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An Invariant Subspace Approach in M/G/1 and G/M/1 Type Markov Chains

Summary: An Invariant Subspace Approach
in M/G/1 and G/M/1 Type Markov Chains
Nail Akar and Khosrow Sohraby
Computer Science Telecommunications
University of Missouri ­ Kansas City
Kansas City, MO 64110
Abstract Let A k ; k – 0, be a sequence of m \Theta m nonnegative matrices
and let A(z) = P 1
k=0 A k z k be such that A(1) is an irreducible stochastic
matrix. The unique power­bounded solution of the nonlinear matrix equa­
tion G = P 1
k=0 A k G k has been shown to play a key role in the analysis of
Markov chains of M/G/1 type. Assuming that the matrix A(z) is rational,
we show that the solution of this matrix equation reduces to finding an
invariant subspace of a certain matrix. We present an iterative method
for computing this subspace which is globally convergent. Moreover, the
method can be implemented with quadratic or higher convergence rate
matrix sign function iterations, which brings in a new dimension to the
analysis of M/G/1 type Markov chains for which the existing algorithms
may suffer from low linear convergence rates. The method can be viewed


Source: Akar, Nail - Department of Electrical and Electronics Engineering, Bilkent University


Collections: Engineering