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Markovian Arrival Process Subject to Renewal Generated Binomial Catastrophes

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  • Nitin Kumar

    (Indian Institute of Technology)

  • Umesh Chandra Gupta

    (Indian Institute of Technology)

Abstract

This paper investigates a population model which grows as per the Markovian arrival process and is influenced by binomial catastrophes that occur according to renewal process. That is, when a catastrophe attacks, an individual (element) of the population survives with probability p or dies with probability $$1-p$$ 1 - p , independent of anything else. Using the supplementary variable technique, the steady-state vector generating function (VGF) of the population size distribution at post-catastrophe epoch is obtained in terms of the infinite product of matrices. Further, the VGF of the population size distribution at arbitrary and pre-catastrophe epochs are also deduced. To make the model valuable for practitioners, a step-wise computing process for evaluation of the distribution of population size at various epochs is given. A recursive formula to compute factorial moments of the population size is also presented. Finally, some numerical results are included to illustrate the impact of parameters on the behavior of the model.

Suggested Citation

  • Nitin Kumar & Umesh Chandra Gupta, 2022. "Markovian Arrival Process Subject to Renewal Generated Binomial Catastrophes," Methodology and Computing in Applied Probability, Springer, vol. 24(4), pages 2287-2312, December.
    • Handle: RePEc:spr:metcap:v:24:y:2022:i:4:d:10.1007_s11009-022-09929-2
    DOI: 10.1007/s11009-022-09929-2
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    References listed on IDEAS

    as
    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, 2021. "Analysis of a population model with batch Markovian arrivals influenced by Markov arrival geometric catastrophes," Communications in Statistics - Theory and Methods, Taylor & Francis Journals, vol. 50(13), pages 3137-3158, July.
    3. F. P. Barbhuiya & Nitin Kumar & U. C. Gupta, 2019. "Batch Renewal Arrival Process Subject to Geometric Catastrophes," Methodology and Computing in Applied Probability, Springer, vol. 21(1), pages 69-83, March.
    4. Economou, Antonis, 2003. "On the control of a compound immigration process through total catastrophes," European Journal of Operational Research, Elsevier, vol. 147(3), pages 522-529, June.
    5. Economou, Antonis & Fakinos, Demetrios, 2003. "A continuous-time Markov chain under the influence of a regulating point process and applications in stochastic models with catastrophes," European Journal of Operational Research, Elsevier, vol. 149(3), pages 625-640, September.
    6. Epaminondas G. Kyriakidis & Theodosis D. Dimitrakos, 2005. "Computation of the Optimal Policy for the Control of a Compound Immigration Process through Total Catastrophes," Methodology and Computing in Applied Probability, Springer, vol. 7(1), pages 97-118, March.
    7. Kyriakidis, E. G., 1993. "A Markov decision algorithm for optimal pest control through uniform catastrophes," European Journal of Operational Research, Elsevier, vol. 64(1), pages 38-44, January.
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