Infinite Buffer Queueing Model with Time-phased Arrivals

A single-server queueing model with infinite buffer and batch arrival of customers is considered. In contrast to the standard batch arrival when a whole batch arrives into the system at one epoch, we assume that the customers of an accepted batch arrive one by one in exponentially distributed times. Service time is exponentially distributed. Flow of batches is the stationary Poisson arrival process. Batch size distribution is geometric. The number of batches, which can be admitted into the system simultaneously, is subject of control. The problem of maximizing the throughput of the system under the fixed value of the upper bound for admitted batch average sojourn time is considered. Analysis of the joint distribution of the number of batches and customers in the system and sojourn time distribution is implemented by means of the matrix technique and method of catastrophes.