Cost and sensitivity analysis of transient Markovian queue with waiting server, heterogeneous vacation policy and reneged customers

The unavailability of server during complete vacation (CV) and the reduced service pace during working vacation (WV) may pose a critical challenge to the service and manufacturing organizations. This study focuses on analyzing the performance of a queueing system operating under a hybrid vacation policy while considering customer impatience. The server has the option to choose either CV or WV, immediately after finishing all the jobs. During the WV period, the server renders service in slow pace as such some impatient customers may abandon the system before served. Moreover, the joining rate of the customers is state dependent i.e. the arriving customers joins the system with different rates for the different status of the server. The transient analytical expressions for the queue size distribution using continued fraction (CF) and probability generating function (PGF) methods are established which are further used to formulate various performance metrics. The practical applicability of the system is demonstrated through numerical simulation Moreover, cost optimization is performed to obtain minimum cost and corresponding optimal service rate.