Stationary queue-length and waiting-time distributions in single-server feedback queues

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Abstract
For M/GI/1/∞ queues with instantaneous Bernoulli feedback time- and customer-stationary characteristics of the number of customers in the system and of the waiting time are investigated. Customer-stationary characteristics are thereby obtained describing the behaviour of the queueing processes, for example, at arrival epochs, at feedback epochs, and at times at which an arbitrary (arriving or fed-back) customer enters the waiting room. The method used to obtain these characteristics consists of simple relationships between them and the time-stationary distribution of the number of customers in the system at an arbitrary point in time. The latter is obtained from the wellknown Pollaczek–Khinchine formula for M/GI/1/∞ queues without feedback.