Abstract:
In this paper, the renewal equation is studied. It is the Volterra integral convolution equation of the second kind with a difference kernel. This equation is considered both for the renewal density and for its primitive, the renewal function. The renewal function is essential in the theory of technical systems reliability not only as a descriptive characteristic, but also for operational strategies optimization in the preventive maintenance management, assuming the implementation of the recurrent recovery flows model. A certain analytical method is suggested for obtaining an asymptotic representation of the recovery equation solution for the special class of distributions under some given conditions. The validity of the stated expansion was checked for the exponential distribution, which is basic in the reliability mathematical theory. To show that the class of the described distributions is not an empty set, as an example, the two-parameter Weibull-Gnedenko distribution was considered, which is a natural generalization of the exponential distribution. The apparatus of series theory and the generating moment function method are used. The last is a Laplace transform of non-negative continuous random variable density distributions. The Chebyshev-Markov-Stieltjes moment problem is also highlighted. It means the possibility of the unique distribution restoration by the sequence of its moments. This problem is significant for the mentioned expansion. The expression for the renewal equation solution in the case of the renewal density has the form of Gram-Charlier’s type series in terms of probability moments.
Fulltext:
PDF file (1005 kB)