability. In this article, we give an efficient algorithm to derive an exact solution for the steady state system size probabilities. Our approach is based on developing an efficient polynomial convolution method to compute the transition probabilities of a birth process over node 2 service times and solving an imbedded Markov chain at node 2 service completion epochs. This is a significant improvement over the exponential algorithm given in an earlier paper. Numerical examples are given to demonstrate the performance of our method.

Suggested Citation

  • Muhammad El-Taha & Thomas Michaud, 2021. "An Efficient Convolution Algorithm for the Non‐Markovian Two‐Node Cyclic Network," Journal of Applied Mathematics, John Wiley & Sons, vol. 2021(1).
    • Handle: RePEc:wly:jnljam:v:2021:y:2021:i:1:n:5550363
    DOI: 10.1155/2021/5550363
    as

    Download full text from publisher

    File URL: https://doi.org/10.1155/2021/5550363
    Download Restriction: no
    File URL: https://libkey.io/10.1155/2021/5550363?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    References listed on IDEAS

    as
    1. Gupta, U. C. & Srinivasa Rao, T. S. S., 1996. "On the M/G/1 machine interference model with spares," European Journal of Operational Research, Elsevier, vol. 89(1), pages 164-171, February.
    2. Hans Daduna, 1986. "Two-Stage Cyclic Queues with Nonexponential Servers: Steady-State and Cyclic Time," Operations Research, INFORMS, vol. 34(3), pages 455-459, June.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Haque, Lani & Armstrong, Michael J., 2007. "A survey of the machine interference problem," European Journal of Operational Research, Elsevier, vol. 179(2), pages 469-482, June.
    2. Kennedy, W. J. & Wayne Patterson, J. & Fredendall, Lawrence D., 2002. "An overview of recent literature on spare parts inventories," International Journal of Production Economics, Elsevier, vol. 76(2), pages 201-215, March.
    3. Shengli Lv, 2021. "Multi-Machine Repairable System with One Unreliable Server and Variable Repair Rate," Mathematics, MDPI, vol. 9(11), pages 1-16, June.
    4. Ke, Jau-Chuan & Wang, Kuo-Hsiung, 2002. "A recursive method for the N policy G/M/1 queueing system with finite capacity," European Journal of Operational Research, Elsevier, vol. 142(3), pages 577-594, November.
    5. K.-H. Wang & C.-C. Kuo & W. L. Pearn, 2007. "Optimal Control of an M/G/1/K Queueing System with Combined F Policy and Startup Time," Journal of Optimization Theory and Applications, Springer, vol. 135(2), pages 285-299, November.
    6. Nitzan Carmeli & Galit B. Yom-Tov & Onno J. Boxma, 2023. "State-Dependent Estimation of Delay Distributions in Fork-Join Networks," Manufacturing & Service Operations Management, INFORMS, vol. 25(3), pages 1081-1098, May.

    More about this item

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:wly:jnljam:v:2021:y:2021:i:1:n:5550363. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Wiley Content Delivery (email available below). General contact details of provider: https://onlinelibrary.wiley.com/journal/4185 .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.