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### Failure frequency of repairable monotone multi-state systems.Part 1. General formulae and conversion rules

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Basic reliability indices of repairable systems are availability and failure frequency. These indices may be defined for a fixed instant of time t, or for the steady state, when t tends to infinity. The instantaneous availability A(t) of a system is defined as the probability that the system is in an up state at time t. The instantaneous failure frequency w(t) of a system (also referred to as the rate of occurrence of failures) is defined as the mean number of the system failures per unit time, so that by integrating it over the time interval [0,T], we obtain the mean number of the system failures in the interval. The steady state (or asymptotic, limiting) availability A = A(∞) and failure frequency w = w(∞) are defined as the limiting values of A(t) and w(t), respe[...]

### Failure frequency of repairable monotone multi-state systems. Part 2.Application of dual number algebra

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In part 1 of this paper [8], general formulae and conversion rules for calculating steady state failure frequency of repairable monotone multi-state systems (MMS) were presented. According to these conversion rules, the calculation of failure frequency is performed in two stages. In the first stage, an expression for steady state availability is obtained. In the second stage, the steady state failure frequency is calculated by applying the conversion rules to availability expression. Although the conversion rules are very simple, the main disadvantage of this two-stage procedure is that the availability should be given in symbolic algebraic expression. Therefore, it is difficult and/or impractical to implement this two-stage procedure using the Universal Generating Function (UG[...]

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