Distribution Properties of Discrete Characteristics in M/G/1 and Gi/M/1 Queues

It has been shown by many authors that distribution properties of some characteristics in queues are inherited from those of the service times. For instances. Keilson (1978) showed that the length of a busy period in an M/G/1 queue has a completely monotone density if so does the service time, Shanthikumar (1988) proved that the waiting time distribution in a GI/G/1 queue is DFR if so is the service time distribution, etc. This paper studies such inheritance of distribution properties of discrete characteristics in M/G/1 and GI/M/1 queues. To do so, uniformly monotone discrete time Markov chains are first investigated. Various first passage times which are of independent interest for such Markov chains are considered. By showing that some characteristics in M/G/1 and GI/M/1 queues are expressible as the first passage times, distribution properties of those characteristics are studied.