On Ramanujan's Q-function

This paper presents new Gaussian approximations for the cumulative distribution function P(A λ ≤ s) of a Poisson random variable A λ with mean λ. Using an integral transformation, we first bring the Poisson distribution into quasiGaussian 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(A λ ≤ s).The results for P(A λ ≤ s) 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 → ∞. Changing β into α turns out to increase precision for small and moderately large values of s.The results for P(A λ ≤ 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.

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“…This was finally proved by Flajolet et al [10] in 1995 using singularity analysis. The connection with our framework is easily seen from…”

Section: A Conjecture Of Ramanujanmentioning

confidence: 79%

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

2008

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“…An asymptotic series for Q(n) was first derived by Ramanujan [1], and tight estimates are obtained in [4].…”

Section: The Tree Function and The Q-functionsmentioning

confidence: 99%

On the Analysis of Linear Probing Hashing

Flajolet

Poblete

Viola³

1998

Algorithmica

126

189

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Abstract. This paper presents moment analyses and characterizations of limit distributions for the construction cost of hash tables under the linear probing strategy. Two models are considered, that of full tables and that of sparse tables with a fixed filling ratio strictly smaller than one. For full tables, the construction cost has expectation O(n 3/2 ), the standard deviation is of the same order, and a limit law of the Airy type holds. (The Airy distribution is a semiclassical distribution that is defined in terms of the usual Airy functions or equivalently in terms of Bessel functions of indices − 1 3 , 2 3 .) For sparse tables, the construction cost has expectation O(n), standard deviation O( √ n), and a limit law of the Gaussian type. Combinatorial relations with other problems leading to Airy phenomena (like graph connectivity, tree inversions, tree path length, or area under excursions) are also briefly discussed.

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“…The classical problem is the following: what is the expected number of balls that you will need, if you throw these balls randomly into n bins, until one bins gets two balls? The answer to this question is given by Ramanujan's QFunction [34], and is equal to q(n) where q(n) = + . .…”

Section: Replicationmentioning

confidence: 99%

Fault-Tolerance Techniques for High-Performance Computing

2015

Computer Communications and Networks

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This report provides an introduction to resilience methods. The emphasis is on checkpointing, the de-facto standard technique for resilience in High Performance Computing. We present the main two protocols, namely coordinated checkpointing and hierarchical checkpointing. Then we introduce performance models and use them to assess the performance of theses protocols. We cover the Young/Daly formula for the optimal period and much more! Next we explain how the efficiency of checkpointing can be improved via fault prediction or replication. Then we move to application-specific methods, such as ABFT. We conclude the report by discussing techniques to cope with silent errors (or silent data corruption).This report is a slightly modified version of the first chapter of the monograph Fault tolerance techniques for high-performance computing edited by Thomas Herault and Yves Robert, and to be published by Springer Verlag.

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On Ramanujan's Q-function

Cited by 77 publications

References 7 publications

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

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

On the Analysis of Linear Probing Hashing

Fault-Tolerance Techniques for High-Performance Computing

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