Analytically explicit results for the GI/C-MSP/1/\(\infty \) queueing system using roots (Q2909825)

The paper under review establishes a closed form for system states stationary distribution at moments immediately before arrivals in the GI/C-MSP/1 queueing system in terms of roots of matrix equations. Here, the notion C-MSP stands for continuous-time Markov service process, which is the analogue of C-MAP, continuous-time Markov arrival process see [\textit{A. Pacheco, L. C. Tang} and \textit{N. U. Prabhu}, Markov-modulated processes and semi-regenerative phenomena. Hackensack, NJ: World Scientific (2009; Zbl 1181.60005)]. The analysis of the paper is based on roots of the associated characteristic equation of the vector-generating function of the system-state distribution. The paper also provides the steady-state system queue-length distribution at an arbitrary time by using arguments of Markov renewal theory. The sojourn-time distribution is also investigated. The method of roots, which is used in the paper, is an alternative to the matrix-geometric and spectral methods. It was also used in [the first author, \textit{G. Singh} and \textit{U. C. Gupta}, ``A simple and complete computational analysis of MAP/R/1 queue using roots'', Methodol. Comput. Appl. Probab., to appear, \url{doi:10.1007/s11009-011-9266-3}].