Queues with Instantaneous Feedback

Queueing problems in which a customer having received a unit of service, returns to the waiting line, under some decision rule, to receive another unit of service occur often in applications. Inspection procedures provide such a framework for units that must be reworked. A large class of such problems appears in computer modelling under the name of round-robin models and foreground-background models. In the present paper, such a system is referred to as a queue with instantaneous feedback. Throughout the text it is assumed that there are two independent Poisson arrival processes giving two types of customers. Each type of customer has its own service time distribution. The decision to feedback (to receive another unit of service) or not is based on the type of customer completing service. Conditions for the existence of a steady state queue length are found and some joint and marginal distribution of these queue lengths are given. Moreover it is shown that several earlier results in queueing and computer modelling can be obtained simply from the results given here. A particular case of the foreground-background model in computer systems analysis serves as an example.

A Characterization of M/G/1 Queues with Renewal Departure Processes

Burke [Burke, P. J. 1956. The output of a queueing system. Oper. Res. 4 699-704.] showed that the departure process from an M/M/1 queue with infinite capacity was in fact a Poisson process. Using methods from semi-Markov process theory, this paper extends this result by determining that the departure process from an M/G/1 queue is a renewal process if and only if the queue is in steady state and one of the following four conditions holds: (1) the queue is the null queue--the service times are all 0; (2) the queue has capacity (excluding the server) 0; (3) the queue has capacity 1 and the service times are constant (deterministic); or (4) the queue has infinite capacity and the service times are negatively exponentially distributed (M/M/1/\infty queue).