Technical Note—The Expected Remaining Service Time in a Single Server Queue

For the G/G/1 queueing system, let (p n ), n = 0, 1, 2, …, and (r n ), n = 0, 1, 2, … be the limiting probability distributions of the number of customers in the system, when the system is considered “at any time” and when it is considered at arrival epochs respectively. Also let b n (n = 1, 2, …) be the mean remaining duration of the service in progress at the epoch of an arrival which finds n customers in the system. In this note, a relation between the sequences (p n ), (r n ) and (b n ) is given and it is used to provide alternative derivations for two well-known results in the theory of queues.