Stability for Fluid Queues: Characteristic Inequalities

Maria Govorun · Guy Latouche · Marie-Ange Remiche

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Advanced Queuing Theory Analysis Stochastic processes and statistical mechanics Probability and Risk Models

Abstract

We explore the conditions under which a fluid queue is positive recurrent, null recurrent, or transient. We consider the simple system where fluid rates are independent of the level, and exploit the similarity found by Ramaswami between fluid queues and Quasi-Birth-and-Death (QBD) processes. Inspired by the Wiener-Hopf factorization obtained by Rogers for a key matrix, and by an earlier work by Latouche and Taylor on drift conditions for QBDs, we determine in this note two pairs of eigenvectors which combine to give four characteristic inequalities. This is illustrated on a simple example.

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References

1999Introduction to Matrix Analytic Methods in Stochastic Modeling 1994Fluid Models in Queueing Theory and Wiener-Hopf Factorization of Markov Chains 1996Spectral analysis ofM/G/1 andG/M/1 type Markov chains 1978Markov chains with applications in queueing theory, which have a matrix-geometric invariant probability vector 2005Matrix-analytic methods for fluid queues with finite buffers 2003Fluid Flow Models and Queues—A Connection by Stochastic Coupling 1990Probabilistic interpretations of some duality results for the matrix paradigms in queueing theory 1998Matrix-geometric invariant measures for G/M/l type Markov chains

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