Modeling and analysis of an infinite-buffer batch-arrival queue with batch-size-dependent service: [formula omitted

Abstract

This paper considers a single-server infinite-buffer queue with batch-arrival, batch-service, and generally distributed batch-size-dependent service time. Based on supplementary variable technique by considering supplementary variable as remaining service time of batch in service, a bivariate probability generating function, the entire spectrum of new contributions, of queue content and number in a served batch at departure epoch is derived. We extract the joint distribution of these quantities and present them in terms of the roots of the associated characteristic equation. In addition, departure epoch probabilities are utilized to obtain joint distribution at arbitrary epoch while the later one are employed to acquire joint distribution at pre-arrival epoch. Finally, some numerical results demonstrate the proposed procedure where occurrence of multiple roots is included. Moreover, some graphical representations show that batch-size-dependent service is more potent as compared to batch-size-independent service.