A comparative analysis of the successive lumping and the lattice path counting algorithms
TL;DR: This paper provides a comparison of the successive lumping (SL) methodology developed in Katehakis et al. (2015) with the popular lattice path counting (Mohanty (1979)) in obtaining rate matrices for queueing models, satisfying the specific quasi birth and death structure as in Van Leeuwaarden et al (2009).
Abstract: This paper provides a comparison of the successive lumping (SL) methodology developed in Katehakis et al. (2015) with the popular lattice path counting (Mohanty (1979)) in obtaining rate matrices for queueing models, satisfying the specific quasi birth and death structure as in Van Leeuwaarden et al. (2009) and Van Leeuwaarden and Winands (2006). The two methodologies are compared both in terms of applicability requirements and numerical complexity by analyzing their performance for the same classical queueing models considered in Van Leeuwaarden et al. (2009). The main findings are threefold. First, when both methods are applicable, the SL-based algorithms outperform the lattice path counting algorithm (LPCA). Second, there are important classes of problems (for example, models with (level) nonhomogenous rates or with finite state spaces) for which the SL methodology is applicable and for which the LPCA cannot be used. Third, another main advantage of SL algorithms over lattice path counting is that the former includes a method to compute the steady state distribution using this rate matrix.
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01 Jan 2007
TL;DR: In this paper, the authors considered a class of quasi-birth-and-death processes for which explicit solutions can be obtained for the rate matrix R and the associated matrix G. The probabilistic interpretations of these matrices allow us to describe their elements in terms of paths on the two-dimensional lattice.
Abstract: This paper considers a class of quasi-birth-and-death processes for which explicit solutions can be obtained for the rate matrix R and the associated matrix G. The probabilistic interpretations of these matrices allow us to describe their elements in terms of paths on the two-dimensional lattice. Then determining explicit expressions for the matrices becomes equivalent to solving a lattice path counting problem, the solution of which is derived using path decomposition, Bernoulli excursions, and hypergeometric functions. A few applications are provided, including classical models for which we obtain some new results.
22 citations
01 Jul 1980
13 citations
Journal Article•
TL;DR: It is observed that (discrete-time) Quasi-Birth-Death Processes (QBDs) are equivalent, in a precise sense, to probabilistic 1-Counter Automata (p1CAs).
12 citations
TL;DR: In this article, a sub-class of two-dimensional homogeneous nearest neighbor (simple) random walk restricted on the lattice using the matrix geometric approach is analyzed. In particular, they first pre...
Abstract: In this paper, we analyze a sub-class of two-dimensional homogeneous nearest neighbor (simple) random walk restricted on the lattice using the matrix geometric approach. In particular, we first pre...
12 citations
Cites background or methods from "A comparative analysis of the succe..."
...In Refs.[19,20], the authors discuss how the results for discrete-time Markov chains extend to semi-Markov processes and continuous-time Markov processes....
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...In Refs.[19,20] and the references therein, the authors investigate stochastic processes with a special structure regarding the allowed transitions satisfying the so-called successive lumpability property....
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...Furthermore, in Ref.[20], the authors compare the successive lumping methodology developed with the lattice path counting approach[28] for the calculation of the rate matrix of a queueing model....
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TL;DR: The primary focus of this work is on deriving Laplace transforms of transition functions, and analogous results can be derived for the stationary distribution; these results seem to yield the most explicit expressions known to date.
Abstract: We analyze the time-dependent behavior of an M / M / c priority queue having two customer classes, class-dependent service rates, and preemptive priority between classes. More particularly, we develop a method that determines the Laplace transforms of the transition functions when the system is initially empty. The Laplace transforms corresponding to states with at least c high-priority customers are expressed explicitly in terms of the Laplace transforms corresponding to states with at most $$c - 1$$c-1 high-priority customers. We then show how to compute the remaining Laplace transforms recursively, by making use of a variant of Ramaswami's formula from the theory of M / G / 1-type Markov processes. While the primary focus of our work is on deriving Laplace transforms of transition functions, analogous results can be derived for the stationary distribution; these results seem to yield the most explicit expressions known to date.
11 citations
Cites methods from "A comparative analysis of the succe..."
...[20], where they explain how the successive lumping technique can be used to study M/M/1 2-class priority models when both customer classes experience the same service rate....
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References
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TL;DR: This chapter discusses quasi-Birth-and-Death Processes, a large number of which are based on the Markovian Point Processes and the Matrix-Geometric Distribution, as well as algorithms for the Rate Matrix.
Abstract: Preface Part I. Quasi-Birth-and-Death Processes. 1. Examples Part II. The Method of Phases. 2. PH Distributions 3. Markovian Point Processes Part III. The Matrix-Geometric Distribution. 4. Birth-and-Death Processes 5. Processes Under a Taboo 6. Homogeneous QBDs 7. Stability Condition Part IV. Algorithms. 8. Algorithms for the Rate Matrix 9. Spectral Analysis 10. Finite QBDs 11. First Passage Times Part V. Beyond Simple QBDs. 12. Nonhomogeneous QBDs 13. Processes, Skip-Free in One Direction 14. Tree Processes 15. Product Form Networks 16. Nondenumerable States Bibliography Index.
1,940 citations
Book•
01 Jan 1981
1,913 citations
"A comparative analysis of the succe..." refers methods in this paper
...Algorithms for solving this equation were given in [20] and [24]....
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...This is done with the aid of a rate matrix R, which is the basis of the matrix-geometric solution introduced by Neuts [24]....
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Book•
01 Jan 1964
TL;DR: In this article, a very special class of random processes, namely to random walk on the lattice points of ordinary Euclidean space, is studied, and the author considered this high degree of specialization worth while because of the theory of such random walks is far more complete than that of any larger class of Markov chains.
Abstract: This book is devoted exclusively to a very special class of random processes, namely to random walk on the lattice points of ordinary Euclidean space. The author considered this high degree of specialization worth while, because of the theory of such random walks is far more complete than that of any larger class of Markov chains. The book will present no technical difficulties to the readers with some solid experience in analysis in two or three of the following areas: probability theory, real variables and measure, analytic functions, Fourier analysis, differential and integral operators. There are almost 100 pages of examples and problems.
1,605 citations
19 May 2012
TL;DR: An automated approach for designing matrix multiplication algorithms based on constructions similar to the Coppersmith-Winograd construction is developed and a new improved bound on the matrix multiplication exponent ω<2.3727 is obtained.
Abstract: We develop an automated approach for designing matrix multiplication algorithms based on constructions similar to the Coppersmith-Winograd construction. Using this approach we obtain a new improved bound on the matrix multiplication exponent ω
1,098 citations
TL;DR: The history of these fomulas is presented and various applications to statistics, networks, structural analysis, asymptotic analysis, optimization, and partial differential equations are discussed.
Abstract: The Sherman–Morrison–Woodbury formulas relate the inverse of a matrix after a small-rank perturbation to the inverse of the original matrix. The history of these fomulas is presented and various applications to statistics, networks, structural analysis, asymptotic analysis, optimization, and partial differential equations are discussed. The Sherman-Morrison-Woodbury formulas express the inverse of a matrix after a small rank perturbation in terms of the inverse of the original matrix. This paper surveys the history of these formulas and we examine some applications where these formulas are helpful
1,026 citations
"A comparative analysis of the succe..." refers methods in this paper
...In that case it might be beneficial to use other fast matrix inversion algorithms, such as in [12] and [33]....
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