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.