A logarithmic reduction algorithm for quasi-birth-death processes

Quasi-birth-death processes are commonly used Markov chain models in queueing theory, computer performance, teletraffic modeling and other areas. We provide a new, simple algorithm for the matrix-geometric rate matrix. We demonstrate that it has quadratic convergence. We show theoretically and through numerical examples that it converges very fast and provides extremely accurate results even for almost unstable models.

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Markov chains with applications in queueing theory, which have a matrix-geometric invariant probability vector

Advances in Applied Probability ◽

10.1017/s0001867800029542 ◽

1978 ◽

Vol 10 (01) ◽

pp. 185-212 ◽

Cited By ~ 18

Author(s):

Marcel F. Neuts

Keyword(s):

Spectral Radius ◽

Queueing Theory ◽

Minimal Solution ◽

Linear Matrix ◽

Probability Vector ◽

Stochastic Matrices ◽

Geometric Invariant ◽

Rate Matrix ◽

Linear Matrix Equation ◽

The Matrix

It is shown that a class of infinite, block-partitioned, stochastic matrices has a matrix-geometric invariant probability vector of the form (x 0, x 1,…), where xk = x 0 Rk , for k ≧ 0. The rate matrix R is an irreducible, non-negative matrix of spectral radius less than one. The matrix R is the minimal solution, in the set of non-negative matrices of spectral radius at most one, of a non-linear matrix equation. Applications to queueing theory are discussed. Detailed explicit and computationally tractable solutions for the GI/PH/1 and the SM/M/1 queue are obtained.

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Markov chains with applications in queueing theory, which have a matrix-geometric invariant probability vector

Advances in Applied Probability ◽

10.2307/1426725 ◽

1978 ◽

Vol 10 (1) ◽

pp. 185-212 ◽

Cited By ~ 87

Author(s):

Marcel F. Neuts

Keyword(s):

Spectral Radius ◽

Queueing Theory ◽

Minimal Solution ◽

Linear Matrix ◽

Probability Vector ◽

Stochastic Matrices ◽

Geometric Invariant ◽

Rate Matrix ◽

Linear Matrix Equation ◽

The Matrix

It is shown that a class of infinite, block-partitioned, stochastic matrices has a matrix-geometric invariant probability vector of the form (x0, x1,…), where xk = x0Rk, for k ≧ 0. The rate matrix R is an irreducible, non-negative matrix of spectral radius less than one. The matrix R is the minimal solution, in the set of non-negative matrices of spectral radius at most one, of a non-linear matrix equation.Applications to queueing theory are discussed. Detailed explicit and computationally tractable solutions for the GI/PH/1 and the SM/M/1 queue are obtained.

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A Numerical Approach for Evaluating the Time-Dependent Distribution of a Quasi Birth-Death Process

Methodology And Computing In Applied Probability ◽

10.1007/s11009-021-09882-6 ◽

2021 ◽

Author(s):

Michel Mandjes ◽

Birgit Sollie

Keyword(s):

Continuous Time ◽

Real Life ◽

Numerical Approach ◽

Time Dependent ◽

Death Process ◽

Continuous Time Markov Chain ◽

Likelihood Estimator ◽

Real Life Data ◽

The Matrix ◽

Birth Death

AbstractThis paper considers a continuous-time quasi birth-death (qbd) process, which informally can be seen as a birth-death process of which the parameters are modulated by an external continuous-time Markov chain. The aim is to numerically approximate the time-dependent distribution of the resulting bivariate Markov process in an accurate and efficient way. An approach based on the Erlangization principle is proposed and formally justified. Its performance is investigated and compared with two existing approaches: one based on numerical evaluation of the matrix exponential underlying the qbd process, and one based on the uniformization technique. It is shown that in many settings the approach based on Erlangization is faster than the other approaches, while still being highly accurate. In the last part of the paper, we demonstrate the use of the developed technique in the context of the evaluation of the likelihood pertaining to a time series, which can then be optimized over its parameters to obtain the maximum likelihood estimator. More specifically, through a series of examples with simulated and real-life data, we show how it can be deployed in model selection problems that involve the choice between a qbd and its non-modulated counterpart.

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A New Iterative Algorithm for Solving a Class of Matrix Nearness Problem

ISRN Computational Mathematics ◽

10.5402/2012/126908 ◽

2012 ◽

Vol 2012 ◽

pp. 1-6

Author(s):

Xuefeng Duan ◽

Chunmei Li

Keyword(s):

Iterative Algorithm ◽

Projection Algorithm ◽

Frobenius Norm ◽

Matrix Equations ◽

Von Neumann ◽

Numerical Examples ◽

Matrix Nearness Problem ◽

Least Frobenius Norm Solution ◽

Alternating Projection ◽

The Matrix

Based on the alternating projection algorithm, which was proposed by Von Neumann to treat the problem of finding the projection of a given point onto the intersection of two closed subspaces, we propose a new iterative algorithm to solve the matrix nearness problem associated with the matrix equations AXB=E, CXD=F, which arises frequently in experimental design. If we choose the initial iterative matrix X0=0, the least Frobenius norm solution of these matrix equations is obtained. Numerical examples show that the new algorithm is feasible and effective.

