• ANALYSIS OF SINGLE SERVER BATCH SERVICE QUEUEING SYSTEM UNDER MULTIPLE VACATIONS WITH SECOND OPTIONAL SERVICE

Abstract

Consider a single server fixed batch service queueing system under multiple vacations with second optional service in which the arrival rate follows a Poisson process with parameter λ. The concept of Second optional service is introduced in this paper and server provides two types of services namely Essential Service and Second Optional Service. The essential service will be given first to all batches and the second optional service will be extended only if the batch demands optional service with probability p. Both essential and optional services follow an exponential distribution with parameters μ₁ and µ₂. Further assume that the length of the multiple vacations follows an exponential distribution with parameter α. Assume that the system initially contains k customers when the server enters in to the system and starts the essential service immediately in a batch of size k. After completion of an essential service, if the batch demands optional service with probability p then the server starts optional service and after completion of this optional service if he finds less than k customers in the queue, then the server goes for a multiple vacation of a length α. 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 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 for various values of λ, µ₁, µ_2, p, α and k and also various particular cases of this model have been discussed.