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.

The concept of relaxing a Markov process is introduced; this is the creation of additional transitions between ergodic classes of the process in such a way as to conserve the existing equilibrium distribution within ergodic classes. The ‘open' version of a ‘closed' model of migration, polymerisation etc. often has this character. As further examples, generalized versions of Jackson networks and networks with clustering nodes are given.

This paper obtains the stationary joint distribution of a customer's sojourn times along an overtake-free path in a closed multiclass Jackson network. The distribution has a simple representation in terms of the product form distribution for the state of the network at an arrival instant.

Consider a path in a multiclass Gordon–Newell network such that a customer present in a node of this path cannot be overtaken by any other customer behind him in a node of this path or by probabilistic influences created by such customers. The passage time through such a path is a mixture of Erlangian distributions, where the mixing distribution is given by the steady state of the network.