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 λ ≤ s ) of a Poisson random variable A λ with mean λ. Using an integral transformation, we first bring the Poisson distribution into quasi-Gaussian form, which permits evaluation in terms of the normal distribution function Φ. 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 λ ≤ s ). The results for P( A λ ≤ s ) differ from most classical results related to the central limit theorem in that the leading term Φ( β ), with is replaced by Φ( α ), where α is a simple function of s that converges to β as s tends to ∞. Changing β into α turns out to increase precision for small and moderately large values of s . The results for P( A λ ≤ 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.