Analysis of a queue-length-dependent vacation queue with bulk service, N-policy, set-up time and cost optimization

Abstract

Due to the extensive applications of bulk service vacation queues in manufacturing industries, inventory systems, wireless sensor networks for deducing energy consumption etc., in this article, we analyze the steady-state behavior of an infinite-buffer group arrival bulk service queue with vacation scenario, set-up time and $N$-threshold policy. Here the customers arrive according to the compound Poisson process and the server originates the service process with minimum ‘$a$’ customers and can give service to maximum ‘$b$’ customers at a time. We adopt batch-size-dependent service time as well as queue-length-dependent vacation duration which improve the system’s performance significantly. The $N$-threshold policy is proposed to awaken the server from a vacation/dormant state where the service station starts the set-up procedure after the accumulation of pre-decided ‘$N$’ customers. Using the supplementary variable technique, firstly, we derive the set of system equations in the steady-state. After that, we obtain the bivariate probability generating functions (pgfs) of queue content and size of the departing batch, the queue content and type of vacation taken by the server at vacation completion epoch and also the single pgf of queue content at the end of set-up time. We extract the joint distribution from those generating functions using the roots method and derive a simple algebraic relation between the probabilities at departure and arbitrary epoch. We also provide assorted numerical results to validate our proposed methodology and obtained theoretical results. The impact of the system parameters on the performance measures is presented through tables and graphs. Finally, a cost optimization function is provided for the benefit of system designers.