Markovian Arrival Process Subject to Renewal Generated Binomial Catastrophes
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 prob...
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| Published in | Methodology and computing in applied probability Vol. 24; no. 4; pp. 2287 - 2312 |
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| Main Authors | , |
| Format | Journal Article |
| Language | English |
| Published |
New York
Springer US
01.12.2022
Springer Nature B.V |
| Subjects | |
| Online Access | Get full text |
| ISSN | 1387-5841 1573-7713 |
| DOI | 10.1007/s11009-022-09929-2 |
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| 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
, 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. |
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| AbstractList | 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, 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. 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 , 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. |
| Author | Gupta, Umesh Chandra Kumar, Nitin |
| Author_xml | – sequence: 1 givenname: Nitin surname: Kumar fullname: Kumar, Nitin email: nitinkumar7276@gmail.com organization: Department of Mathematics, Indian Institute of Technology – sequence: 2 givenname: Umesh Chandra surname: Gupta fullname: Gupta, Umesh Chandra organization: Department of Mathematics, Indian Institute of Technology |
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| CitedBy_id | crossref_primary_10_1080_03610926_2023_2261059 crossref_primary_10_1080_16843703_2024_2313241 crossref_primary_10_1088_1742_5468_acf8bc |
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| Keywords | Catastrophes 92D25 Renewal process MSC 65C40 Binomial distribution Markovian arrival process Population MSC 60J22 |
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| SubjectTerms | Business and Management Catastrophes Economics Electrical Engineering Life Sciences Mathematics and Statistics Size distribution State vectors Statistics |
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| Title | Markovian Arrival Process Subject to Renewal Generated Binomial Catastrophes |
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