Analysis of Balking & Probabilistically Modified Reneging in a M/M/1 Waiting Line under Multiple Differentiated Vacations

Abstract

This paper discusses and investigates transient behavior of a M/M/1 waiting line undergoing multiple differentiated vacations in conjunction with impatient customers- manifested in the form of balking and probabilistically modified reneging. Special function, namely, confluent hypergeometric function, continued fractions, technique of probability generating function and Laplace transform are our mathematical tools to obtain relevant system size probabilities. Time varying mean, variance, throughput and delay time are the performance measures for which we have obtained explicit expressions. Steady-state expressions for system size probabilities and performance measures are also obtained. This paper is a generalized form of a model of Mishra et al. (2022).