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Matrix Geometric Approach for Random Walks Matrix geometric approach for random walks Citation for published version (APA): Kapodistria, S., & Palmowski, Z. B. (2017). Matrix geometric approach for random walks: stability condition and equilibrium distribution. Stochastic Models, 33(4), 572-597. https://doi.org/10.1080/15326349.2017.1359096 Document license: CC BY-NC-ND DOI: 10.1080/15326349.2017.1359096 Document status and date: Published: 02/10/2017 Document Version: Publisher’s PDF, also known as Version of Record (includes final page, issue and volume numbers) Please check the document version of this publication: • A submitted manuscript is the version of the article upon submission and before peer-review. There can be important differences between the submitted version and the official published version of record. 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If the publication is distributed under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license above, please follow below link for the End User Agreement: www.tue.nl/taverne Take down policy If you believe that this document breaches copyright please contact us at: [email protected] providing details and we will investigate your claim. Download date: 23. Sep. 2021 Stochastic Models ISSN: 1532-6349 (Print) 1532-4214 (Online) Journal homepage: http://www.tandfonline.com/loi/lstm20 Matrix geometric approach for random walks: Stability condition and equilibrium distribution Stella Kapodistria & Zbigniew Palmowski To cite this article: Stella Kapodistria & Zbigniew Palmowski (2017) Matrix geometric approach for random walks: Stability condition and equilibrium distribution, Stochastic Models, 33:4, 572-597, DOI: 10.1080/15326349.2017.1359096 To link to this article: https://doi.org/10.1080/15326349.2017.1359096 © 2017 The Author(s). Published with license by Taylor & Francis© Stella Kapodistria and Zbigniew Palmowski Published online: 31 Aug 2017. Submit your article to this journal Article views: 145 View related articles View Crossmark data Full Terms & Conditions of access and use can be found at http://www.tandfonline.com/action/journalInformation?journalCode=lstm20 Download by: [Eindhoven University of Technology] Date: 13 December 2017, At: 06:14 STOCHASTIC MODELS , VOL. , NO. , – https://doi.org/./.. Matrix geometric approach for random walks: Stability condition and equilibrium distribution Stella Kapodistriaa and Zbigniew Palmowskib aDepartment of Mathematics and Computer Science, Eindhoven University of Technology, Eindhoven, The Netherlands; bFaculty of Pure and Applied Mathematics, Wroclaw University of Science and Technology, Wroclaw, Poland ABSTRACT ARTICLE HISTORY In this paper, we analyze a sub-class of two-dimensional homo- Received April geneous nearest neighbor (simple) random walk restricted on the Accepted July lattice using the matrix geometric approach. In particular, we first KEYWORDS present an alternative approach for the calculation of the stability [30] Boundary value problem condition, extending the result of Neuts drift conditions and method; compensation connecting it with the result of Fayolle et al. which is based on approach; equilibrium Lyapunov functions.[13] Furthermore, we consider the sub-class distribution; matrix of random walks with equilibrium distributions given as series of geometric approach; random product forms and, for this class of random walks, we calculate walks; spectrum; stability the eigenvalues and the corresponding eigenvectors of the infi- condition R nite matrix appearing in the matrix geometric approach. This MATHEMATICS SUBJECT result is obtained by connecting and extending three existing CLASSIFICATION approaches available for such an analysis: the matrix geometric K; B approach, the compensation approach and the boundary value problem method. In this paper, we also present the spectral prop- erties of the infinite matrix R. 1. Introduction The objective of this work is to demonstrate how to obtain the stability condition and the equilibrium distribution of the state of a two-dimensional homogeneous nearest neighbor (simple) random walk restricted on the lattice using its underly- ing Quasi-Birth–Death (QBD) structure and the matrix geometric approach. This type of random walk can be modeled as a QBD process with the characteris- tic that both the levels and the phases are countably infinite. Then, based on the Downloaded by [Eindhoven University of Technology] at 06:14 13 December 2017 matrix geometric approach, if πn = (πn,0 πn,1 ···) denotes the vector of the equi- librium distribution at level n, n = 0, 1,...,itisknownthatπn+1 = πnR, n ≥ 1. Thisisaverywell-knownresult,butthecomplexityofthesolutionliesinthe calculation of the infinite-dimension matrix R. In this paper, we investigate how the matrix geometric approach can be extended to the case of countably infinite CONTACT Stella Kapodistria [email protected] Department of Mathematics and Computer Science, Eindhoven University of Technology, P.O. Box , MB Eindhoven, The Netherlands. Color versions of one or more of the figures in the article can be found online at www.tandfonline.com/lstm. Published with license by Taylor & Francis © Stella Kapodistria and Zbigniew Palmowski. This is an Open Access article distributed under the terms of the Creative Commons Attribution-NonCommercial-NoDerivatives License (http://creativecommons.org/licenses/by-nc-nd/./), which permits non-commercial re-use, distribution, and reproduction in any medium, provided the original work is properly cited, and is not altered, transformed, or built upon in any way. STOCHASTIC MODELS 573 phases and we discuss the challenges that arise with such an extension. Further- more, we propose an approach for the calculation of the eigenvalues and eigenvec- tors of matrix R, that complements the existing approaches for this type of random walks. Moreover, this approach can be numerically used for the approximation of the matrix R by considering spectral truncation instead of the usual state space truncation. 1.1. Literature overview ThemainbodyofliteratureonthetopicofcountablyinfinitephasesofQBDsis devoted to either stochastic processes in which the matrix R has a simple structure or to asymptotic results concerning the decay rate. Regarding QBDs with a simple structure, the literature is mainly devoted to stochastic processes for which the rate matrix has a simple property, see, e.g.,[11] and the references therein, or to stochastic process with a special structure in the allowed transitions, see, e.g.,[19,20] and the references therein. In Ref.[11],theauthors consider an infinite rate matrix R that can be written as the product of a vector col- umn times a row vector. This simple structure permits, under the assumptions of homogeneity and irreducibility, to show that the stationary distributions of these processes have a product form structure as a function of the level. Additionally, they apply their results in the case of the Cox(k)/MY /1 queue. In Refs.[19,20] and the references therein, the authors investigate stochastic processes with a special struc- ture regarding the allowed transitions satisfying the so-called successive lumpability property. This class of Markov chains is specified through its property of cal- culating the stationary probabilities of the system by successively computing the stationary probabilities of a propitiously constructed sequence of Markov chains. Each of the latter chains has a (typically much) smaller state space and this yields significant computational improvements. Such models are a special class of the GI/M/1 or the M/G/1 type of QBD processes. In Refs.[19,20],theauthorsdiscuss how the results for discrete-time Markov chains extend to semi-Markov processes and continuous-time Markov processes. Furthermore, in Ref.[20], the authors com- pare the successive lumping methodology developed with the lattice path count- ing approach[28] for the calculation of the rate matrix of a queueing model. These two methodologies are compared both in terms of applicability requirements and Downloaded by [Eindhoven University of Technology] at 06:14 13 December 2017 numerical complexity by analyzing their performance for some classical queueing models. Their main findings are (i) when both methods are applicable, the suc- cessive lumping-based algorithms outperform the lattice path counting algorithm, (ii) the successive lumping algorithms, contrary to the lattice path counting algo- rithm, include a method to compute the steady-state distribution using this rate matrix. Regarding the work devoted on the decay rate of QBDs with
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