A Renewal Generated Geometric Catastrophe Model with Discrete-Time Markovian Arrival Process

Abstract

Any event that results in sudden change of the normal functioning of a system may be thought of as a catastrophe. Stochastic processes involving catastrophes have very rich application in modeling of a dynamic population in areas of ecology, marketing, queueing theory, etc. When the size of the population reduces abruptly as a whole, due to a catastrophe, it is termed as the total catastrophe. However, in many real-life circumstances the catastrophes have a mild influence on the population and have a sequential effect on the individuals. This paper presents a discrete-time catastrophic model in which the catastrophes occur according to renewal process, and it eliminates each individual of the population in sequential order with probability p until the one individual survives or the entire population wipes out. The individuals arrive according to the discrete-time Markovian arrival process. Using the supplementary variable technique, we obtain the steady-state vector generating function (VGF) of the population size at various epochs. Further using the inversion method of VGF, the population size distribution is expressed in terms of the roots of the associated characteristic equation. We further give a detailed computational procedure by considering inter-catastrophe time distributions as discrete phase-type as well as arbitrary. Finally, a few numerical results in form of tables and graphs are presented. Moreover, the impact of the correlation of arrival process on the mean population size is also investigated.