Bat algorithm for optimizing a working vacation queue with impatient clients and secondary service

This article investigates a Markovian working vacations queue with impatient clients and optional service. An arriving client can choose either to enter or not to enter the queue with a certain probability. Due to impatience, he may renege after joining the queue as per Poisson distribution. Two different kinds of services are offered by the server, namely, mandatory service which is required by all the clients entering the system and optional service which is opted by only few clients with certain probability after mandatory service. Just at the moment the system gets depleted, the server temporarily exits from the system for a short span called working vacations. Rather than shutting down completely throughout the working vacation duration, service is delivered at a different rate. The times between successive arrivals, vacation durations, service durations during mandatory service, during optional service and during working vacations are exponentially distributed and independent. Probability generating functions are used to evaluate the model’s steady-state probabilities. The optimum service rate during mandatory service is obtained using bat algorithm. A table and a few graphs provide the numerical interpretations of the model for the impact of various model parameters on the system characteristics.