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Title: Kronecker product-forms of steady-state probabilities with Ck/Cm/1 by matrix polynomial approaches
Authors: Luh, Hsing Paul
Liu, Hsin Yi
Contributors: 應數系
Keywords: Matrix polynomials;Phase-type distributions;Quasi-birth-and-death process
Date: 2011-11
Issue Date: 2015-06-22 14:26:31 (UTC+8)
Abstract: In this paper, we analyze a single server queueing system Ck/Cm/1. We construct a general solution space of vector product-forms for steady-state probability and express it in terms of singularities and vectors of the fundamental matrix polynomial Q(ω). It is shown that there is a strong relation between the singularities of Q(ω) and the roots of the characteristic polynomial involving the Laplace transforms of the inter-arrival and service times distributions. Thus, some steady-state probabilities may be written as a linear combination of vectors derived in expression of these roots. In this paper, we focus on solving a set of equations of matrix polynomials in the case of multiple roots. As a result, we give a closed-form solution of unboundary steady-state probabilities of Ck/Cm/1, thereupon considerably reducing the computational complexity of solving a complicated problem in a general queueing model.
Relation: Numerical Algebra, Control and Optimization, 1(4), 691-711
Data Type: article
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