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Simple derivations of the invariance relations and their applications

In the literature, various methods have been studied for obtaining invariance relations, for example, L = λW (Little's formula), in queueing models. Recently, it has become known that the theory of point processes provides a unified approach to them (cf. Franken (1976), König et al. (1978), Miyazawa (1979)). This paper is also based on that theory, and we derive a general formula from the inversion formula of point processes. It is shown that this leads to a simple proof for invariance relations in G/G/c queues. Using these results, we discuss a condition for the distribution of the waiting time vector of a G/G/c queue to be identical with that of an M/G/c queue.