Advances in Applied Probability

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

A. J. E. M. Janssen, J. S. H. van Leeuwaarden, and B. Zwart

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This paper presents new Gaussian approximations for the cumulative distribution function P(Aλs) 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(Aλs). The results for P(Aλs) 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(Aλs) 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.

Article information

Adv. in Appl. Probab. Volume 40, Number 1 (2008), 122-143.

First available in Project Euclid: 16 April 2008

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 34E05: Asymptotic expansions 60K25: Queueing theory [See also 68M20, 90B22] 62E20: Asymptotic distribution theory

Erlang B formula Erlang loss model Poisson distribution normal distribution Gaussian integrals Lambert W-function Ramanujan's conjecture asymptotic expansion bounds


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.

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