This paper develops the properties of the Erlang hss 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 dB(x, a)/dx.

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Blocking is considered for an N‐trunk group of exponential servers with Poisson‐offered load whose rate parameter varies with time. The infinite trunk case is solved by means of a rapidly convergent series of Poisson‐Charlier polynomials. This solution is used to obtain practical approximations of blocking probability, transition probabilities, and recovery function for general time‐variable offered load in the finite trunk‐group case. An integral equation is derived satisfied by the blocking probability in the general case. In the situation of constant offered load, two additional methods are derived for providing easily computable approximations; one based on the integral equation, the other based on an approximate inversion of the Laplace transform. To aid in the latter approximation, bounds on the roots of Poisson‐Charlier polynomials are obtained; in particular, an approximation is obtained for the dominant root. The inversion of the integral equation is studied with the purpose of providing the basis for future investigations of errors of approximation. Curves are provided for a number of examples permitting comparison of exact and approximate solutions.

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