SUFFICIENT CONDITIONS FOR A GEOMETRIC TAIL IN A QBD PROCESS WITH MANY COUNTABLE LEVELS AND PHASES

Lani Haque · Yiqiang Q. Zhao · Liming Liu

37 citations0/0 files storedStochastic Models

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

Abstract

In this paper, we present sufficient conditions, under which the stationary probability vector of a QBD process with both infinite levels and phases decays geometrically, characterized by the convergence norm η and the 1/η-left-invariant vector x of the rate matrix R. We also present a method to compute η and x based on spectral properties of the censored matrix of a matrix function constructed with the repeating blocks of the transition matrix of the QBD process. What makes this method attractive is its simplicity; finding η reduces to determining the zeros of a polynomial. We demonstrate the application of our method through a few interesting examples.

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References

1984Matrix-Geometric Solutions to Stochastic Models 1985Matrix-geometric solutions in stochastic models 1984Two Parallel Queues Created by Arrivals with Two Demands I 2001Join the shortest queue: stability and exact asymptotics 1981Computation of Steady-State Probabilities for M/M/1 Priority Queues 1996Some properties of the rate perators in level dependent uasi-birth-and-death processes with countable number of phases 2004The stationary tail asymptotics in the GI/G/1-type queue with countably many background states 2001Queueing Models with Multiple Waiting Lines

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