Analysis of Single Server Fixed Batch Service Queueing System under Multiple Vacation with Catastrophe

Abstract

Consider a single server fixed batch service queueing system under multiple vacation with a possibility of catastrophe in which the arrival rate ? follows a Poisson process and the service time follows an exponential distribution with parameter ?. Further we assume that the catastrophe occur at the rate of ? which follows a Poisson process and the length of time the server in vacation follows an exponential distribution with parameter ?.  Assume that the system initially contains k customers when the server enters in to the system and starts the service immediately in a batch of size k. After completion of a service, if he finds less than k customers in the queue, then the server goes for a multiple vacation of length ?. If there are more than k customers in the queue then the first k customers will be selected from the queue and service will be given as a batch. We are analyzing the possibility of catastrophe that is whenever a catastrophe occurs in the system, all the customers who are in the system will be completely destroyed and system becomes an empty and server goes for a multiple vacation. This model is completely solved by constructing the generating function  and we have derived the closed form solutions for probability of number of customers in the queue during the server busy and in vacation. Further we are providing the analytical solution for mean number of customers and variance of the system. Numerical studies have been done for analysis of mean and variance of number of customers in the system for various values of ?, µ, ? and k and also various particular cases of this model have been discussed.

Keywords: Single server queue , Fixed batch service , Catastrophe, Multiple vacation, Steady state distribution