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

Published online by Cambridge University Press:  01 July 2016

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: [email protected]
∗∗ Postal address: EURANDOM, PO Box 513, 5600 MB Eindhoven, The Netherlands. Email address: [email protected]
∗∗∗ Postal address: Georgia Institute of Technology, H. Milton Stewart School of Industrial and Systems Engineering, 765 Ferst Drive, Atlanta, GA 30332, USA. Email address: [email protected]
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Abstract

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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 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.

Type
General Applied Probability
Copyright
Copyright © Applied Probability Trust 2008 

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