Clearing Analysis on Phases: Exact Limiting Probabilities for Skip-free, Unidirectional, Quasi-birth-death Processes

TL;DR

The paper introduces the Clearing Analysis on Phases (CAP) method for exactly computing limiting probabilities in infinite, skip-free, unidirectional quasi-birth-death Markov chains, common in computing and manufacturing systems.

Contribution

The paper presents a novel analytical procedure, CAP, that efficiently computes exact limiting probabilities for a class of Markov chains with infinite levels and finite phases.

Findings

CAP provides exact limiting probabilities as linear combinations of scalar bases.

The weights are determined by solving a finite linear system.

Applicable to models in computing, service, and manufacturing systems.

Abstract

Many problems in computing, service, and manufacturing systems can be modeled via infinite repeating Markov chains with an infinite number of levels and a finite number of phases. Many such chains are quasi-birth-death processes (QBDs) with transitions that are skip-free in level, in that one can only transition between consecutive levels, and unidirectional in phase, in that one can only transition from lower-numbered phases to higher-numbered phases. We present a procedure, which we call Clearing Analysis on Phases (CAP), for determining the limiting probabilities of such Markov chains exactly. The CAP method yields the limiting probability of each state in the repeating portion of the chain as a linear combination of scalar bases raised to a power corresponding to the level of the state. The weights in these linear combinations can be determined by solving a finite system of linear equations.