Analysis of a geometric catastrophe model with discrete-time batch renewal arrival process

Abstract

Discrete-time stochastic models have been extensively studied since the past few decades due to its huge application in areas of computer-communication networks and telecommunication systems. However, the growing use of the internet often makes these systems vulnerable to catastrophe/ virus attack leading to the removal of some or all the elements from the system. Taking note of this, we consider a discrete-time model where the population (in the form of packets, data, etc.) is assumed to grow in batches according to renewal process and is likely to be affected by catastrophes which occur according to Bernoulli process. The catastrophes have a sequential impact on the population and it destroys each individual at a time with probabilityp. This destruction process stops as soon as an individual survives or when the entire population becomes extinct. We analyze both late and early arrival systems independently and using supplementary variable and shift operator methods obtain explicit expressions of steady-state population size distribution at pre-arrival and arbitrary epochs. We deduce some important performance measures and further show that for both the systems the tail probabilities at pre-arrival epoch can be well approximated using a single root of the characteristic equation. In order to illustrate the computational procedure, we present some numerical results and also investigate the change in the behavior of the model with the change in parameter values.