Stationary Distribution Vector Research Articles

A Study on Dynamically Reconfigurable Robetic System. (7th Report. A Method of Determining a Master Cell for Cellular Robotics on the Network Energy Estimation Method betweel Cells).

In this paper, we propose a master cell for "Cellular Robotics (CEBOT)" as a director to decide CEBOT's activities. Since the CEBOT system can be considered to be one of the intelligent network systems, we determine the master cell of the system based on network energy estimation method (NEEM) between cells. Then, we describe both a method of "mastering", which implies the determination of a master cell, and a method of "mastering" by local energy-sensitive estimation (LESE) as a method of determining the master cell. LESE is based on local network energy only considered for cells adjacent to the cell. In this paper, we assume that the CEBOT structure is a binary tree structure because the CEBOT employs the hierarchical structure for control and communication. The master cell is difined as a window for users and as an information propagation source for the CEBOT. We can derive this network energy from both the transition probability matrix led by the communication transition probability and the stationary distribution vector. We also propose the connection load matrix to consider the channel capacity between cells. Simulation results of "mastering" both by NEEM and by LESE are shown in this paper.

Bordered Matrices, Singular Systems, and Ergodic Markov Chains

Systems of linear equations involving submatrices of singular M-matrices have been considered in the calculation of stationary distribution vectors of ergodic Markov chains. In this paper, an alternative appproach is suggested, using bordered matrices instead, motivated by the stability analysis of the first approach. The conditioning of this approach, which applies to general as well as M-matrices, is considered. Two numerical examples with small matrices are included.

Numerical solution of sparse singular systems of equations arising from ergodic markov chains

The stationary probability distribution vector, x, associated with an ergodic finite Markov chain satisfies a homogeneous singular system of equations , where A is a real and generally unsymmetric square matrix of the form . Here I is the identity matrix and T is the chain's column stochastic matrix. In many applications A is very large and sparse, and in such cases it is desirable to exploit this property in computing x. In this paper we review some of the literature dealing with sparse techniques for solving the above system of equations, and in so doing attempt to present a variety of methods from a unified point of view

Convergent Iterations for Computing Stationary Distributions of Markov Chains

Classical iterative schemes such as the Gauss–Seidel method and its variations constitute powerful tools for computing stationary distribution vectors for large-scale Markov process, such as those arising in queueing network analysis. The coefficient matrix A in these processes in a Q-matrix, i.e., a singular irreducible M-matrix with zero column sums and, unlike the nonsingular case, the classical iterations for A do not always converge. The purpose of this paper is to survey the recent literature and to analyze the behavior of these methods completely in terms of the graph structure of A. The results given here hold under somewhat weaker assumptions on A.

On the Smallest Positive Singular Value of a Singular M-Matrix with Applications to Ergodic Markov Chains

Let A be a singular $n \times n$M-matrix of rank $n - 1$ and let B be an $( n - 1 ) \times ( n - 1 )$ nonsingular matrix that results from deleting row j and column k from A. An open question discussed by Harrod and Plemmons [SIAM J. Sci. Stat. Comput., 5 (1984), pp. 453–469] is whether there is a choice of j and k such that $\sigma _{n - 1}^{( A )} \approx \sigma _{n - 1}^{( B )} $ where $\sigma _{n - 1}^{( A )} $ is the smallest positive singular value of A and $\sigma _{n - 1}^{( B )} $ is the smallest singular value of B. In this paper, we resolve this conjecture by showing that there is always such a choice. This conjecture is important when finding the stationary distribution of an ergodic Markov chain. This can be posed as the problem of finding the $n \times 1$ vector p such that $Ap = 0$ and $e^T p = 1$ where $e = ( 1, \cdots ,1 )^T $. Here $A = I - Q^T $ where Q is a row stochastic matrix and p is the stationary distribution vector of the chain. This problem can be reduced to the problem of solving a system of linear equations with coefficient matrix B. If $\sigma _{n - 1}^{( A )} \approx \sigma _{n - 1}^{( B )} $ then this system of linear equations is about as well conditioned as the original problem.

Sensitivity of the stationary distribution vector for an ergodic Markov chain

Stationary distribution vectors p ∞ for Markov chains with associated transition matrices T are important in the analysis of many models in the mathematical sciences, such as queuing networks, input-output economic models, and compartmental tracer analysis models. The purpose of this paper is to provide insight into the sensitivity of p ∞ to perturbations in the transition probabilities of T and to understand some of the difficulties in computing an accurate p ∞. The group inverse A # of I − T is shown to be of fundamental importance in understanding sensitivity or conditioning of p ∞. The main result shows that if there is a state that is accessible from every other state and the corresponding column of T has no small off-diagonal elements, then p ∞ cannot be sensitive to small perturbations in T. Ecological examples are given. A new algorithm for calculating A # is described.

Computing the stationary distribution for infinite Markov chains

In a situation where the unique stationary distribution vector of an infinite irreducible positive-recurrent stochastic matrix P is not analytically determinable, numerical approximations are needed. This paper partially synthesizes and extends work on finite-vector approximative solutions obtained from n x n northwest corner truncations ( n ) P of P , from the standpoints of (pointwise convergence) algorithms as n →∞, and the manner of their computer implementation with a view to numerical stability and conditioning. The problem for finite n is connected with that of finding the unique stationary distribution of the finite stochastic matrix ( n ) P̃ obtained from ( n ) P by augmenting a column.

Computing the stationary distribution for infinite Markov chains

Abstract In a situation where the unique stationary distribution vector of an infinite irreducible positive-recurrent stochastic matrix P is not analytically determinable, numerical approximations are needed. This paper partially synthesizes and extends work on finite-vector approximative solutions obtained from nxn northwest corner truncations (n)P of P, from the standpoints of (pointwise convergence) algorithms as n→∞, and the manner of their computer implementation with a view to numerical stability and conditioning. The problem for finite n is connected with that of finding the unique stationary distribution of the finite stochastic matrix (n)P obtained from (n)P by augmenting a column.

Convergent nonnegative matrices and iterative methods for consistent linear systems

In this paper we study linear stationary iterative methods with nonnegative iteration matrices for solving singular and consistent systems of linear equationsAx=b. The iteration matrices for the schemes are obtained via regular and weak regular splittings of the coefficients matrixA. In certain cases when only some necessary, but not sufficient, conditions for the convergence of the iterations schemes exist, we consider a transformation on the iteration matrices and obtain new iterative schemes which ensure convergence to a solution toAx=b. This transformation is parameter-dependent, and in the case where all the eigenvalues of the iteration matrix are real, we show how to choose this parameter so that the asymptotic convergence rate of the new schemes is optimal. Finally, some applications to the problem of computing the stationary distribution vector for a finite homogeneous ergodic Markov chain are discussed.