Advances in Applied Probability
- Adv. in Appl. Probab.
- Volume 40, Number 1 (2008), 122-143.
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(Aλ ≤ s) of a Poisson random variable Aλ 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(Aλ ≤ s). The results for P(Aλ ≤ s) differ from most classical results related to the central limit theorem in that the leading term Φ(β), with β = (s - λ)/√λ, 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(Aλ ≤ 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.
Adv. in Appl. Probab. Volume 40, Number 1 (2008), 122-143.
First available in Project Euclid: 16 April 2008
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Janssen, A. J. E. M.; van Leeuwaarden, J. S. H.; Zwart, B. Gaussian expansions and bounds for the Poisson distribution applied to the Erlang B formula. Adv. in Appl. Probab. 40 (2008), no. 1, 122--143. doi:10.1239/aap/1208358889. https://projecteuclid.org/euclid.aap/1208358889.