Algebraic criteria for extended product form in generalised semi-Markov processes

It is well known that partial balance of the corresponding Markov process is necessary and sufficient for a generalised semi-Markov process to have a product form supplemented stationary distribution and hence possess the property of insensitivity with respect to sets of distributions with fixed mean. In this paper we give algebraic criteria, similar to partial balance of the corresponding Markov process, for a generalised semi-Markov process to have a supplemented stationary distribution which is an extension of the usual product form and an insensitivity property which is an extension of the standard insensitivity.