PARTIAL BALANCE, INSENSITIVITY AND WEAK COUPLING

The idea is developed of imbedding a given Markov process in a more general Markov process in a definite sense, the sense of weak coupling. It is shown that this gives a natural theory of insensitivity in that, the stronger the balance conditions satisfied by the imbedded process, the weaker the conditions that are required of the imbedding process. The imbeddings associated with a range of balance conditions are discussed. IMBEDDING; JACKSON NETWORKS 1. Extensions of the partial balance concept In an earlier paper (Whittle (1985)) I demonstrated the equivalence of partial balance and insensitivity, but dealt more convincingly with the case of a single partially-balancing set than with that of several. I would claim that this paper presents the natural general formulation of insensitivity, and demonstrates how the strength of one's conclusions corresponds exactly to the degree of balance assumed. As in the earlier paper we work with a Markov process in continuous time whose states x belong to a discrete space X. In order to avoid irrelevant complications we shall even suppose X finite. The intensity of the transition