Lumpable continuous-time stochastic automata networks

Cataloged from PDF version of article.The generator matrix of a continuous-time stochastic automata network (SAN) is a sum of tensor products of smaller matrices, which may have entries that are functions of the global state space. This paper specifies easy to check conditions for a class of ordinarily lumpable partitionings of the generator of a continuous-time SAN in which aggregation is performed automaton by automaton. When there exists a lumpable partitioning induced by the tensor representation of the generator, it is shown that an efficient aggregation-iterative disaggregation algorithm may be employed to compute the steady-state distribution. The results of experiments with two SAN models show that the proposed algorithm performs better than the highly competitive block Gauss-Seidel in terms of both the number of iterations and the time to converge to the solution. © 2002 Elsevier Science B.V. All rights reserved

Equalities and Inequalities for Norms of Block Imaginary Circulant Operator Matrices

Circulant, block circulant-type matrices and operator norms have become effective tools in solving networked systems. In this paper, the block imaginary circulant operator matrices are discussed. By utilizing the special structure of such matrices, several norm equalities and inequalities are presented. The norm τ in consideration is the weakly unitarily invariant norm, which satisfies τA=τ(UAV). The usual operator norm and Schatten p-norm are included. Furthermore, some special cases and examples are given

Iterative disaggregation for a class of lumpable discrete-time stochastic automata networks

Cataloged from PDF version of article.Stochastic automata networks (SANs) have been developed and used in the last 15 years as a modeling formalism for large systems that can be decomposed into loosely connected components. In this work, we concentrate on the not so much emphasized discrete-time SANs. First, we remodel and extend an SAN that arises in wireless communications. Second, for an SAN with functional transitions, we derive conditions for a special case of ordinary lumpability in which aggregation is done automaton by automaton. Finally, for this class of lumpable discrete-time SANs we devise an efficient aggregation–iterative disaggregation algorithm and demonstrate its performance on the SAN model of interest. © 2002 Elsevier Science B.V. All rights reserved

An Extension of Two Conjugate Direction Methods to Markov Chain Problems

Motivated by the recent applications of the conjugate residual method to nonsymmetric linear systems by Sogabe, Sugihara and Zhang [An extension of the conjugate residual method to nonsymmetric linear systems. J. Comput. Appl. Math., Vol. 266, 2009, pp. 103--113], this paper describes two conjugate direction methods, BiCR and BiCG, and attempts to extend their applications to compute the stationary probability distribution for an irreducible Markov chain with the aim of finding an alternative basic solver. Numerical experiments show the feasibility of the BiCR and BiCG to some extent, with applications to several practical Markov chain problems

Block SOR preconditioned projection methods for Kronecker structured Markovian representations

Kronecker structured representations are used to cope with the state space explosion problem in Markovian modeling and analysis. Currently, an open research problem is that of devising strong preconditioners to be used with projection methods for the computation of the stationary vector of Markov chains (MCs) underlying such representations. This paper proposes a block successive overrelaxation (BSOR) preconditioner for hierarchical Markovian models (HMMs1) that are composed of multiple low-level models and a high-level model that defines the interaction among low-level models. The Kronecker structure of an HMM yields nested block partitionings in its underlying continuous-time MC which may be used in the BSOR preconditioner. The computation of the BSOR preconditioned residual in each iteration of a preconditioned projection method becomes the problem of solving multiple nonsingular linear systems whose coefficient matrices are the diagonal blocks of the chosen partitioning. The proposed BSOR preconditioner solves these systems using sparse LU or real Schur factors of diagonal blocks. The fill-in of sparse LU factorized diagonal blocks is reduced using the column approximate minimum degree (COLAMD) ordering. A set of numerical experiments is presented to show the merits of the proposed BSOR preconditioner. © 2005 Society for Industrial and Applied Mathematics

A model for FMS of unreliable machines

Fulltext link (Conference paper): http://www.wseas.us/e-library/conferences/athens2000/Papers2000/170.pdfDocument Type: ArticleThis paper studies Markovian queueing model for flexible manufacturing system. The manufacturing system consists of multiple unreliable machines. Hedging point policy is applied to the system as production control. We model the machine states and inventory levels of the system as a multi-server queueing system. Fast numerical algorithm is presented to solve the steady state probability distribution of the system. Using the probability distribution, the system performance and the effect of machine reliability and maintainability can be evaluated.link_to_subscribed_fulltex

Boundary value methods for transient solutions of Markovian queueing networks.

