IDEAS home Printed from https://ideas.repec.org/a/spr/annopr/v317y2022i2d10.1007_s10479-017-2727-5.html
   My bibliography  Save this article

Parallel computing for Markov chains with islands and ports

Author

Listed:
  • Amod J. Basnet

    (University of North Carolina at Charlotte)

  • Isaac M. Sonin

    (University of North Carolina at Charlotte)

Abstract

We develop an algorithm to calculate invariant distributions of large Markov chains whose state spaces are partitioned into “islands” and “ports”. An island is a group of states (cluster) with potentially many connections inside of the island but a relatively small number of connections between islands. The states connecting different islands are called ports. Our algorithm is developed in the framework of the “state reduction approach”, but the special structure of the state space allows calculation of the invariant distribution to be done in parallel. Additional problems such as computation of fundamental matrices and optimal stopping problems are also analyzed for such Markov chains.

Suggested Citation

  • Amod J. Basnet & Isaac M. Sonin, 2022. "Parallel computing for Markov chains with islands and ports," Annals of Operations Research, Springer, vol. 317(2), pages 335-352, October.
    • Handle: RePEc:spr:annopr:v:317:y:2022:i:2:d:10.1007_s10479-017-2727-5
    DOI: 10.1007/s10479-017-2727-5
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10479-017-2727-5
    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/s10479-017-2727-5?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. Isaac M. Sonin & Constantine Steinberg, 2016. "Continue, quit, restart probability model," Annals of Operations Research, Springer, vol. 241(1), pages 295-318, June.
    2. Sonin, Isaac M., 2008. "A generalized Gittins index for a Markov chain and its recursive calculation," Statistics & Probability Letters, Elsevier, vol. 78(12), pages 1526-1533, September.
    3. Theodore J. Sheskin, 1985. "Technical Note—A Markov Chain Partitioning Algorithm for Computing Steady State Probabilities," Operations Research, INFORMS, vol. 33(1), pages 228-235, February.
    4. Isaac Sonin, 1999. "The Elimination algorithm for the problem of optimal stopping," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 49(1), pages 111-123, March.
    5. Winfried K. Grassmann & Michael I. Taksar & Daniel P. Heyman, 1985. "Regenerative Analysis and Steady State Distributions for Markov Chains," Operations Research, INFORMS, vol. 33(5), pages 1107-1116, October.
    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. Isaac M. Sonin & Constantine Steinberg, 2016. "Continue, quit, restart probability model," Annals of Operations Research, Springer, vol. 241(1), pages 295-318, June.
    2. Yiqiang Q. Zhao & W. John Braun & Wei Li, 1999. "Northwest corner and banded matrix approximations to a Markov chain," Naval Research Logistics (NRL), John Wiley & Sons, vol. 46(2), pages 187-197, March.
    3. Sonin, Isaac M., 2008. "A generalized Gittins index for a Markov chain and its recursive calculation," Statistics & Probability Letters, Elsevier, vol. 78(12), pages 1526-1533, September.
    4. Tijms, H.C. & Coevering, M.C.T. van de, 1990. "How to solve numerically the equilibrium equations of a Markov chain with infinitely many states," Serie Research Memoranda 0046, VU University Amsterdam, Faculty of Economics, Business Administration and Econometrics.
    5. S. K. Samanta & R. Nandi, 2021. "Queue-Length, Waiting-Time and Service Batch Size Analysis for the Discrete-Time GI/D-MSP (a,b) / 1 / ∞ $^{\text {(a,b)}}/1/\infty $ Queueing System," Methodology and Computing in Applied Probability, Springer, vol. 23(4), pages 1461-1488, December.
    6. 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.
    7. Soren Christensen & Albrecht Irle & Julian Peter Lemburg, 2021. "Flexible forward improvement iteration for infinite time horizon Markovian optimal stopping problems," Papers 2111.13443, arXiv.org.
    8. 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.
    9. Nicolas Gast & Bruno Gaujal & Kimang Khun, 2023. "Testing indexability and computing Whittle and Gittins index in subcubic time," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 97(3), pages 391-436, June.
    10. Sonin, Konstantin & Sonin, Isaac, 2020. "A Continuous-Time Model of Financial Clearing," CEPR Discussion Papers 15117, C.E.P.R. Discussion Papers.
    11. Souvik Ghosh & A. D. Banik & Joris Walraevens & Herwig Bruneel, 2022. "A detailed note on the finite-buffer queueing system with correlated batch-arrivals and batch-size-/phase-dependent bulk-service," 4OR, Springer, vol. 20(2), pages 241-272, June.
    12. A. D. Banik & M. L. Chaudhry & U. C. Gupta, 2008. "On the Finite Buffer Queue with Renewal Input and Batch Markovian Service Process: GI/BMSP/1/N," Methodology and Computing in Applied Probability, Springer, vol. 10(4), pages 559-575, December.
    13. Senthil K. Veeraraghavan & Laurens G. Debo, 2011. "Herding in Queues with Waiting Costs: Rationality and Regret," Manufacturing & Service Operations Management, INFORMS, vol. 13(3), pages 329-346, July.
    14. Edmundo de Souza e Silva & Rosa M. M. Leão & Raymond Marie, 2013. "Efficient Transient Analysis of Markovian Models Using a Block Reduction Approach," INFORMS Journal on Computing, INFORMS, vol. 25(4), pages 743-757, November.
    15. Zhenya Li & Zulfiqar Ali & Tong Cui & Sadia Qamar & Muhammad Ismail & Amna Nazeer & Muhammad Faisal, 2022. "A comparative analysis of pre- and post-industrial spatiotemporal drought trends and patterns of Tibet Plateau using Sen slope estimator and steady-state probabilities of Markov Chain," Natural Hazards: Journal of the International Society for the Prevention and Mitigation of Natural Hazards, Springer;International Society for the Prevention and Mitigation of Natural Hazards, vol. 113(1), pages 547-576, August.
    16. Grassmann, Winfried K., 1996. "Optimizing steady state Markov chains by state reduction," European Journal of Operational Research, Elsevier, vol. 89(2), pages 277-284, March.
    17. Malekipirbazari, Milad & Çavuş, Özlem, 2024. "Index policy for multiarmed bandit problem with dynamic risk measures," European Journal of Operational Research, Elsevier, vol. 312(2), pages 627-640.
    18. Esther Frostig & Gideon Weiss, 2016. "Four proofs of Gittins’ multiarmed bandit theorem," Annals of Operations Research, Springer, vol. 241(1), pages 127-165, June.
    19. 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.
    20. Pala, Ali & Zhuang, Jun, 2018. "Security screening queues with impatient applicants: A new model with a case study," European Journal of Operational Research, Elsevier, vol. 265(3), pages 919-930.

    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:317:y:2022:i:2:d:10.1007_s10479-017-2727-5. 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.