Alternative approach based on roots for computing the stationary queue-length distributions in GIX/M(1,b)/1 single working vacation queue

Abstract

The purpose of this paper is to present an alternative algorithm for computing the stationary queue-length and system-length distributions of a single working vacation queue with renewal input batch arrival and exponential holding times. Here we assume that a group of customers arrives into the system, and they are served in batches not exceeding a specific number b. Because of batch arrival, the transition probability matrix of the corresponding embedded Markov chain for the working vacation queue has no skip-free-to-the-right property. Without considering whether the transition probability matrix has a special block structure, through the calculation of roots of the associated characteristic equation of the generating function of queue-length distribution immediately before batch arrival, we suggest a procedure to obtain the steady-state distributions of the number of customers in the queue at different epochs. Furthermore, we present the analytic results for the sojourn time of an arbitrary customer in a batch by utilizing the queue-length distribution at the pre-arrival epoch. Finally, various examples are provided to show the applicability of the numerical algorithm.