Statistical analysis of redundant systems with ” warm ” stand - by units

To warrant high reliability of key components of systems, stand-by units are used. If such a component fails and at least one not failed stand-by unit remains, it operates instead of the key component. If this one fails, the second stand-by unit is used and so on until the last stand-by unit fails. We suppose that switching is instantaneous and there are no repairs. If the stand-by units are functioning in the same ”hot” conditions as the main unit then usually after switching the reliability of the stand-by units does not change. But ”hot” redundancy has disadvantages because any one of the stand-by units fails earlier than the main one with the probability 0.5. If the stand-by units are not operating until the failure of the main unit (”cold” reserving), it is possible that during and after switching the failure rate increases because the stand-by unit is not ”warmed” enough [1]. So ”warm” reserving is sometimes used [2]: stand by units function under lower stress than the main one. In such a case the probability of the failure of the stand-by unit is smaller than that of the main unit and it is also possible that switching is fluent, i.e. switching from ”warm” to ”hot” conditions does not do any damage to units. What does it mean mathematically? Let us consider a system of m units: one main unit and m − 1 stand-by units. We shall use notation S(1,m − 1) for such systems. Denote by T1, F1 and f1 the failure time, the c.d.f. and the probability density function of the main unit. The failure times of the stand-by units are denoted by T2, . . . , Tm. In ”hot” conditions their distribution functions are also F1. In ”warm”