On Steady State Analysis of an Infinite Capacity Mᕽ/G⁽ᵃ˒ʸ⁾/1 Queue with Optional Service and Queue Length Dependent Single (Multiple) Vacation

Abstract

This paper analyzes an infinite buffer bulk arrival batch size dependent bulk ser- vice queue with server’s vacation. Customers enter the system in groups of random size following Poisson manner. Single server provides two kinds of services, first one is essen- tial for all the joining customers and is termed as first essential service (FES) and the second service is optional to the customers and is termed as second optional service (SOS). The server serves the customers in batches following versatile bulk service (VBS) rule in FES. At the end of FES, the entire batch of customers served in FES will either join SOS with certain probability or leave the system. While server is providing SOS, the FES can not be resumed. The service time distribution for FES and SOS both are generally distributed while FES time distribution is considered to be batch size dependent. At the end of a round of ser- vice (which includes FES and/or SOS), upon looking into the queue length, the server will decide to go for vacation (SV or MV) with general vacation time distribution which depends on the queue length at vacation initiation epoch. We analyze the model mathematically by using the supplementary variable technique (SVT), embedded Markov chain technique, and bivariate generating function technique. We obtained the joint probabilities of the queue length and server content at the service completion (arbitrary) epoch. We also obtained the joint probabilities of the queue length and vacation type at the vacation completion (arbi- trary) epoch. Several qualitative performance measures are obtained. Finally, Numerical results are also presented to see the behavior of the considered model.