A Fast Newton-Shamanskii Iteration for a Matrix Equation Arising from M/G/1-Type Markov Chains

Pei-Chang Guo

doi:10.1155/2017/4018239

Abstract

For the nonlinear matrix equations arising in the analysis of M/G/1-type and GI/M/1-type Markov chains, the minimal nonnegative solution G or R can be found by Newton-like methods. We prove monotone convergence results for the Newton-Shamanskii iteration for this class of equations. Starting with zero initial guess or some other suitable initial guess, the Newton-Shamanskii iteration provides a monotonically increasing sequence of nonnegative matrices converging to the minimal nonnegative solution. A Schur decomposition method is used to accelerate the Newton-Shamanskii iteration. Numerical examples illustrate the effectiveness of the Newton-Shamanskii iteration.

Highlights

Some necessary notation for this article is as follows
For a large number of stochastic models, most steady-state and certain transient characteristics of the process can be expressed in terms of the minimal nonnegative solution described above
This problem arises in the analysis of queues with phase type service times, as well as in queues that can be represented as quasi-birth-death processes [3]

Summary

Introduction

Some necessary notation for this article is as follows. For any matrix B = [bij] ∈ Rn×n, B ≥ 0 (B > 0) if bij ≥ 0 (bij > 0) for all i, j; for any matrices A, B ∈ Rn×n, A ≥ B (A > B) if aij ≥ bij (aij > bij) for all i, j; the vector with all entries equal to one is denoted by e; that is, e = (1, 1, . . . , 1)T; and the identity matrix is denoted by I. The steady-state probability vector of an M/G/1-type MC, if it exists, can be expressed in terms of a matrix G that is the element-wise minimal nonnegative solution to the nonlinear matrix equation [1]. For a large number of stochastic models, most steady-state and certain transient characteristics of the process can be expressed in terms of the minimal nonnegative solution described above. This problem arises in the analysis of queues with phase type service times, as well as in queues that can be represented as quasi-birth-death processes [3].

Newton-Shamanskii Iteration

The Case of Low-Rank Downward Transitions

The Case of Low-Rank Local and Upward Transitions

Convergence Analysis

Numerical Experiments