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

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.