Performance Characteristics and Optimization in Bulk-Service Queues: M/G(a,Y)/1/∞

This paper studies a single-server queue with Poisson arrivals and an infinite buffer, where customers arrive individually according to a Poisson process and are served in batches following a flexible batch-service rule, the (a,Y)-rule. A service is initiated only when at least a customer is present, at which time a maximum batch size is chosen according to the distribution of Y. The batch service times follow a general distribution and are independent of each other and the arrival process. The probability generating function (pgf) of the queue-length distributions at both an arbitrary time and just after a batch-service completion are derived using an embedded Markov chain and a level-crossing argument. The Laplace-Stieltjes transform (LST) of the waiting-time distribution for an arbitrary customer in the queue is derived. Tail probabilities for the queue-length distribution at a batch-service completion epoch have been derived when service times follow a power-law distribution. We obtain the probability density function of the sojourn-time distribution for an arbitrary customer. Finally, we study a reward maximization problem with two decision variables: the customer arrival rate and the reward for serving a customer using particle swarm optimization (PSO).