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

Part of: Special processes Asymptotic theory Distribution theory

Published online by Cambridge University Press:  01 July 2016

A. J. E. M. Janssen ,
J. S. H. van Leeuwaarden  and
B. Zwart
A. J. E. M. Janssen*
Affiliation:
Philips Research
J. S. H. van Leeuwaarden*
Affiliation:
Eindhoven University of Technology and EURANDOM
B. Zwart*
Affiliation:
Georgia Institute of Technology
*
∗ Postal address: Philips Research, Digital Signal Processing Group, HTC-36, 5656 AE Eindhoven, The Netherlands. Email address: a.j.e.m.janssen@philips.com
∗∗ Postal address: EURANDOM, PO Box 513, 5600 MB Eindhoven, The Netherlands. Email address: j.s.h.v.leeuwaarden@tue.nl
∗∗∗ Postal address: Georgia Institute of Technology, H. Milton Stewart School of Industrial and Systems Engineering, 765 Ferst Drive, Atlanta, GA 30332, USA. Email address: bertzwart@gatech.edu
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Abstract

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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 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.

Keywords

Erlang B formulaErlang loss modelPoisson distributionnormal distributionGaussian integralsLambert W-functionRamanujan's conjectureasymptotic expansionbounds

MSC classification

Primary: 34E05: Asymptotic expansions 60K25: Queueing theory 62E20: Asymptotic distribution theory
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 40 , Issue 1 , March 2008 , pp. 122 - 143
Copyright
Copyright © Applied Probability Trust 2008 

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