An M/M/1 queue with single vacation in random environment and multiple vacations in deterministic environment

Abstract

The model under consideration is a single server multiple vacation queueing system where the vacation is classified into two categories namely type I vacation and type II vacation. The server opts type I vacation following non-empty busy period of providing service to at the minimum of one customer. On returning from type I vacation if the server finds the system empty, it goes for type II vacation. In type I vacation, depending on the environment, there are n distinct kinds of vacations. All types of vacation are vulnerable to Interruption. Each type of vacation can be interrupted when the number of customers in the system reaches predefined thresholds, where each vacation has a different threshold. The long run system probabilities, mean and variance of the number of customers in the system, etc. are computed. Using Little’s formula the expression for waiting time is obtained and numerically illustrated. An optimization problem is discussed with numerical illustration.