Analyses of priority queues with server control (Q761343)

This paper is concerned with waiting time distributions in single server queues with several non-preemptive priority classes of customers. The arrival pattern is Poissonian for all classes. Such queueing systems have been analysed in the literature and this paper considers two models where the server may take a vacation when the queue is empty. In the first model the server scans the queue after a time period T once a busy period ends. Service is suspended for this interval of time. It starts after the scan if customers are present. If the queue is empty at the time of the scan another scan is made after a time T. The duration T of the server vacation is also assumed to be a random variable. The second model is a variation of the first where the server makes only one scan and stays alert till the start of the next busy period. This alert period is a waiting period for the server. At the end of every busy period a server vacation occurs. The author derives the Laplace-Stieltjes transforms of the waiting time and its first two moments for both these models. A recursive relation is given for the higher moments. The method makes use of level crossing analyses for regenerative processes.