Numerical Algebra, Control and Optimization (NACO)

Kronecker product-forms of steady-state probabilities with $C_k$/$C_m$/$1$ by matrix polynomial approaches

Pages: 691 - 711, Volume 1, Issue 4, December 2011      doi:10.3934/naco.2011.1.691

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Hsin-Yi Liu - Department of Mathematical Science, National Chengchi University, Taipei, Taiwan (email)
Hsing Paul Luh - Department of Mathematical Science, National Chengchi University, Taipei, Taiwan (email)

Abstract: In this paper, we analyze a single server queueing system $C_k/C_m/1$. 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 $\textbf{Q}(\omega)$. It is shown that there is a strong relation between the singularities of $\textbf{Q}(\omega)$ 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 $C_k/C_m/1$, thereupon considerably reducing the computational complexity of solving a complicated problem in a general queueing model.

Keywords:  Matrix polynomials, phase-type distributions, quasi-birth-and-death process
Mathematics Subject Classification:  Primary: 15A16, 60K25, 40C05.

Received: June 2011;      Revised: August 2011;      Available Online: November 2011.