Decay rates for quasi-birth-and-death processes with countably many phases and tridiagonal block generators

Allan Motyer · Peter G. Taylor

34 citations0/0 files storedAdvances in Applied Probability

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Advanced Queuing Theory Analysis Probability and Statistical Research Stochastic processes and statistical mechanics

Abstract

We consider the class of level-independent quasi-birth-and-death (QBD) processes that have countably many phases and generator matrices with tridiagonal blocks that are themselves tridiagonal and phase independent. We derive simple conditions for possible decay rates of the stationary distribution of the ‘level’ process. It may be possible to obtain decay rates satisfying these conditions by varying only the transition structure at level 0. Our results generalize those of Kroese, Scheinhardt, and Taylor, who studied in detail a particular example, the tandem Jackson network, from the class of QBD processes studied here. The conditions derived here are applied to three practical examples.

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References

1999Introduction to Matrix Analytic Methods in Stochastic Modeling 1984Matrix-Geometric Solutions to Stochastic Models 1985Matrix-geometric solutions in stochastic models 1984Two Parallel Queues Created by Arrivals with Two Demands I 1996Spectral analysis ofM/G/1 andG/M/1 type Markov chains 2001Join the shortest queue: stability and exact asymptotics 1982Operator-geometric stationary distributions for markov chains, with application to queueing models 1996Some properties of the rate perators in level dependent uasi-birth-and-death processes with countable number of phases

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