Transient and Steady-State Analysis of an <i>M</i>/<i>PH</i>₂/1 Queue with Catastrophes

Abstract

In the paper, we consider the PH2-distribution, which is a particular case of the PH-distribution. In other words, The first service phase is exponentially distributed, and the service rate is μ. After the first service phase, the customer can to go away with probability p or continue the service with probability (1−p) and service rate μ′. We study an analysis of an M/PH2/1 queue model with catastrophes, which is regarded as a generalization of an M/M/1 queue model with catastrophes. Whenever a catastrophe happens, all customers will be cleaned up immediately, and the queuing system is empty. The customers arrive at the queuing system based on a Poisson process, and the total service duration has two phases. Transient probabilities and steady-state probabilities of this queuing system are considered using practical applications of the modified Bessel function of the first kind, the Laplace transform, and probability-generating function techniques. Moreover, some important performance measures are obtained in the system. Finally, numerical illustrations are used to discuss the system’s behavior, and conclusions and future directions of the model are given.

Keywords

• catastrophes
• Bessel function
• Laplace transform
• transient and steady-state probabilities
• performance measures