A Computational Approach for Evaluating the Transient Behaviour of M/M (a, b)/1 Bulk Service Queueing System with Working Vacation

Abstract

A new computational technique is used to evaluate the Transient behaviour of Single Server Bulk Service Queueing System with Working Vacation with arrival rate \(\lambda\) which follows a Poisson process and the service will be in bulk. In this model the server provides two types of services namely normal service and lower service. The normal service time follows an exponential distribution with parameter \(\mu\)1. The lower service rate follows an exponential distribution with parameter \(\mu\)2. The vacation time follows an exponential distribution with parameter \(\alpha\). According to Neuts, the server begins service only when a minimum of ‘a’ customers in the waiting room and a maximum service capacity is ‘b’. An infinitesimal generator matrix is formed for all transitions. Time dependent solutions and Steady state solutions are acquired by using Cayley Hamilton theorem. Numerical studies have been done for Time dependent average number of customers in the queue, Transient probabilities of server in vacation and server busy for several values of t, \(\lambda\), µ1, \(\mu\)2, \(\alpha\), a and b. In this model we have provided transient probability distribution of number of customers in the queue at time t and also time dependent system measures.