Markovian Arrival Process Subject to Renewal Generated Binomial Catastrophes

• Published in: Methodology and computing in applied probability Vol. 24; no. 4; pp. 2287 - 2312
• Main Authors: Kumar, Nitin, Gupta, Umesh Chandra
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.12.2022; Springer Nature B.V
• Subjects: Business and Management; Catastrophes; Economics; Electrical Engineering; Life Sciences; Mathematics and Statistics; Size distribution; State vectors; Statistics; Catastrophes; 92D25; Renewal process; MSC 65C40; Binomial distribution; Markovian arrival process; Population; MSC 60J22
• ISSN: 1387-5841; 1573-7713
• DOI: 10.1007/s11009-022-09929-2

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.