The main objective of this work is to present a process to compute the Markov renewal matrix R(t) for Markov renewal processes with countable infinite spaces, which semi-Markov matrixes Q(t) are immigration and death type and assume a tridiagonal form. These processes occur often in practical applications. And the difficulty in obtaining friendly results for R(t)is a great obstacle to its application in practical cases modelling. It is considered the application to the M|M|? queue particular case.

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Markov renewal process; Tridiagonal matrix; M|M|? queue