A renewal input single and batch service queues with accessibility to batches

This paper presents a renewal input single and batch service infinite (finite) buffer queueing system with accessibility to the batches. The inter-arrival times of successive arrivals are assumed to be independent and arbitrarily distributed. The customers are served by a single-server either one at a time or in batches with accessibility to the batches depending upon the number of customers present in the system. The service times are assumed to be independent and exponentially distributed. We present a recursive method, using the supplementary variable technique and treating the supplementary variable as the remaining inter-arrival time, to obtain the steady-state probability distribution of the number of customers in the system/queue. A cost model is developed to determine the optimum service rates which minimizes the total expected cost for assumed numerical values of system parameters. Various performance measures are provided based on this analysis.