A Multi-Component General Discrete System Subject to Different Types of Failures with Loss of Units

Discrete systems are used in several fields such as reliability and computing and electronics in digital systems. On the other hand, there are systems that cannot be continuously monitored and they can be observed only at certain periods of time, via inspections, for example. In this paper a repairable multi-component system subject to internal and accidental external failures with loss of units is developed. The system is composed of a finite number of units, including the main one and the others disposed in cold standby. If a repairable failure occurs, the main unit enters the repair channel. On the other hand, the unit is removed if the failure is non-repairable. A repairman is considered. The distribution of the lifetime of the main unit is a general one and its phase-type representation is considered. Accidental failures occur according to a general discrete renewal process. The model is developed in detail and the up period is worked out up to no units and up to total failure of the system. Some reliability measures of interest such as the conditional probability of different types of failures are calculated. The operating of the system is analysed according to rewards introduced in the model. We have built complex algorithms for calculating the measures defined in this paper. We have introduced the RG-factorization method to work out these measures by means of matrices with low order. The results have been implemented computationally with Matlab. An example illustrates the model and the number of units is optimised according to the average net reward.