IDEAS home Printed from https://ideas.repec.org/a/spr/annopr/v252y2017i1d10.1007_s10479-015-2026-y.html
   My bibliography  Save this article

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 / ∞

Author

Listed:
  • M. L. Chaudhry

    (Royal Military College of Canada)

  • A. D. Banik

    (Indian Institute of Technology
    Universidade de Lisboa)

  • A. Pacheco

    (Universidade de Lisboa)

Abstract

We consider a batch arrival infinite-buffer single-server queue with generally distributed inter-batch arrival times with arrivals occurring in batches of random sizes. The service process is correlated and its structure is governed by a Markovian service process in continuous time. The proposed analysis is based on roots of the associated characteristic equation of the vector-generating function of system-length distribution at a pre-arrival epoch. We also obtain the steady-state probability distribution at an arbitrary epoch using the classical argument based on Markov renewal theory. Some important performance measures such as the average number of customers in the system and the mean sojourn time have also been obtained. Later, we have established heavy- and light-traffic approximations as well as an approximation for the tail probabilities at pre-arrival epoch based on one root of the characteristic equation. Numerical results for some cases have been presented to show the effect of model parameters on the performance measures.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:annopr:v:252:y:2017:i:1:d:10.1007_s10479-015-2026-y
    DOI: 10.1007/s10479-015-2026-y
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10479-015-2026-y
    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/s10479-015-2026-y?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. A. Banik & U. Gupta, 2007. "Analyzing the finite buffer batch arrival queue under Markovian service process: GI X /MSP/1/N," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 15(1), pages 146-160, July.
    2. António Pacheco & Helena Ribeiro, 2008. "Consecutive customer losses in oscillating GI X /M//n systems with state dependent services rates," Annals of Operations Research, Springer, vol. 162(1), pages 143-158, September.
    3. 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.
    4. John F. Shortle & Percy H. Brill & Martin J. Fischer & Donald Gross & Denise M. B. Masi, 2004. "An Algorithm to Compute the Waiting Time Distribution for the M/G/1 Queue," INFORMS Journal on Computing, INFORMS, vol. 16(2), pages 152-161, May.
    5. 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.
    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. Abhijit Datta Banik & Souvik Ghosh & M. L. Chaudhry, 2020. "On the optimal control of loss probability and profit in a GI/C-BMSP/1/N queueing system," OPSEARCH, Springer;Operational Research Society of India, vol. 57(1), pages 144-162, March.
    2. Souvik Ghosh & A. D. Banik, 2018. "Computing conditional sojourn time of a randomly chosen tagged customer in a $$\textit{BMAP/MSP/}1$$ BMAP / MSP / 1 queue under random order service discipline," Annals of Operations Research, Springer, vol. 261(1), pages 185-206, February.

    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. 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.
    2. 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.
    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. 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.
    5. 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.
    6. 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.
    7. 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.
    8. 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.
    9. 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.
    10. 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.
    11. 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.
    12. 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.
    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. Abhijit Datta Banik & Souvik Ghosh & M. L. Chaudhry, 2020. "On the optimal control of loss probability and profit in a GI/C-BMSP/1/N queueing system," OPSEARCH, Springer;Operational Research Society of India, vol. 57(1), pages 144-162, March.
    15. Amir Ahmadi-Javid & Pooya Hoseinpour, 2022. "Convexification of Queueing Formulas by Mixed-Integer Second-Order Cone Programming: An Application to a Discrete Location Problem with Congestion," INFORMS Journal on Computing, INFORMS, vol. 34(5), pages 2621-2633, September.
    16. A. D. Banik & M. L. Chaudhry, 2017. "Efficient Computational Analysis of Stationary Probabilities for the Queueing System BMAP / G /1/ N With or Without Vacation(s)," INFORMS Journal on Computing, INFORMS, vol. 29(1), pages 140-151, February.
    17. John F. Shortle & Martin J. Fischer & Percy H. Brill, 2007. "Waiting-Time Distribution of M/D N /1 Queues Through Numerical Laplace Inversion," INFORMS Journal on Computing, INFORMS, vol. 19(1), pages 112-120, February.
    18. Andrzej Chydzinski & Pawel Mrozowski, 2016. "Queues with Dropping Functions and General Arrival Processes," PLOS ONE, Public Library of Science, vol. 11(3), pages 1-23, March.
    19. Pawel Mrozowski & Andrzej Chydzinski, 2018. "Queues with Dropping Functions and Autocorrelated Arrivals," Methodology and Computing in Applied Probability, Springer, vol. 20(1), pages 97-115, March.
    20. Le-Duc, Tho & de Koster, Rene M.B.M., 2007. "Travel time estimation and order batching in a 2-block warehouse," European Journal of Operational Research, Elsevier, vol. 176(1), pages 374-388, January.

    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:252:y:2017:i:1:d:10.1007_s10479-015-2026-y. 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.