Уравнение восстановления для процессов Кижима-Сумиты

Aim. The article analyses the properties of Kijima-Sumita incomplete renewal models. These processes generalize standard renewal processes and heterogeneous Poisson processes. Being able to sufficiently simply model incomplete renewal, these models allow calculating near-real dependability indicators of technical systems. Complete, minimal renewal, “worse-than-before-the-failure” situation are modeled with the choice of the single parameter q that essentially characterizes the incompleteness of renewal. This paper is the continuation of [1], it conducts a research based on the assumption that the time to first failure has the Weibull distribution that is widely used in the dependability theory. The Kijima-Sumita incomplete renewal models appeared relatively recently ant their properties remain largely understudied. In [1], in particular, a numerical solution was obtained of the leading flow function (of renewal function) of the first Kijima process model represented as a series of functions. The aim of this paper is to derive an integral renewal equation that would associate the failure flow parameter (or renewal function) with the first time to failure distribution. Additionally, some analytical solutions are given for specific cases, a numerical solution of the resulting renewal equations is suggested. The paper analyzes the effect of the incomplete renewal coefficient on the characteristics of the Kijima model’s failure flow. An interesting property of Kijima processes with a decreasing rate function of first operation time is discovered. Despite the expectations, the growth of the incompleteness of renewal in this case causes the reduction of the failure rate.  Methods. The calculations were performed in the R language and various numerical methods of finding integrals and solving integral equations, including the non-uniform mesh trapeze method and grand total method (GTM).  Conclusions. The paper deduces the renewal equation for the Kijima incomplete renewal processes. It also identifies some analytical solutions that demonstrate that the traditional renewal process and heterogeneous Poisson process are particular cases of the Kijima process. The results of numerical solutions for the Weibull distribution of first operation time are provided.