## Hazard Risk Assessment Calculations

A series systell1 is one in which the entire systetll fails to operate if any one of its cOlllponents fails to operate. If such a system consists of n conlponents that function independently, then the reliability of the systeln is the product of the reliabilities of the individual cOlnponents. If Rs denotes the reliability of a series systelll and Ri denotes the reliability of the ith component; i = 1, ... , 11, tllen

(20.2.1)

A parallel system is one that fails to operate only if all its components fail to operate. If Ri is the reliability of the i th cOlllponent, then (I-Ri) is the probability that the ith cOlllponent fails~ i = 1, ... , n. Assunung that all n cOlllponents function independently, the probability that all n cOlllponellts fail is (l-R})(l-Rz)... (l-Rn). Subtracting this product frolll unity yields the following forlnula for Rp, tlle reliability of a parallel systell1. }

Rp =l-TI(l-R i );=} (20.2.2)

The reliability formulas for series and parallel systems can be used to obtain the reliability of a systell1 tlllt cOll1bines features of a series and a parallel systell1. Consider, for eX3111ple, tlle systell1 diagrmnnled in Fig. 20.2.1. COlllponents A, B, C, mId D have for their respective reliabilities 0.90, 0.80, 0.80, and 0.90. The systenI fails to operate if A fails, if B mId C both fail, or if D fails. COll1pOnent B and C constitute a parallel subsystem connected in series to cOll1ponents A and D. The reliability of the parallel subsystem is obtained by applying Eq. (20.2.2), which yields

Rp = 1-(1 - 0.80)(1 - 0.80) = 0.96

The reliability of the systetll is then obtained by applying Eq. (20.2.1), which yields

Rs =(0.90)(0.96)(0.90) =0.78

0.80

0.90 0.90

0.80 Figure 20.2.1. Syste111 \vith parallel and series components.