Equivalences between Markov Renewal Processes.
VIRGINIA POLYTECHNIC INST AND STATE UNIV BLACKSBURG DEPT OF INDUSTRIAL ENGINEERING AND OPERATIONS RESEARCH
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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 Qt and Z sub n, S sub n is a Markov-renewal process with kernel Yt, 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 Qt, t is an element 0, infinity and Yt, 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
- Statistics and Probability