K-out-of-ДО parallel systems

This is a parallel system of M component for which the system would function if K (K < M) or more components function. This type of system also is called a partially redundant system. The general reliability formula for this system is rather cumbersome. For components having an identical reliability function, that is, ps, m(t) = ps(t), the system reliability and unreliability, when component performances are independent, are

M

Ps, sys(t) = 53 Cm, j [Ps(t)]j [1 — Ps(t)]M-j (7.58a)

j =k

K-1

and pf, sys(t) = ]T Cm, j [Ps(t)]j [1 — Ps(t)]M-j (7.58b)

j =0

in which CM, j = M!/[j!(M — j)!] is a binomial constant. Computationally, whether to calculate ps, sys(t) or pf, sys(t) is dictated by the number of terms
involved in the summation. Furthermore, if the failure density function is an exponential distribution, the system reliability can be expressed as

M

ps, Sys(t) = £ Cm, j (e-Xt)j (1 — e-u)M-j (7.59)

j =k

The failure density function for the system f sys(t) based on the system reli­ability in Eq. (7.58a) is

The availability and unavailability of the system can be obtained from sub­stituting component availability for component reliability in Eqs. (7.58a) and (7.58b), respectively.

Example 7.10 As an example of a K-out-of-M system, consider a pumping system with three pumps, one of which is on standby, all with constant failure rates of X = 0.0005 failures/h. The system reliability for t = 1000 h, M = 3, and K = 2 is

ps, sys(t = 1000) = c3,2(e-(a0005>(1000>)2(1 — е-(0-0005)(шю)) + Сз, з(е-(0’0005)(1000))3

= 3(e-(0.0005)(1000))2 _ 2(e-(0.0005)(1000))3

= 1.1036 — 0.4463 = 0.6573