A generalization of the Erlang formula of traffic engineering

Consider a node in a communication network with n outgoing links grouped into k trunks of n1 ...., nk links respectively. n1+...+nk = n. Calls arrive in a Poisson stream of rate λ The state of the node is specified by the number of idle links in each trunk. A policy is a rule by which a call, finding the node in some state, is assigned to one of the available links in one of the available outgoing trunks. The links are assumed to have exponential holding times with mean 1/µ which are independent, and are independent of the arrival process. Further, a call assigned to trunk α, 1;α ⩽α ⩽k is immediately lost with probability (1--ϵ;α)--this feature models the nature of the links and the congestion downstream of the node along that route. A call is said to be blocked if all the outgoing links are busy when it arrives. It is known that the blocking probability is independent of the assignment policy. We give an explicit closed form formula for the blocking probability Pb = 1/Σn1j1 = 0 .. ΣΣnkjk = 0 [n1, j1] ... [nk, jk](j1 + ... + jk)! (µ/λ1)j1 ... (µ/λk)jk where λ1 = ϵ1 . λ ..., λk = λk . λ. This generalizes the well-known Erlang formula of traffic engineering.