IDEAS home Printed from https://ideas.repec.org/a/spr/metcap/v15y2013i3d10.1007_s11009-011-9266-3.html
   My bibliography  Save this article

A Simple and Complete Computational Analysis of MAP/R/1 Queue Using Roots

Author

Listed:
  • M. L. Chaudhry

    (Royal Military College of Canada)

  • Gagandeep Singh

    (Indian Institute of Technology)

  • U. C. Gupta

    (Indian Institute of Technology)

Abstract

In this paper, we present (in terms of roots) a simple closed-form analysis for evaluating system-length distribution at three epochs of time (arbitrary, pre-arrival, and post-departure) and queueing-time distribution (virtual and actual) of the MAP/R/1 queue, where R represents the class of distributions whose Laplace–Stieltjes transforms are rational functions. Our analysis is based on roots of the associated characteristic equations of the (i) vector-generating function of system-length distribution and (ii) Laplace–Stieltjes transform of the virtual queueing-time distribution. The proposed method for evaluating boundary probabilities is an alternative to the matrix-analytic method as well as spectral method. Numerical aspects have been tested for a variety of arrival and service-time (including matrix-exponential (ME)) distributions and a sample of numerical outputs is presented. The method is analytically quite simple and easy to implement. It is hoped that the results obtained would prove to be beneficial to both theoreticians and practitioners.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:metcap:v:15:y:2013:i:3:d:10.1007_s11009-011-9266-3
    DOI: 10.1007/s11009-011-9266-3
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s11009-011-9266-3
    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/s11009-011-9266-3?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. Joseph Abate & Ward Whitt, 2006. "A Unified Framework for Numerically Inverting Laplace Transforms," INFORMS Journal on Computing, INFORMS, vol. 18(4), pages 408-421, November.
    2. 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.
    3. Ho Woo Lee & Jong Min Moon & Jong Keun Park & Byung Kyu Kim, 2003. "A spectral approach to compute the mean performance measures of the queue with low-order BMAP input," International Journal of Stochastic Analysis, Hindawi, vol. 16, pages 1-12, January.
    4. Sadrac K. Matendo, 1994. "Some performance measures for vacation models with a batch Markovian arrival process," International Journal of Stochastic Analysis, Hindawi, vol. 7, pages 1-14, January.
    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. S. Pradhan & U. C. Gupta, 2019. "Analysis of an infinite-buffer batch-size-dependent service queue with Markovian arrival process," Annals of Operations Research, Springer, vol. 277(2), pages 161-196, June.
    2. Nitin Kumar & U. C. Gupta, 2020. "A Renewal Generated Geometric Catastrophe Model with Discrete-Time Markovian Arrival Process," Methodology and Computing in Applied Probability, Springer, vol. 22(3), pages 1293-1324, September.
    3. 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.
    4. 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.
    5. 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.
    6. H. Bruneel & W. Rogiest & J. Walraevens & S. Wittevrongel, 2015. "Analysis of a discrete-time queue with general independent arrivals, general service demands and fixed service capacity," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 82(3), pages 285-315, December.
    7. Michiel Muynck & Herwig Bruneel & Sabine Wittevrongel, 2020. "Analysis of a queue with general service demands and correlated service capacities," Annals of Operations Research, Springer, vol. 293(1), pages 73-99, October.

    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. 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.
    2. 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.
    3. 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.
    4. 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.
    5. 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.
    6. 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.
    7. 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.
    8. 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.
    9. 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.
    10. Mohan Chaudhry & Veena Goswami, 2022. "The Geo / G a , Y /1/ N Queue Revisited," Mathematics, MDPI, vol. 10(17), pages 1-17, September.
    11. 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.
    12. 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.
    13. 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.
    14. David Landriault & Bin Li & Hongzhong Zhang, 2014. "On the Frequency of Drawdowns for Brownian Motion Processes," Papers 1403.1183, arXiv.org.
    15. Leippold, Markus & Vasiljević, Nikola, 2017. "Pricing and disentanglement of American puts in the hyper-exponential jump-diffusion model," Journal of Banking & Finance, Elsevier, vol. 77(C), pages 78-94.
    16. Runhuan Feng & Hans W. Volkmer, 2015. "Conditional Asian Options," Papers 1505.06946, arXiv.org.
    17. Mor Harchol-Balter, 2021. "Open problems in queueing theory inspired by datacenter computing," Queueing Systems: Theory and Applications, Springer, vol. 97(1), pages 3-37, February.
    18. Markus Leippold & Nikola Vasiljević, 2020. "Option-Implied Intrahorizon Value at Risk," Management Science, INFORMS, vol. 66(1), pages 397-414, January.
    19. Landriault, David & Shi, Tianxiang, 2015. "Occupation times in the MAP risk model," Insurance: Mathematics and Economics, Elsevier, vol. 60(C), pages 75-82.
    20. Donatien Hainaut & Yang Shen & Yan Zeng, 2018. "How do capital structure and economic regime affect fair prices of bank’s equity and liabilities?," Annals of Operations Research, Springer, vol. 262(2), pages 519-545, March.

    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:metcap:v:15:y:2013:i:3:d:10.1007_s11009-011-9266-3. 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.