Some Properties of the Erlang Loss Function

T h e Erlang loss function (1) is fundamental to the study of telephone trunking problems. A. K. Erlang 1 used B(N, a) to express the probability t h a t a call, which is a member of a Poisson stream of parameter a, arriving at a group of N telephone trunks will be rejected. Later studies of trunking problems have shown the desirability of enlarging the scope of applications of the loss function. For example, the consideration of t r u n k groups with nonintegral number of trunks arises in determining the equivalent number of trunks in Wilkinson's "equivalent random m e t h o d . " 2 Methods for accomplishing the computation by interpolation are given by Rapp 3 while continued fraction procedures for accurate computation arc given by Levy-Soussan 4 and Burke. 5 Derivatives with respect to N and a arise in optimal trunk group size apportionment problems. See, for example, Akimaru and Nishimura 6 , 7 who studied such models 525 and prepared tables of derivatives. In some investigations, rapid and accurate approximate computations of B(N, a) for very large trunk groups are needed. This occurred in the study of certain satellite telephonic communication problems. 8,9 The need thus arises of enlarging the definition of B(N, a) as given in (1). Of course, t h a t is done implicitly in the above investigations. It has been customary to extend the definition of B(N, a) by use of an integral formula (Theorem 3) ascribed to Fortet. This integral formula is used in (23) to define a transcendent, B (z, a), for complex z and a.