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Regenerative Analysis and Steady State Distributions for Markov Chains

Author

Listed:
  • Winfried K. Grassmann

    (University of Saskatchewan, Saskatoon, Saskatchewan)

  • Michael I. Taksar

    (Florida State University, Tallahassee, Florida)

  • Daniel P. Heyman

    (Bell Communications Research, Holmdel, New Jersey)

Abstract

We apply regenerative theory to derive certain relations between steady state probabilities of a Markov chain. These relations are then used to develop a numerical algorithm to find these probabilities. The algorithm is a modification of the Gauss-Jordan method, in which all elements used in numerical computations are nonnegative; as a consequence, the algorithm is numerically stable.

Suggested Citation

  • 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.
    • Handle: RePEc:inm:oropre:v:33:y:1985:i:5:p:1107-1116
    DOI: 10.1287/opre.33.5.1107
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    Cited by:

    1. K. Sikdar & S. K. Samanta, 2016. "Analysis of a finite buffer variable batch service queue with batch Markovian arrival process and server’s vacation," OPSEARCH, Springer;Operational Research Society of India, vol. 53(3), pages 553-583, September.
    2. 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.
    3. 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.
    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. 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.
    6. 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.
    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. 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.
    9. Isaac M. Sonin & Constantine Steinberg, 2016. "Continue, quit, restart probability model," Annals of Operations Research, Springer, vol. 241(1), pages 295-318, June.
    10. 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.
    11. Shaler Stidham, 2002. "Analysis, Design, and Control of Queueing Systems," Operations Research, INFORMS, vol. 50(1), pages 197-216, February.
    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. 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.
    14. Apurva Jain, 2006. "Priority and dynamic scheduling in a make‐to‐stock queue with hyperexponential demand," Naval Research Logistics (NRL), John Wiley & Sons, vol. 53(5), pages 363-382, August.
    15. 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.
    16. V. Ramaswami & David Poole & Soohan Ahn & Simon Byers & Alan Kaplan, 2005. "Ensuring Access to Emergency Services in the Presence of Long Internet Dial-Up Calls," Interfaces, INFORMS, vol. 35(5), pages 411-422, October.
    17. 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.
    18. 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.
    19. 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.
    20. Jianyu Cao & Weixin Xie, 2017. "Stability of a two-queue cyclic polling system with BMAPs under gated service and state-dependent time-limited service disciplines," Queueing Systems: Theory and Applications, Springer, vol. 85(1), pages 117-147, February.
    21. Pekergin, Nihal & Dayar, Tugrul & Alparslan, Denizhan N., 2005. "Componentwise bounds for nearly completely decomposable Markov chains using stochastic comparison and reordering," European Journal of Operational Research, Elsevier, vol. 165(3), pages 810-825, September.

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