On the Analysis of a M/M/1 Queue with Bulk Services

The analysis of a queue that serves batches of customers with a novel service policy is presented in this paper showing that a closed steady state distribution of the number of customers in the queue can be derived for a very general setting of its parameters. Two output processes are flowing out of this queue depending on the fact that single customers are removed from the queue upon completion of a service, or bulks of customers of a fixed size are departing from the queue at service's completion. Based on the expression of the steady state distribution which assumes a Poisson arrival process, specific parameter configurations are identified that make the queue quasi-reversible depending on the output process of interest. Since quasi-reversible queues are very important in the context of product form queueing networks, these results have relevant impacts on their own as well as when considered as the basis for possible computationally efficient approximations. Comparisons among the results obtained for different parameter settings are provided using both stochastic order arguments and numerical experiments. Future research directions are proposed considering also the many practical applications of this model ranging from flexible manufacturing, to logistics, to transportation systems.