Gaussian expansions and bounds for the Poisson distribution applied to the Erlang B formula

This paper presents new Gaussian approximations for the cumulative distribution function P(A¿ = s) of a Poisson random variable A¿ with mean ¿. Using an integral transformation, we first bring the Poisson distribution into quasi-Gaussian form, which permits evaluation in terms of the normal distribution function F. The quasi-Gaussian form contains an implicitly defined function y, which is closely related to the Lambert W-function. A detailed analysis of y leads to a powerful asymptotic expansion and sharp bounds on P(A¿ = s). The results for P(A¿ = s) differ from most classical results related to the central limit theorem in that the leading term F(ß), with ß = (s - ¿)/v¿, is replaced by F(a), where a is a simple function of s that converges to ß as s tends to 8. Changing ß into a turns out to increase precision for small and moderately large values of s. The results for P(A¿ = s) lead to similar results related to the Erlang B formula. The asymptotic expansion for Erlang's B is shown to give rise to accurate approximations; the obtained bounds seem to be the sharpest in the literature thus far.