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

Full-text: Access denied (no subscription detected) We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

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

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

Dates
First available in Project Euclid: 16 April 2008

Permanent link to this document
https://projecteuclid.org/euclid.aap/1208358889

Digital Object Identifier
doi:10.1239/aap/1208358889

Mathematical Reviews number (MathSciNet)
MR2411817

Zentralblatt MATH identifier
1137.62006

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

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

Citation

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.


Export citation

References

  • Atar, R., Mandelbaum, A. and Shaikhet, G. (2006). Queueing systems with many servers: null controllability in heavy traffic. Ann. Appl. Prob. 16, 1764--1804.
  • Barndorff-Nielsen, O. E. and Cox, D. R. (1990). Asymptotic Techniques for Use in Statistics. Chapman and Hall, London.
  • Bhattacharya, R. N. and Rao, R. R. (1976). Normal Approximations and Asymptotic Expansions. John Wiley, New York.
  • Borst S., Mandelbaum, A. and Reiman, M. (2004). Dimensioning large call centers. Operat. Res. 52, 17--34.
  • Brockmeyer, E., Halstrøm, H. L. and Jensen, A. (1948). The life and works of A. K. Erlang. Trans. Danish Acad. Tech. Sci. 1948, 277pp.
  • Corless, R. M. et al. (1996). On the Lambert $W$-function. Adv. Comput. Math. 5, 329--359.
  • Feller, W. (1966). An Introduction to Probability Theory and Its Applications, Vol. II. John Wiley, New York.
  • Flajolet, P., Grabner, P. J., Kirschenhofer, P. and Prodinger, H. (1995). On Ramanujan's Q-function. J. Comput. Appl. Math. 58, 103--116.
  • Gans, N., Koole, G. and Mandelbaum, A. (2003). Telephone call centers: tutorial, review and research prospects. Manufacturing Service Operat. Manag. 5, 79--141.
  • Halfin, S. and Whitt, W. (1981). Heavy-traffic limits for queues with many exponential servers. Operat. Res. 29, 567--588.
  • Hwang, H. K. (1997). Asymptotic estimates of elementary probability distributions. Studies Appl. Math. 4, 339--417.
  • Jagerman, D. (1974). Some properties of the Erlang loss function. Bell System Tech. J. 53, 525--551.
  • Janssen, A. J. E. M., van Leeuwaarden, J. S. H. and Zwart, B. (2007). Corrected asymptotics for a multi-server queue in the Halfin--Whitt regime. Submitted.
  • Janssen, A. J. E. M., van Leeuwaarden, J. S. H. and Zwart, B. (2007). Corrected server staffing by expanding Erlang C. Submitted.
  • Jeffrey, D. J., Corless, R. M., Hare, D. E. G. and Knuth, D. E. (1995). Sur l'inversion de $y^\alpha \re^y$ au moyen des nombres de Stirling associés. C. R. Acad. Sci. Paris 320, 1449--1452.
  • Johnson, N. L., Kotz, S. and Kemp, A. W. (1992). Univariate Discrete Distributions, 2nd edn. John Wiley, New York.
  • Kelly, F. P. (1991). Loss networks. Ann. Appl. Prob. 1, 319--378.
  • Michel, R. (1993). On Berry--Esseen bounds for the compound Poisson distribution. Insurance Math. Econom. 13, 35--37.
  • Petrov, V. V. (1995). Limit Theorems of Probability Theory. Clarendon Press, Oxford.
  • Reed, J. (2006). The G/GI/N queue in the Halfin--Whitt regime. Submitted.
  • Temme, N. M. (1979). The asymptotic expansion of the incomplete gamma functions. SIAM J. Math. Anal. 10, 757--766.
  • Temme, N. M. (1992). Asymptotic inversion of incomplete gamma functions. Math. Comput. 58, 755--764.
  • Temme, N. M. (1996). Special Functions. John Wiley, New York.