Extended analysis and computationally efficient results for the GI/Ma,b/1 queueing system

We consider an infinite waiting space GI/Ma,b/1 queueing system in which customers arrive according to a renewal process. The server serves the customers in batches under general bulk-service rule. The successive service times of batches are mutually independent and have a common exponential distribution. To obtain the queue-length distributions at prearrival and random epochs, we have used the displacement operator which assists to solve simultaneous non-homogeneous difference equations. An analytically simple and computationally efficient procedures have developed to compute the waiting-time distribution of an arrival customer. Our formula to determine the waiting-time distribution is also useful even when multiple poles occur in the Laplace-Stieltjes transform of the interarrival time distribution. We have determined closed-form analytical expression for the distribution of size of a service batch of an arrived customer. We also have derived the results of some particular well-known queueing models from our model. Some numerical results are provided in the form of tables for a variety of interarrival-time distributions to demonstrate the variation of performance measures of the system.