Heterogeneous multiserver queues with a general input

In this paper, we derive equilibrium solutions for certain queueing systems which can be formulated as two-dimensional Markov chains. The first dimension corresponds to the level, and the second dimension to the phase. There are infinitely many levels, but only a finite number of phases. It is assumed that for almost all states, a consistent change in the level will not affect the transition rates. The proposed solution is based on difference equations defined on vectors, an approach which requires the determination of the null values and null vectors of a lambda-matrix. Methods to find these will be discussed. Once the null values and null vectors have been determined, one uses the boundary conditions to find the equilibrium probabilities. Complications arise if a certain matrix is singular. For this case, we present transformations to make the matrix in question non-singular without affecting the solution vector. The determination of null values and the solution of the boundary conditions will be demonstrated by two models, namely the one server queue in a randomly changing environments, and the shortest queue problem. However, the method is applicable in many other situations.
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