On the Distribution of Sums of Independent Exponential Random Variables Via Wilks’ Integral Representation

Favaro, Stefano; Walker, Stephen G.

doi:10.1007/s10440-008-9357-5

Cited by 13 publications

(6 citation statements)

References 15 publications

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“…Note that P d can be written as the sum of exponential random variables, for which the statistical properties are well investigated [11]. The probability density function of P d in this case can be described as [12]:…”

Section: Optimization Frameworkmentioning

confidence: 99%

Energy-efficient cooperative transmission in wireless networks with limited channel state information

Habibi

Ghrayeb

Aghdam

2015

2015 54th IEEE Conference on Decision and Control (CDC)

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This paper tackles the problem of minimumenergy cooperative transmission in wireless networks under the assumption of limited channel state information at the transmitters. It is assumed that only the average statistics of the fading channels are known to the transmitters. The objective here is to jointly optimize the set of relays and the transmission powers for both the broadcasting and cooperative transmission phases while satisfying probabilistic signal-to-noise ratio (SNR) constraints at the relays and at the destination node. Increasing the broadcasting power expands the set of potential relays and decreases the required power for cooperative transmission. Hence, there is a trade-off in the selection of the power values, which is addressed in this work by using a chanceconstrained optimization framework. Simulations are presented to demonstrate the efficacy of the proposed approach.

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Section: Optimization Frameworkmentioning

confidence: 99%

Energy-efficient cooperative transmission in wireless networks with limited channel state information

Habibi

Ghrayeb

Aghdam

2015

2015 54th IEEE Conference on Decision and Control (CDC)

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“…It can be easily checked that the firing interval D is computed similarly. However, the sum of Erlang distributed random variables with different parameters has been studied in [24], [25] and [26] gives an easily computable expression of the CDF for such a distribution. In that way, the firing interval of the sum of Erlang distributed random variables can be computed using standard approximation techniques of the quantile function (like bisections).…”

Section: Sequence Of Actionsmentioning

confidence: 99%

Tuning Temporal Features within the Stochastic π-Calculus

Paulevé

Magnin

Roux

2011

IIEEE Trans. Software Eng.

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The stochastic π-calculus is a formalism that have been shown of interest for modelling systems where the stochasticity and the delay of transitions are important features, such as the biochemical reactions. Commonly, duration of transitions within stochastic π-calculus models follow an exponential random variable. Underlying dynamics of such distributed models are expressed in terms of continuous-time Markov chains and can then be efficiently simulated and model-checked. However, the exponential law comes with a huge variance making difficult the modelling of systems with accurate temporal constraints. In this report, a technique for tuning temporal features within the stochastic π-calculus is presented. This method relies on the introduction of a stochasticity absorption factor by replacing the exponential distribution by the Erlang distribution, that is the sum of exponential random variables. A construction of this stochasticity absorption factor in the classical exponentially distributed stochastic π-calculus is provided. This report also offers tools for manipulating the stochasticity absorption factor and its link with timed intervals for firing transitions. Finally, the model-checking of such designed models is tackled through the support for the stochasticity absorption factor in the stochastic π-calculus translation to the probabilistic model checker PRISM.

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“…For a particular case with constraints on the values of the k i s, Van Khuong and Kong [3] provide the probability distribution function by inverting its Fourier transform. Using the Wilk's integral representation of the distribution of the product of independent beta random variables, Favaro and Walker [4] provide an alternative formula for FðÁÞ. We also refer the reader for more details to the review by Nadarajah [5].…”

Section: Introductionmentioning

confidence: 99%

A linear algebraic approach for the computation of sums of Erlang random variables

Legros

Jouini

2015

Applied Mathematical Modelling

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a b s t r a c tWe propose a matrix analysis approach to analytically provide the cumulative distribution function of the sum of independent Erlang random variables. This reduces to the characterization of the exponential of the involved generator matrix. We propose a particular basis of vectors in which we write the generator matrix. We find, in the new basis, a Jordan-Chevalley decomposition allowing to simplify the calculation of the exponential of the generator matrix. This is a simpler alternative approach to the existing ones in the literature.

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On the Distribution of Sums of Independent Exponential Random Variables Via Wilks’ Integral Representation

Cited by 13 publications

References 15 publications

Energy-efficient cooperative transmission in wireless networks with limited channel state information

Energy-efficient cooperative transmission in wireless networks with limited channel state information

Tuning Temporal Features within the Stochastic π-Calculus

A linear algebraic approach for the computation of sums of Erlang random variables

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