A reversible Erlang loss system with multi-type customers and multi-type servers

We consider a memoryless loss system with servers S = {1,...,J}, and with customer types C = {1,...I}. Servers are multi-type, so that server j can serve a subset of customer types C(j). We show that the probabilities of assigning arriving customers to idle servers can be chosen in such a way that the Markov process describing the system is reversible, with a simple product form stationary distribution. Furthermore, the system is insensitive; these properties are preserved for general service time distributions.