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A Simple Algorithm for Finding a Non-negative Basic Solution of a System of Linear Algebraic Equations

Modeling and Analysis of Information Systems ◽

10.18255/1818-1015-2021-3-234-237 ◽

2021 ◽

Vol 28 (3) ◽

pp. 234-237

Author(s):

Gleb D. Stepanov

Keyword(s):

Simplex Method ◽

Gaussian Elimination ◽

Simple Algorithm ◽

Basic Solution ◽

Algebraic Equations ◽

Linear Algebraic Equations ◽

The Matrix ◽

Linear Programming Problems ◽

Two Stages ◽

Computational Resources

This article describes an algorithm for obtaining a non-negative basic solution of a system of linear algebraic equations. This problem, which undoubtedly has an independent interest, in particular, is the most time-consuming part of the famous simplex method for solving linear programming problems.Unlike the artificial basis Orden’s method used in the classical simplex method, the proposed algorithm does not attract artificial variables and economically consumes computational resources.The algorithm consists of two stages, each of which is based on Gaussian exceptions. The first stage coincides with the main part of the Gaussian complete exclusion method, in which the matrix of the system is reduced to the form with an identity submatrix. The second stage is an iterative cycle, at each of the iterations of which, according to some rules, a resolving element is selected, and then a Gaussian elimination step is performed, preserving the matrix structure obtained at the first stage. The cycle ends either when the absence of non-negative solutions is established, or when one of them is found.Two rules for choosing a resolving element are given. The more primitive of them allows for ambiguity of choice and does not exclude looping (but in very rare cases). Use of the second rule ensures that there is no looping.

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The Elastic Field of an Elliptic Inclusion With a Slipping Interface

Journal of Applied Mechanics ◽

10.1115/1.3171693 ◽

1986 ◽

Vol 53 (1) ◽

pp. 103-107 ◽

Cited By ~ 30

Author(s):

E. Tsuchida ◽

T. Mura ◽

J. Dundurs

Keyword(s):

Closed Form ◽

Infinite Series ◽

Elastic Field ◽

Elliptic Inclusion ◽

Numerical Examples ◽

Elastic Fields ◽

Bonded Interface ◽

The Matrix ◽

Displacement Potentials ◽

Uniform Expansion

The paper analyzes the elastic fields caused by an elliptic inclusion which undergoes a uniform expansion. The interface between the inclusion and the matrix cannot sustain shear tractions and is free to slip. Papkovich–Neuber displacement potentials are used to solve the problem. In contrast to the perfectly bonded interface, the solution cannot be expressed in closed form and involves infinite series. The results are illustrated by numerical examples.

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A Markov chain associated with the minimal quasi-stationary distribution of birth–death chains

Journal of Applied Probability ◽

10.1017/s0021900200102542 ◽

1995 ◽

Vol 32 (01) ◽

pp. 25-38

Author(s):

Servet Martínez ◽

Maria Eulália Vares

Keyword(s):

Markov Chain ◽

Stationary Distribution ◽

Conditional Distribution ◽

Transition Probabilities ◽

The Matrix ◽

Quasi Stationary Distribution ◽

Birth Death

We show that if the limiting conditional distribution for an absorbed birth–death chain exists, then the chain conditioned to non-absorption converges to a Markov chain with transition probabilities given by the matrix associated with the minimal quasi-stationary distribution.

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COUPLED MAGNETIC FIELD AND VISCOELASTICITY OF FERROGEL

International Journal of Applied Mechanics ◽

10.1142/s175882511100097x ◽

2011 ◽

Vol 03 (02) ◽

pp. 259-278 ◽

Cited By ~ 24

Author(s):

YI HAN ◽

WEI HONG ◽

LEANN FAIDLEY

Keyword(s):

Magnetic Field ◽

Magnetic Fields ◽

Material Model ◽

Finite Element Code ◽

Equilibrium Thermodynamics ◽

Numerical Examples ◽

Rate Dependent ◽

The Matrix ◽

Specific Material ◽

Soft Polymer

Composed of a soft polymer matrix and magnetic filler particles, ferrogel is a smart material responsive to magnetic fields. Due to the viscoelasticity of the matrix, the behaviors of ferrogel are usually rate-dependent. Very few models with coupled magnetic field and viscoelasticity exist in the literature, and even fewer are capable of reliable predictions. Based on the principles of non-equilibrium thermodynamics, a field theory is developed to describe the magneto-viscoelastic property of ferrogel. The theory provides a guideline for experimental characterizations and structural designs of ferrogel-based devices. A specific material model is then selected and the theory is implemented in a finite element code. Through numerical examples, the responses of a ferrogel in uniform and non-uniform magnetic fields are analyzed. The dynamic response of a ferrogel to cyclic magnetic fields is also studied, and the prediction agrees with our experimental results. In the reversible limit, our theory recovers existing models for elastic ferrogel, and is capable of capturing some instability phenomena.

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Numerical Solution of Duffing Equation by Using an Improved Taylor Matrix Method

Journal of Applied Mathematics ◽

10.1155/2013/691614 ◽

2013 ◽

Vol 2013 ◽

pp. 1-6 ◽

Cited By ~ 3

Author(s):

Berna Bülbül ◽

Mehmet Sezer

Keyword(s):

Matrix Method ◽

Numerical Approach ◽

Duffing Equation ◽

Matrix Equations ◽

Algebraic Equations ◽

Solution Function ◽

Numerical Examples ◽

Collocation Points ◽

Nonlinear Algebraic Equations ◽

The Matrix

We have suggested a numerical approach, which is based on an improved Taylor matrix method, for solving Duffing differential equations. The method is based on the approximation by the truncated Taylor series about center zero. Duffing equation and conditions are transformed into the matrix equations, which corresponds to a system of nonlinear algebraic equations with the unknown coefficients, via collocation points. Combining these matrix equations and then solving the system yield the unknown coefficients of the solution function. Numerical examples are included to demonstrate the validity and the applicability of the technique. The results show the efficiency and the accuracy of the present work. Also, the method can be easily applied to engineering and science problems.

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