A Finite Capacity Queue with Nonrenewal Input and Exponential Dynamic Group Services

In this article we consider a finite capacity queuing model in which jobs (or customers) arrive according to a nonrenewal process. The jobs are processed by a single server in groups of varying size, between a predetermined threshold value and the buffer size. A dynamic probability rule is associated with the service mechanism. The services are assumed to be exponential whose parameter may depend on the group size. The steady-state analysis of this queuing model is performed using Markov chain theory. It is shown that the idle period of the server and the stationary waiting time of an admitted job are of phase type and that the departure process can be modeled using a versatile Markovian point process. Efficient algorithms for computing various performance measures such as throughput, mean number served, job overflow probability, server idle probability, the stationary mean waiting time, and the stationary mean idle time of the server, useful in qualitative and quantitative interpretations are developed. Some illustrative numerical examples are discussed.