Some Properties of the Erlang Loss Function

This paper develops the properties of the Erlang loss function, B (N, a), used in telephone traffic engineering. The extension to a transcendental function of two complex variables is constructed, thus permitting the methods of complex analysis to be employed for the further study of its properties. Exact representations, Rodrigues formulas, and addition theorems are given both for the loss function and for the related Poisson-Charlier polynomials. Asymptotic formulas and approximations are developed for the loss function and also for its derivatives. A table of coefficients is included which, together with one of the asymptotic formulas, permits computation of B (N, a) by simple means even when the number of trunks, N, is very large. This same table is used to obtain ∂B(x, a)/∂x.