IDEAS home Printed from https://ideas.repec.org/a/spr/annopr/v310y2022i2d10.1007_s10479-020-03776-5.html
   My bibliography  Save this article

Approximations for the performance evaluation of a discrete-time two-class queue with an alternating service discipline

Author

Listed:
  • Arnaud Devos

    (Department of Telecommunications and Information Processing)

  • Joris Walraevens

    (Department of Telecommunications and Information Processing)

  • Dieter Fiems

    (Department of Telecommunications and Information Processing)

  • Herwig Bruneel

    (Department of Telecommunications and Information Processing)

Abstract

We consider a discrete-time queueing system with two queues and one server. The server is allocated in each slot to the first queue with probability $$\alpha $$ α and to the second queue with probability $$1-\alpha $$ 1 - α . The service times are equal to one time slot. The queues have exponentially bounded, but general, arrival distributions. The mathematical description of this system leads to a single functional equation for the joint probability generating function of the stationary system contents. As the joint stochastic process of the system contents is not amenable for exact analysis, we focus on an efficient approximation of the joint probability generating function. In particular, first we prove that the partial probability generating functions, present in the functional equation, have a unique dominant pole. Secondly, we use this information to approximate these partial probability generating functions by truncating an infinite sum. The remaining finite number of unknowns are estimated from a noise perturbed linear system. We illustrate our approach by various numerical examples and verify the accuracy by means of simulation.

