Equivalences between Markov Renewal Processes.
Burton Simon · 1978
We define a form of equivalence between Markov-renewal processes that includes strong and weak lumpability as special cases, and examine its properties. If (X sub n, T sub n) is a Markov-renewal process with kernel Q(t) and (Z sub n, S sub n) is a Markov-renewal process with kernel Y(t), then it is shown that (X sub n, T sub n) and (Z sub n, S sub n) are equivalent if and only if there is a certain homomorphism between the matrix rings generated by Q(t), t is an element (0, infinity) and Y(t), t is an element (0, infinity). The equivalence is identical to weak lumpability in the case where (Z sub n, S sub n) is a renewal process. Although the conditions for strong lumpability can be written in an attractive form, they are too restrictive to be of any real interest. Weak lumpability is of more interest since (as will be shown) it occurs in less trivial examples, but the necessary conditions are very complicated. The equivalence defined herin has the advantage of having simple necessary and sufficient conditions. (Author)