Moment formulas for the Markov renewal branching process

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.