Detecting Block GI/M/1 and Block M/G/1 Matrices from Model Specifications

Abstract

Given a high level specification of a Markov chain, it is desirable to be able to determine if the state space of the Markov chain is finite. If the state space is finite, the entire transition matrix can be generated. If the state space of the Markov chain is infinite, although it is impossible to generate the entire transition matrix, it still may be possible to solve the Markov chain. One interesting and useful class of denumerable state space Markov chains that can be solved consists of those that have a block GI/M/1 or a block M/G/1 form. Block GI/M/1 and block M/G/1 matrices, though infinite, have a repeating structure that allows analytical solution. If it can be detected from the specification that the Markov chain is block GI/M/1 or block M/G/1, then our system can automatically apply matrix analytical methods to obtain a solution. In this paper, a high level specification language for Markov chains is proposed such that the existence of block GI/M/1 or block M/G/1 form may be verified directly from the specification. An algorithm is given to determine whether a Markov chain with a 2 dimensional state space specified in this language is infinite and, if so, whether it can be represented in block GI/M/1 or block M/G/1 form. It is shown that in general, it is undecidable whether a matrix specified in this language with a state space vector of dimension greater than 2 can be represented as a block GI/M/1 or block M/G/1 matrix. Finally a second, less powerful specification language, is provided for which the question is decidable for a specification with a state space vector of any finite dimension.