Structured Markov chains solver

Бини, Дарио Андреа; Meini, Beatrice; Steffé, S.; Houdt, Benny Van

doi:10.1145/1190366.1190378

Cited by 21 publications

(16 citation statements)

References 19 publications

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“…From (5), the relation between h and m can be established as h = σ/ √ mκ. This means that the step size h is not a free variable once m is given (the total number of states is 2m + 1).…”

Section: The Discrete Counterpart: Dgmr(m)-mppmentioning

confidence: 99%

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Computational analysis of a Markovian queueing system with geometric mean-reverting arrival process

Miao

Lin

Chao

2016

Computers & Operations Research

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Section: The Discrete Counterpart: Dgmr(m)-mppmentioning

confidence: 99%

“…Since the basic iteration of (11) is less efficient, we test a number of advanced algorithms available in the literature. For these algorithms we refer to [4] for the theory and [5] [6] for the software package (Structural Markov Chain Solver, or SMC-Solver).…”

Section: A Comparison Of the Algorithms For Computing R In The Smc-somentioning

confidence: 99%

Computational analysis of a Markovian queueing system with geometric mean-reverting arrival process

Miao

Lin

Chao

2016

Computers & Operations Research

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“…Any algorithm, including the logarithmic or cyclic reduction algorithm, to solve this type of equation can be used to obtain Y 1 (Bini et al 2006). Rewriting (4) for V k , for k = 2, .…”

Section: Steady State Analysismentioning

confidence: 99%

“…However, to improve the flexibility of the framework, the range of the variable N t may depend on the rightmost integer of X t . Apart from discussing the ergodicity condition of such a Markov chain, we also develop an algorithm with quadratic convergence to determine its steady state vector, by showing that the matrices needed to characterize this vector can be obtained by solving a series of d quadratic matrix equations, for which various algorithms and software tools exist (e.g., Bini et al 2006).…”

mentioning

confidence: 99%

QBD Markov chains on binomial-like trees and its application to multilevel feedback queues

2007

Self Cite

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A matrix analytic paradigm, termed Quasi-Birth-Death Markov chains on binomial-like trees, is introduced and a quadratically converging algorithm to assess its steady state is presented. In a bivariate Markov chain {(X t , N t ), t ≥ 0}, the values of the variable X t are nodes of a binomial-like tree of order d, where the ith child has i children of its own. We demonstrate that it suffices to solve d quadratic matrix equations to yield the steady state vector, the form of which is matrix geometric. We apply this framework to analyze the multilevel feedback scheduling discipline, which forms an essential part in contemporary operating systems.Keywords QBD Markov chains · Tree-like processes · Matrix-analytic methods · Multilevel feedback queues Over the last few decades, broad classes of frequently encountered queueing models have been analyzed by matrix-analytic methods (Bini et al. 2005;Latouche and Ramaswami 1999;Neuts 1981Neuts , 1989. The embedded Markov chains in these models are two-dimensional generalizations of the classic M/G/1 and GI/M/1 queues, and quasi-birthdeath (QBD) processes. Matrix-analytic models include notions such as the Markovian arrival process (MAP) and the phase-type (PH) distribution, both in discrete and continuous time. Considerable efforts have been put into the development of efficient and numerically stable methods for their analysis (Bini et al. 2005). There is also an active search for accurate matching algorithms for both PH distributions and MAP arrivals when modeling communication systems (e.g., Asmussen et al. 1996;Bobbio et al. 2003;Breuer 2002;Horváth et al. 2005;Ryden 1996).One of the more recent paradigms within the field of matrix-analytic methods has been its generalization to discrete-time bivariate Markov chains {(X t , N t ), t ≥ 0}, in which the values of X t are the nodes of a d-ary tree (and N t takes integer values between 1 and h).

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“…Concerning applications, CR has become the method of choice in the solution of queuing problems from engineering and telecommunications where some effective implementations in Matlab and Fortran 95 have been designed by Bini, Meini, Steffé and Van Houdt [29,30] and are publically available.…”

mentioning

confidence: 99%

The cyclic reduction algorithm: from Poisson equation to stochastic processes and beyond

Бини

Meini

2008

Numer Algor

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Cyclic reduction is an algorithm invented by G. H. Golub and R. W. Hockney in the mid 1960s for solving linear systems related to the finite differences discretization of the Poisson equation over a rectangle. Among the algorithms of Gene Golub, it is one of the most versatile and powerful ever created. Recently, it has been applied to solve different problems from different applicative areas. In this paper we survey the main features of cyclic reduction, relate it to properties of analytic functions, recall its extension to solving more general finite and infinite linear systems, and different kinds of nonlinear matrix equations, including algebraic Riccati equations, with applications to Markov chains, queueing models and transport theory. Some new results concerning the convergence properties of cyclic reduction and its applicability are proved under very weak assumptions. New formulae for overcoming breakdown are provided.

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Structured Markov chains solver

Abstract: We analyze the problem of the numerical solution of structured Markov chains encountered in queuing models: we describe the main computational problems and present the most advanced algorithms currently available for their solutions.

Cited by 21 publications

References 19 publications

Computational analysis of a Markovian queueing system with geometric mean-reverting arrival process

Computational analysis of a Markovian queueing system with geometric mean-reverting arrival process

QBD Markov chains on binomial-like trees and its application to multilevel feedback queues

The cyclic reduction algorithm: from Poisson equation to stochastic processes and beyond

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