Quasi-Birth-and-Death Processes with Rational Arrival Process Components

Nigel Geoffrey Bean · Bo Friis Nielsen

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Advanced Queuing Theory Analysis Probability and Statistical Research Probability and Risk Models

Abstract

This paper introduces the concept of a Quasi-Birth-and-Death process (QBD) with Rational Arrival Process (RAP) components. We use the physical interpretation of the prediction process of the RAP, developed by Asmussen and Bladt, and develop an analysis that parallels the analysis of a traditional QBD. Further, we present an algorithm for the numerical evaluation of the matrix G. As an example, we consider two queues where the arrival process and the sequence of service times are taken from two dependent RAPs, that are not Markovian Arrival Processes.

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References

1999Introduction to Matrix Analytic Methods in Stochastic Modeling 1984Matrix-Geometric Solutions to Stochastic Models 2005The BMAP/G/1 queue: A tutorial 1993Algorithms for Infinite Markov Chains with Repeating Columns 2003Matrix‐Exponential Distributions: Calculus and Interpretations via Flows 1999Point processes with finite-dimensional conditional probabilities

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2011On minimal representations of Rational Arrival Processes 2008Q-MAM: A Tool for Solving Infinite Queues using Matrix-Analytic Methods

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