A generalization of the Erlang formula of traffic engineering

Calls arrive at a switch, where they are assigned to any one of the available idle outgoing links. A call is blocked if all the links are busy. A call assigned to an idle link may be immediately lost with a probability which depends on the link. For exponential holding times and an arbitrary arrival process we show that the conditional distribution of the time to reach the blocked state from any state, given the sequence of arrivals, is independent of the policy used to route the calls. Thus the law of overflow traffic is independent of the assignment policy. An explicit formula for the stationary probability that an arriving call sees the node blocked is given for Poisson arrivals. We also give a simple asymptotic formula in this case.Work on this paper was done while the author was at Bellcore and at Berkeley.