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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