Recovery function and optimization strategies of exploitation of technical systems which operation time distribution is a mixture of distributions

The paper discusses three typical problems of mathematical theory of reliability of technical systems restored. These are the choice of the distribution function of operation time of elements to failure in the recovery process, the finding of recovery function and determination of the optimal operating strategy function on the criterion of minimum maintenance cost intensity. For many classical distribution laws, for example exponential, Veybull-Gnedenko, Erlang, Gamma distribution, normal, truncated normal, lognormal, inverse Gaussian, Rayleigh these tasks are well investigated, At the same time these laws cannot describe a variety of distributions of operation time of elements of technical system. For example, probability densities of the listed laws are no more than unimodal, though density of operation time can be bimodal and even polymodal or when the distribution function of operation time is mixture of two or larger numbers of distribution functions from a set of the known laws of distributions. In this regard in work the listed tasks are studied for a case when operation time is distributed in the form of mix of functions of distributions. Special attention is paid to mix of exponential distributions. This result is from the fact that failure rate of such mix has a running-in period which is characteristic of an initial stage of operation of many technical systems after which failure rate is almost constant. This is important difference from a widely applicable exponential distribution in a reliability theory at which failure rate is constant - the period of a running-in is absent. For a simple recovery process explicitly recovery function (the expectation of the number of failures in the interval from zero to t) for mixtures of two exponential and two normal distributions has been obtained. For general process, when the first distribution function for operation time - the mixture n, and the second and following - a mixture of two exponential distributions, an explicit recovery function has been also received. For three strategies of operation of technical systems (in two of them preventive recovery held along with the emergency), with operating time distributed a mixture of exponential distributions, we consider the problem of choosing the optimal by criterion of a minimum intensity of the operating costs. Explicit formulas for point estimates of three parameters, which included in the mixture of two Erlang distributions of order n, are obtained by the method of moments.