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Renewal function asymptotics refined à la Feller

100%

Daley D.

Probability and Mathematical Statistics

2017

tom Vol. 37, Fasc. 2

291--298

Feller’s volume 2 shows how to use the Key Renewal Theorem to prove that in the limit x → ∞, the renewal function U(x) of a renewal process with nonarithmetic generic lifetime X with finite mean E(X) = 1/λ and second moment differs from its linear asymptote λx by the quantity 1/2λ2E(X2). His first edition (1966) (but not the second in 1971) asserted that a similar approach would refine this asymptotic result when X has finite higher order moments. The paper shows how higher order moments may justify drawing conclusions from a recurrence relation that exploits a general renewal equation and further appeal to the Key Renewal Theorem.

Sums of Independent Random Variables of the Modified Makeham Distribution

100%

Kosznik-Biernacka S.

Computational Methods in Science and Technology

2011

tom Vol. 17, No. 1-2

35-40

This paper shows how the renewal process in which the sums of independent random variables of the modified Makeham distribution appear can be approximated by the surrogate distribution.

A Simple Discrete Approximation for the Renewal Function

75%

Brezavšček A.

2013

tom 4

nr 1

65-75

Background: The renewal function is widely useful in the areas of reliability, maintenance and spare component inventory planning. Its calculation relies on the type of the probability density function of component failure times which can be, regarding the region of the component lifetime, modelled either by the exponential or by one of the peak-shaped density functions. For most peak-shaped distribution families the closed form of the renewal function is not available. Many approximate solutions can be found in the literature, but calculations are often tedious. Simple formulas are usually obtained for a limited range of functions only. Objectives: We propose a new approach for evaluation of the renewal function by the use of a simple discrete approximation method, applicable to any probability density function. Methods/Approach: The approximation is based on the well known renewal equation. Results: The usefulness is proved through some numerical results using the normal, lognormal, Weibull and gamma density functions. The accuracy is analysed using the normal density function. Conclusions: The approximation proposed enables simple and fairly accurate calculation of the renewal function irrespective of the type of the probability density function. It is especially applicable to the peak-shaped density functions when the analytical solution hardly ever exists.