IDEAS home Printed from https://ideas.repec.org/a/spr/queues/v82y2016i1d10.1007_s11134-015-9469-3.html
   My bibliography  Save this article

Analytically elegant and computationally efficient results in terms of roots for the $$GI^{X}/M/c$$ G I X / M / c queueing system

Author

Listed:
  • Mohan L. Chaudhry

    (Royal Military College of Canada)

  • James J. Kim

    (Royal Canadian Air Force (RCAF))

Abstract

An elegant and simple solution to determine the distributions of queue length at different epochs and the waiting time for the model $$GI^{X}/M/c$$ G I X / M / c is presented. In the past, the model $$GI^{X}/M/c$$ G I X / M / c has been extensively analyzed using various techniques by many authors. The purpose of this paper is to present a simple and effective derivation of the analytic solution for pre-arrival epoch probabilities as a linear combination of specific geometric terms (except for the boundary probabilities when the number of servers is greater than the maximum batch size) involving the roots of the underlying characteristic equation. The solution is then leveraged to compute the waiting-time distributions of both first and arbitrary customers of an incoming batch. Numerical examples with various arrival patterns and batch size distributions are also presented. The method that is being proposed here not only gives an alternate solution to the existing methods, but it is also analytically simple, easy to implement, and computationally efficient. It is hoped that the results obtained will prove beneficial to both theoreticians and practitioners.

Suggested Citation

  • Mohan L. Chaudhry & James J. Kim, 2016. "Analytically elegant and computationally efficient results in terms of roots for the $$GI^{X}/M/c$$ G I X / M / c queueing system," Queueing Systems: Theory and Applications, Springer, vol. 82(1), pages 237-257, February.
    • Handle: RePEc:spr:queues:v:82:y:2016:i:1:d:10.1007_s11134-015-9469-3
    DOI: 10.1007/s11134-015-9469-3
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s11134-015-9469-3
    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/s11134-015-9469-3?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. Mohan L. Chaudhry & Carl M. Harris & William G. Marchal, 1990. "Robustness of Rootfinding in Single-Server Queueing Models," INFORMS Journal on Computing, INFORMS, vol. 2(3), pages 273-286, August.
    2. P. J. Burke, 1975. "Technical Note—Delays in Single-Server Queues with Batch Input," Operations Research, INFORMS, vol. 23(4), pages 830-833, August.
    3. Laoucine Kerbache & G. M. Gontijo & G. S. Atuncar & F.R.B. Cruz, 2011. "Performance Evaluation and Dimensioning of GIX/M/c/N Systems Through Kernel Estimation," Post-Print hal-00796342, HAL.
    4. Ronald W. Wolff, 1982. "Poisson Arrivals See Time Averages," Operations Research, INFORMS, vol. 30(2), pages 223-231, April.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. J. J. Kim & M. L. Chaudhry & V. Goswami & A. D. Banik, 2021. "A New and Pragmatic Approach to the GIX/Geo/c/N Queues Using Roots," Methodology and Computing in Applied Probability, Springer, vol. 23(1), pages 273-289, March.
    2. James J. Kim & Douglas G. Down & Mohan Chaudhry & Abhijit Datta Banik, 2022. "Difference Equations Approach for Multi-Server Queueing Models with Removable Servers," Methodology and Computing in Applied Probability, Springer, vol. 24(3), pages 1297-1321, September.
    3. Miaomiao Yu & Yinghui Tang, 2018. "Analysis of the Sojourn Time Distribution for M/GL/1 Queue with Bulk-Service of Exactly Size L," Methodology and Computing in Applied Probability, Springer, vol. 20(4), pages 1503-1514, December.