by Ma Ka Chun.Thesis (M.Phil.)--Chinese University of Hong Kong, 2004.Includes bibliographical references (leaves 50-52).Abstracts in English and Chinese.Chapter 1 --- Introduction --- p.7Chapter 2 --- Queueing Networks --- p.9Chapter 2.1 --- One-queue Networks --- p.9Chapter 2.2 --- Two-queue Free Networks --- p.12Chapter 2.3 --- Two-queue Overflow Networks --- p.13Chapter 2.4 --- Networks with Batch Arrivals --- p.14Chapter 3 --- ODE Solvers --- p.16Chapter 3.1 --- The Initial Value Methods --- p.16Chapter 3.1.1 --- The Linear System of Ordinary Differential Equations --- p.16Chapter 3.1.2 --- Euler's Method --- p.17Chapter 3.1.3 --- Runge-Kutta Methods --- p.17Chapter 3.1.4 --- The Stability of the IVMs --- p.19Chapter 3.1.5 --- Applications in Queueing Networks --- p.20Chapter 3.2 --- The Boundary Value Methods --- p.20Chapter 3.2.1 --- The Generalized Backward Differentiation For- mulae --- p.21Chapter 3.2.2 --- An example --- p.24Chapter 4 --- The Linear Equation Solver --- p.26Chapter 4.1 --- Iterative Methods --- p.26Chapter 4.1.1 --- The Jacobi method --- p.27Chapter 4.1.2 --- The Gauss-Seidel Method --- p.28Chapter 4.1.3 --- Other Iterative Methods --- p.29Chapter 4.1.4 --- Preconditioning --- p.29Chapter 4.2 --- The Multigrid Method --- p.30Chapter 4.2.1 --- Iterative Refinement --- p.30Chapter 4.2.2 --- Restriction and Prolongation --- p.30Chapter 4.2.3 --- The Geometric Multigrid Method --- p.33Chapter 4.2.4 --- The Algebraic Multigrid Method --- p.38Chapter 4.2.5 --- Higher Dimensional Cases --- p.38Chapter 4.2.6 --- Applications in Queueing Networks --- p.38Chapter 5 --- Numerical Experiments --- p.41Chapter 6 --- Concluding Remarks --- p.49Bibliography --- p.5

Self-Evaluation Applied Mathematics 2003-2008 University of Twente

This report contains the self-study for the research assessment of the Department of Applied Mathematics (AM) of the Faculty of Electrical Engineering, Mathematics and Computer Science (EEMCS) at the University of Twente (UT). The report provides the information for the Research Assessment Committee for Applied Mathematics, dealing with mathematical sciences at the three universities of technology in the Netherlands. It describes the state of affairs pertaining to the period 1 January 2003 to 31 December 2008

Krylov subspace methods and their generalizations for solving singular linear operator equations with applications to continuous time Markov chains

Viele Resultate über MR- und OR-Verfahren zur Lösung linearer Gleichungssysteme bleiben (in leicht modifizierter Form) gültig, wenn der betrachtete Operator nicht invertierbar ist. Neben dem für reguläre Probleme charakteristischen Abbruchverhalten, kann bei einem singulären Gleichungssystem auch ein so genannter singulärer Zusammenbruch auftreten. Für beide Fälle werden verschiedene Charakterisierungen angegeben. Die Unterrauminverse, eine spezielle verallgemeinerte Inverse, beschreibt die Näherungen eines MR-Unterraumkorrektur-Verfahrens. Für Krylov-Unterräume spielt die Drazin-Inverse eine Schlüsselrolle. Bei Krylov-Unterraum-Verfahren kann a-priori entschieden werden, ob ein regulärer oder ein singulärer Abbruch auftritt. Wir können zeigen, dass ein Krylov-Verfahren genau dann für beliebige Startwerte eine Lösung des linearen Gleichungssystems liefert, wenn der Index der Matrix nicht größer als eins und das Gleichungssystem konsistent ist. Die Berechnung stationärer Zustandsverteilungen zeitstetiger Markov-Ketten mit endlichem Zustandsraum stellt eine praktische Aufgabe dar, welche die Lösung eines singulären linearen Gleichungssystems erfordert. Die Eigenschaften der Übergangs-Halbgruppe folgen aus einfachen Annahmen auf rein analytischem und matrixalgebrischen Wege. Insbesondere ist die erzeugende Matrix eine singuläre M-Matrix mit Index 1. Ist die Markov-Kette irreduzibel, so ist die stationäre Zustandsverteilung eindeutig bestimmt