Suggested Citation

  • Arnaud Devos & Joris Walraevens & Dieter Fiems & Herwig Bruneel, 2022. "Approximations for the performance evaluation of a discrete-time two-class queue with an alternating service discipline," Annals of Operations Research, Springer, vol. 310(2), pages 477-503, March.
    • Handle: RePEc:spr:annopr:v:310:y:2022:i:2:d:10.1007_s10479-020-03776-5
    DOI: 10.1007/s10479-020-03776-5
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10479-020-03776-5
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1007/s10479-020-03776-5?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
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Blanc, J.P.C., 1993. "Performance Analysis and Optimization with the Power- Series Algorithm," Papers 9347, Tilburg - Center for Economic Research.
    2. Ahmad Hanbali & Roland Haan & Richard Boucherie & Jan-Kees Ommeren, 2012. "Time-limited polling systems with batch arrivals and phase-type service times," Annals of Operations Research, Springer, vol. 198(1), pages 57-82, September.
    3. Ioannis Dimitriou, 2016. "A queueing model with two classes of retrial customers and paired services," Annals of Operations Research, Springer, vol. 238(1), pages 123-143, March.
    4. H.C. Tijms & D.J. van Vuuren, 2002. "Markov Processes on a Semi-Infinite Strip and the Geometric Tail Algorithm," Annals of Operations Research, Springer, vol. 113(1), pages 133-140, July.
    5. Jasper Vanlerberghe & Tom Maertens & Joris Walraevens & Stijn Vuyst & Herwig Bruneel, 2016. "On the optimization of two-class work-conserving parameterized scheduling policies," 4OR, Springer, vol. 14(3), pages 281-308, September.
    6. Ioannis Dimitriou, 2016. "A queueing model with two classes of retrial customers and paired services," Annals of Operations Research, Springer, vol. 238(1), pages 123-143, March.
    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. Sanga, Sudeep Singh & Jain, Madhu, 2019. "FM/FM/1 double orbit retrial queue with customers’ joining strategy: A parametric nonlinear programing approach," Applied Mathematics and Computation, Elsevier, vol. 362(C), pages 1-1.
    2. Blanc, J.P.C. & Lenzini, L., 1995. "Analysis of communication systems with timed token protocols using the power-series algorithm," Discussion Paper 1995-100, Tilburg University, Center for Economic Research.
    3. van den Hout, W.B. & Blanc, J.P.C., 1994. "The Power-Series Algorithm for a Wide Class of Markov Processes," Other publications TiSEM 54b74f52-9378-47b9-aa0f-5, Tilburg University, School of Economics and Management.
    4. 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.
    5. Vladimir Vishnevsky & Olga Semenova, 2021. "Polling Systems and Their Application to Telecommunication Networks," Mathematics, MDPI, vol. 9(2), pages 1-30, January.
    6. Blanc, J.P.C., 1996. "Optimization of Periodic Polling Systems with Non-Preemptive, Time-Limited Service," Discussion Paper 1996-63, Tilburg University, Center for Economic Research.
    7. Kao, Edward P. C. & Wilson, Sandra D., 1999. "Analysis of nonpreemptive priority queues with multiple servers and two priority classes," European Journal of Operational Research, Elsevier, vol. 118(1), pages 181-193, October.
    8. Koole, Ger & Pot, Auke, 2006. "Workload minimization in re-entrant lines," European Journal of Operational Research, Elsevier, vol. 174(1), pages 216-233, October.
    9. Dieter Fiems & Tuan Phung-Duc, 2019. "Light-traffic analysis of random access systems without collisions," Annals of Operations Research, Springer, vol. 277(2), pages 311-327, June.
    10. Blanc, J.P.C., 1999. "On the Stability of Communication Systems with Timed Token Protocols," Other publications TiSEM bd28ec2d-1fc2-447d-9c0a-c, Tilburg University, School of Economics and Management.
    11. Jan-Kees Ommeren & Ahmad Al Hanbali & Richard J. Boucherie, 2020. "Analysis of polling models with a self-ruling server," Queueing Systems: Theory and Applications, Springer, vol. 94(1), pages 77-107, February.
    12. van den Hout, W.B. & Blanc, J.P.C., 1994. "The power-series algorithm for Markovian queueing networks," Other publications TiSEM 1312893d-c375-4b17-bbbe-3, Tilburg University, School of Economics and Management.
    13. Arnaud Devos & Joris Walraevens & Dieter Fiems & Herwig Bruneel, 2020. "Analysis of a discrete-time two-class randomly alternating service model with Bernoulli arrivals," Queueing Systems: Theory and Applications, Springer, vol. 96(1), pages 133-152, October.
    14. Wanlu Gu & Neng Fan & Haitao Liao, 2019. "Evaluating readmission rates and discharge planning by analyzing the length-of-stay of patients," Annals of Operations Research, Springer, vol. 276(1), pages 89-108, May.
    15. Blanc, J.P.C., 1998. "The power-series algorithm for polling systems with time limits," Other publications TiSEM 3366bad3-964d-4039-82cc-a, Tilburg University, School of Economics and Management.
    16. Chesoong Kim & Alexander Dudin & Olga Dudina & Valentina Klimenok, 2020. "Analysis of Queueing System with Non-Preemptive Time Limited Service and Impatient Customers," Methodology and Computing in Applied Probability, Springer, vol. 22(2), pages 401-432, June.
    17. Arnaud Devos & Joris Walraevens & Dieter Fiems & Herwig Bruneel, 2021. "Heavy-Traffic Comparison of a Discrete-Time Generalized Processor Sharing Queue and a Pure Randomly Alternating Service Queue," Mathematics, MDPI, vol. 9(21), pages 1-25, October.
    18. A. Oblakova & A. Al Hanbali & R. J. Boucherie & J. C. W. Ommeren & W. H. M. Zijm, 2019. "An exact root-free method for the expected queue length for a class of discrete-time queueing systems," Queueing Systems: Theory and Applications, Springer, vol. 92(3), pages 257-292, August.
    19. Shaul K. Bar-Lev & Hans Blanc & Onno Boxma & Guido Janssen & David Perry, 2013. "Tandem Queues with Impatient Customers for Blood Screening Procedures," Methodology and Computing in Applied Probability, Springer, vol. 15(2), pages 423-451, June.
    20. Ger Koole & Olaf Passchier, 1997. "Optimal control in light traffic Markov decision processes," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 45(1), pages 63-79, February.

    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:spr:annopr:v:310:y:2022:i:2:d:10.1007_s10479-020-03776-5. 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: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    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.