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April, 1992 Phase-Type Representations in Random Walk and Queueing Problems
Soren Asmussen
Ann. Probab. 20(2): 772-789 (April, 1992). DOI: 10.1214/aop/1176989805


The distributions of random walk quantities like ascending ladder heights and the maximum are shown to be phase-type provided that the generic random walk increment $X$ has difference structure $X = U - T$ with $U$ phase-type, or the one-sided assumption of $X_+$ being phase-type is imposed. As a corollary, it follows that the stationary waiting time in a GI/PH/1 queue with phase-type service times is again phase-type. The phase-type representations are characterized in terms of the intensity matrix $\mathbf{Q}$ of a certain Markov jump process associated with the random walk. From an algorithmic point of view, the fundamental step is the iterative solution of a fix-point problem $\mathbf{Q} = \psi(\mathbf{Q})$, and using a coupling argument it is shown that the iteration typically converges geometrically fast. Also, a variant of the classical approach based upon Rouche's theorem and root-finding in the complex plane is derived, and the relation between the approaches is shown to be that $\mathbf{Q}$ has the Rouche roots as its set of eigenvalues.


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Soren Asmussen. "Phase-Type Representations in Random Walk and Queueing Problems." Ann. Probab. 20 (2) 772 - 789, April, 1992.


Published: April, 1992
First available in Project Euclid: 19 April 2007

zbMATH: 0755.60049
MathSciNet: MR1159573
Digital Object Identifier: 10.1214/aop/1176989805

Primary: 60J15
Secondary: 60J05 , 60K25

Keywords: coupling , GI/PH/1 queue , ladder height distribution , Markov jump process , nonlinear matrix iteration , PH/G/1 queue , phase-type distribution , Random walk , uniformization , Wiener-Hopf factorization

Rights: Copyright © 1992 Institute of Mathematical Statistics

Vol.20 • No. 2 • April, 1992
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