Insensitivity of blocking probabilities in a circuit-switching network

Consider a network of nodes (switches) and connecting links. Each link consists of a group of channels (trunks). A call instantaneously seizes channels along a route between the originating and terminating node, holds them for a randomly distributed length of time and frees them instantaneously at the end of the call. If no channels are available, the call is blocked. For special networks with exponential call holding times, Erlang has shown that the steady-state probabilities are in product form. In this paper, we extend this work to general networks and show that if for each pair of nodes there is a unique route, then the blocking probabilities are in product form and are insensitive to the call holding-time distribution, which means that they depend on the call duration only through its mean.