Partial Orders and the Matrix R in Matrix Analytic Methods

Authors:
Qi-Ming He
Affiliations:
-
Venue:
SIAM Journal on Matrix Analysis and Applications
Year:
1999

Citing 0
Cited 1

Monotonicity And Convexity Of Some Functions Associated With Denumerable Markov Chains And Their Applications To Queuing Systems

Probability in the Engineering and Informational Sciences

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Abstract

This paper studies the matrix R, which is the minimal nonnegative solution to a nonlinear matrix equation, raised in matrix analytic methods. Based on some partial orders defined on the transition matrix of Markov chains of GI/M/1 type, the monotonicity of the corresponding matrix R and its Perron--Frobenius eigenvalue is investigated. The results are useful in estimating tail probabilities of stationary distributions of Markov chains of GI/M/1 type and constructing upper bounds for the matrix R. Applications to the GI/MAP/1 queue are discussed as well.