On the departure processes of M/M/1/N and GI/G/1/N queues

The purpose of this note is to point out the connection between the invariance property of M/M/1 and GI/G/1 queues (which has been reported in several papers) and the interchangeability and reversibility properties of tandem queues. This enables us to gain new insights for both problems and obtain stronger invariance results for M/M/1, GI/G/1, as well as loss systems M/M/1/N, GI/G/1/N and tandem systems.

Expected number of departures in M/M/1 and G/G/1 queues

For an initially empty M/M/1 queue, it is shown that the transform of the expectation of the number of departures in the interval (0, t] is invariant under an interchange of arrival and service rates. However, in the GI/G/1 queue with an initial single customer, the corresponding transform does not have this symmetric property.

On the use of a fundamental identity in the theory of semi-Markov queues

In the paper a single server semi-Markov queue is considered. The analysis is based on a fundamental matrix identity due to H. D. Miller. A natural method for the solution of semi-Markov queues is indicated; use is also made of a duality relation, which is discussed in the paper.