On single server batch arrival queueing system with balking, three types of heterogeneous service and Bernoulli schedule server vacation

Abstract

This paper investigates a batch arrival queueing system in which customers arrives at the system in a Poisson stream following a compound Poisson process and the system has a single server providing three types of general heterogeneous services. At the beginning of each service, a customer is allowed to choose any one of the three services and as soon as a service of any type gets completed, the server may take a vacation or may continue staying in the system. The vacation time is assumed to follow a general (arbitrary) distribution and the server vacation is based on Bernoulli schedule under a single vacation policy. During the server vacation period, impatient customers are assumed to balk. This paper described the model as a bivariate Markov chain and employed the supplementary variable technique to find closed-form solutions of the steady state probability generating function of number of customers, the steady state probabilities of various states of the system, the average queue size, the average system size, and the average waiting time in the queue as well as the average waiting time in the system. Further, some interesting special cases of the model are also derived.

Keywords Batch Arrivals. Queueing System. Balking. Heterogeneous types of Service. Bernoulli schedule server vacation. Bivariate Markov Processes.

MSC2020-Mathematics Subject Classification 34B07, 60G05, 62E15