Complete Analysis of M/Gr(a,b)/1/N Queue with Second Optional Service

Abstract

The current article is about a finite buffer, group size dependent bulk service queue, in which a single server renders two type of services, first essential service (FES) and second optional service (SoS). Customers arrive at the system in the Poisson fashion. Service is rendered by a server in groups following ‘general bulk service’ (GBS) rule on first come first serve (FCFS) basis for FES. In this article, we allowed a part of a group, served in FES, to join SoS in group following binomial law. The service time distribution is considered to be generally distributed and dependent on the group size under service for both the cases, FES and SoS. The mathematical analysis is performed using the supplementary variable technique (SVT) and the embedded Markov chain technique to obtain the steady state, departure epoch and arbitrary epoch joint probabilities of the count of customers in the queue and with the server when the server is in FES as well as in SoS. Finally, various numerical studies are presented to show the behaviour of the key efficiency metrics, which eventually shows the importance of the current study. © The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature 2024.