    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. Miaomiao Yu & Yinghui Tang, 2022. "Analysis of a renewal batch arrival queue with a fault-tolerant server using shift operator method," Operational Research, Springer, vol. 22(3), pages 2831-2858, July.
    2. S. K. Samanta, 2020. "Waiting-time analysis of D-$${ BMAP}{/}G{/}1$$BMAP/G/1 queueing system," Annals of Operations Research, Springer, vol. 284(1), pages 401-413, January.
    3. Mohan Chaudhry & Veena Goswami, 2022. "The Geo / G a , Y /1/ N Queue Revisited," Mathematics, MDPI, vol. 10(17), pages 1-17, September.
    4. M. L. Chaudhry & Veena Goswami, 2019. "The Queue Geo/G/1/N + 1 Revisited," Methodology and Computing in Applied Probability, Springer, vol. 21(1), pages 155-168, March.
    5. M. L. Chaudhry, 1992. "Computing stationary queueing‐time distributions of GI/D/1 and GI/D/c queues," Naval Research Logistics (NRL), John Wiley & Sons, vol. 39(7), pages 975-996, December.
    6. J. S. H. van Leeuwaarden, 2006. "Delay Analysis for the Fixed-Cycle Traffic-Light Queue," Transportation Science, INFORMS, vol. 40(2), pages 189-199, May.
    7. Pinai Linwong* & Nei Kato* & Yoshiaki Nemoto*, 2004. "A Polynomial Factorization Approach for the Discrete Time GIX/>G/1/K Queue," Methodology and Computing in Applied Probability, Springer, vol. 6(3), pages 277-291, September.
    8. P. Patrick Wang, 1993. "Static and dynamic scheduling of customer arrivals to a single‐server system," Naval Research Logistics (NRL), John Wiley & Sons, vol. 40(3), pages 345-360, April.
    9. James J. Kim & Douglas G. Down & Mohan Chaudhry & Abhijit Datta Banik, 2022. "Difference Equations Approach for Multi-Server Queueing Models with Removable Servers," Methodology and Computing in Applied Probability, Springer, vol. 24(3), pages 1297-1321, September.
    10. S. K. Samanta & M. L. Chaudhry & A. Pacheco, 2016. "Analysis of B M A P/M S P/1 Queue," Methodology and Computing in Applied Probability, Springer, vol. 18(2), pages 419-440, June.
    11. M. L. Chaudhry & Gagandeep Singh & U. C. Gupta, 2013. "A Simple and Complete Computational Analysis of MAP/R/1 Queue Using Roots," Methodology and Computing in Applied Probability, Springer, vol. 15(3), pages 563-582, September.
    12. M. L. Chaudhry & A. D. Banik & A. Pacheco, 2017. "A simple analysis of the batch arrival queue with infinite-buffer and Markovian service process using roots method: $$ GI ^{[X]}/C$$ G I [ X ] / C - $$ MSP /1/\infty $$ M S P / 1 / ∞," Annals of Operations Research, Springer, vol. 252(1), pages 135-173, May.
    13. Mohan Chaudhry & Abhijit Datta Banik & Sitaram Barik & Veena Goswami, 2023. "A Novel Computational Procedure for the Waiting-Time Distribution (In the Queue) for Bulk-Service Finite-Buffer Queues with Poisson Input," Mathematics, MDPI, vol. 11(5), pages 1-26, February.
    14. Tetsuji Hirayama, 2003. "Mean sojourn times in multiclass feedback queues with gated disciplines," Naval Research Logistics (NRL), John Wiley & Sons, vol. 50(7), pages 719-741, October.
    15. Ananth V. Iyer & Apurva Jain, 2003. "The Logistics Impact of a Mixture of Order-Streams in a Manufacturer-Retailer System," Management Science, INFORMS, vol. 49(7), pages 890-906, July.
    16. Yanting Chen & Jingui Xie & Taozeng Zhu, 2023. "Overflow in systems with two servers: the negative consequences," Flexible Services and Manufacturing Journal, Springer, vol. 35(3), pages 838-863, September.
    17. Mohebbi, E., 2008. "A note on a production control model for a facility with limited storage capacity in a random environment," European Journal of Operational Research, Elsevier, vol. 190(2), pages 562-570, October.
    18. Hassin, Refael & Haviv, Moshe & Oz, Binyamin, 2023. "Strategic behavior in queues with arrival rate uncertainty," European Journal of Operational Research, Elsevier, vol. 309(1), pages 217-224.
    19. Huseyin C. Ozmutlu, 2009. "Markovian analysis for automatic new topic identification in search engine transaction logs," Applied Stochastic Models in Business and Industry, John Wiley & Sons, vol. 25(6), pages 737-768, November.
    20. Tom van Woensel & Frederico R B Cruz, 2014. "Optimal Routing in General Finite Multi-Server Queueing Networks," PLOS ONE, Public Library of Science, vol. 9(7), pages 1-15, July.

    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:queues:v:82:y:2016:i:1:d:10.1007_s11134-015-9469-3. 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.