Moment formulas for the Markov renewal branching process

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Abstract
There are many queueing models in which there appears a semi-Markov matrix G(·), whose entries are absorption-time distributions in a Markov renewal branching process. The role of G(·) is similar to that of the busy period in the simple M/G/1 model. The computation of various quantities associated with G(·) is however much more complicated. The moment matrices, and particularly the mean matrix of G(·), are essential in the construction of general and mathematically well-justified algorithms for the steady-state distributions of such queues. This paper discusses the moment matrices of G(·) and algorithms for their numerical computation. Its contents are basic to the algorithmic solutions to several queueing models, which are to be presented in follow-up papers.