On the Quasireversibility of a Multiclass Brownian Service Station

Abstract

The object of study in this paper is a Brownian model of a multiclass service station. Such Brownian models arise as heavy traffic limits of conventional queueing models in which several different types or classes of customers are processed through a common service facility. Assuming that the Brownian service station is initialized with its stationary distribution, four different model characteristics are shown to be equivalent, and the station is said to be quasireversible if these equivalent conditions pertain. Three of the four conditions characterize the vector departure process from the Brownian service station, and our definition of quasireversibility parallels that proposed by F. P. Kelly for conventional queueing models. The last of our four conditions is expressed directly in terms of primitive model parameters, so one may easily determine from basic data whether or not a Brownian station model is quasireversible. Rather than characterizing the complete vector of departure processes from a Brownian service station, we prove a more general theorem expressed in terms of arbitrary linear combinations of the departure processes; this yields a generalized notion of quasireversibility that will play an important role in future work. To be more specific, in a future paper on multiclass Brownian network models, it will be shown that there is an intimate relationship between product form stationary distributions and the generalized notion of quasireversibility developed here.

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