Analysis of Single Server Batch Service Queueing System Under Multiples Vacations with Bernoulli Schedule

Abstract

Consider a single server fixed batch service queueing system under multiples vacation with Bernoulli schedule in which the arrival rate λ follows a Poisson process; the service time follows an exponential distribution with parameter μ. The concept of Bernoulli schedule is incorporated in this paper with regular multiples vacation. Assume that the system initially contain 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 the server finds less than k customers in the queue, then the server goes for a multiples vacation of a length α which follows an exponential distribution. 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 with probability (1 – p) else the server goes for a vacation with probability p. This model is completely solved by constructing the generating function and Rouche’s theorem is applied and we have derived the closed form solutions for probabilities of number of customers in the queue during the server busy and in vacation. Further we are providing the analytical solutions for mean number of customers and variance of the system. Numerical studies have been done for various values of λ, µ, α, p and k and also various particular cases of this model have been discussed. Keywords : Single Server ,Batch Service , Bernoulli schedule , Multiple vacation, Steady state distribution.