The Asymptotic Expansion of the Incomplete Gamma Functions

Temme, Nico

doi:10.1137/0510071

Cited by 82 publications

(77 citation statements)

References 4 publications

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“…We quote here that computations of the functions c i , involved in (16), as well as the methods to get them are detailed in [56], [57]. Now using (16) and letting the value of j (resp.…”

Section: May 5 2017 Draftmentioning

confidence: 99%

Extremal properties of three-dimensional sensor networks with applications

Ravelomanana

2004

IEEE Trans. on Mobile Comput.

149

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In this paper, we analyze various critical transmitting/sensing ranges for connectivity and coverage in three-dimensional sensor networks. As in other large-scale complex systems, many global parameters of sensor networks undergo phase transitions: For a given property of the network, there is a critical threshold, corresponding to the minimum amount of the communication effort or power expenditure by individual nodes, above (resp. below) which the property exists with high (resp. a low) probability.For sensor networks, properties of interest include simple and multiple degrees of connectivity/coverage. First, we investigate the network topology according to the region of deployment, the number of deployed sensors and their transmitting/sensing ranges. More specifically, we consider the following problems:Assume that n nodes, each capable of sensing events within a radius of r, are randomly and uniformly distributed in a 3-dimensional region R of volume V , how large must the sensing range r Sense be to ensure a given degree of coverage of the region to monitor? For a given transmission range r Trans , what is the minimum (resp. maximum) degree of the network? What is then the typical hop-diameter of the underlying network? Next, we show how these results affect algorithmic aspects of the network by designing specific distributed protocols for sensor networks.

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“…We quote here that computations of the functions c i , involved in (16), as well as the methods to get them are detailed in [56], [57]. Now using (16) and letting the value of j (resp.…”

Section: May 5 2017 Draftmentioning

confidence: 99%

Extremal properties of three-dimensional sensor networks with applications

Ravelomanana

2004

IEEE Trans. on Mobile Comput.

149

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“…This gives the same coefficients C k (up to scaling) as derived in [10] for the incomplete gamma functions. The first term in the expansion (30) agrees with the first term in the Lugannani and Rice formula [8] for the sums of i.i.d.…”

Section: Erlang Distribution (α = 0)mentioning

confidence: 55%

A uniform asymptotic expansion for weighted sums of exponentials

Leeuwaarden

Temme

2011

Statistics & Probability Letters

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We consider the random variable Z n,α = Y 1 + 2 α Y 2 + . . . + n α Y n , with α ∈ R and Y 1 , Y 2 , . . . independent and exponentially distributed random variables with mean one. The distribution function of Z n,α is in terms of a series with alternating signs, causing great numerical difficulties. Using an extended version of the saddle point method, we derive a uniform asymptotic expansion for P(Z n,α < x) that remains valid inside (α ≥ −1/2) and outside (α < −1/2) the domain of attraction of the central limit theorem. We discuss several special cases, including α = 1, for which we sharpen some of the results in [7].

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“…Details of this analysis are given in Appendix D. The first coefficient is found to be 14) where C 0 (η) is the first coefficient in the expansion in (5.3).…”

Section: Higher Order Approximationmentioning

confidence: 99%

“…For all other cases with p ∈ (0, 1) we employ a uniform asymptotic expansion for the incomplete gamma function derived in [14], in which the standard normal distribution function (or error function) is the leading approximant. In [15] the first coefficients in the expansion were derived by using a perturbation method for a differential equation, but this approach cannot be transferred directly to the inversion of the Erlang B formula.…”

Section: Outlinementioning

confidence: 99%

“…As pointed out by Jagerman and many others, the Erlang B formula can be expressed in terms of the incomplete gamma function. Temme [15] has considered the inversion problem for the incomplete gamma function based on a uniform asymptotic expansion developed in [14]. We shall derive various asymptotic expansions for the inverse λ of the Erlang B formula using asymptotic expansions for the incomplete gamma function.…”

Section: Introductionmentioning

confidence: 99%

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Asymptotic Inversion of the Erlang B Formula

Leeuwaarden¹,

Temme²

2009

SIAM J. Appl. Math.

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The Erlang B formula represents the steady-state blocking probability in the Erlang loss model or M/M/s/s queue. We derive asymptotic expansions for the offered load that matches, for a given number of servers, a certain blocking probability. In addressing this inversion problem we make use of various asymptotic expansions for the incomplete gamma function. A similar inversion problem is investigated for the Erlang C formula.

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The Asymptotic Expansion of the Incomplete Gamma Functions

Abstract: Abstract. Earlier investigations on uniform asymptotic expansions of the incomplete gamma functions are reconsidered. The new results include estimations for the remainder and the extension of the results to complex variables.

Cited by 82 publications

References 4 publications

Extremal properties of three-dimensional sensor networks with applications

Extremal properties of three-dimensional sensor networks with applications

A uniform asymptotic expansion for weighted sums of exponentials

Asymptotic Inversion of the Erlang B Formula

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