Lower and Upper Reliability Bounds for Consecutive-$k $-Out-of-$ n $:${\rm F}$ Systems

After a comprehensive review of reliability bounds for consecutive- k-out-of- n: F systems with statistically independent components having the same failure probability q (i.i.d. components), we introduce new classes of lower and upper bounds. Our approach is different from previous ones, and relies on alternating summation as well as on the monotony of some particular sequences of real numbers. The starting point is represented by the original formula given by de Moivre; and, by considering partial sums approximating it, new lower and upper bounds on the reliability of a consecutive- k-out-of- n: F system have been established. Simulation results show that all the lower and upper bounds considered here present very similar behaviors, as all of them are exponentially closing in on the exact reliability of a consecutive- k-out-of- n: F system. Additionally, the accuracy of the different bounds depends not only on the particular values of k and n, but also on the particular range of q (some of the new bounds being the most accurate ones over certain ranges).