Basic reliability cookbook
Standby
Reliability
Standby reliability is
a variation of K-of-N reliability and is used when the N-K inactive
elements have a different reliability than the K active elements.
It is typically used when the inactive elements are in warm
standby (power off) and, as a consequence, have a lower failure rate
than the active elements.
Note that inactive elements with power off are often called cold spares,
not to be confused with the reliability term cold standby (standby
elements with zero failure rate.)
Given an element with
primary failure rate
λ, the standby
failure rate
λs
is typically estimated as 0.1 *
λ.
The reliability PP
of a primary (active) element is e-λT
and the reliability PS
of a standby element is e-λsT.
The time interval T is typically the desired lifetime of the
system.
Given a set of N
identical elements with K elements active and N-K elements as cold
spares, the reliability of the K-of-N combination is the sum of the
reliability of K primary elements plus the incremental reliability of
each of the N-K spare elements.
The reliability P0
of K identical active elements is (PP)K.
For spare element J (from 1 to N-K), the incremental reliability
PJ
is (PJ-1
/ J) * (((J-1) *
λS)
+ (K *
λ))
* ((1-PS)
/
λS).
The combined K-of-N
reliability of the N elements is P0
+ P1
+ P2
+ … + PN-K.
For example, a 4-of-6
system has elements with a failure rate of 2 per million hours and a
standby failure rate of 0.2 per million hours.
The reliability of the active elements for a fifteen year
lifetime is e(-2*0.1314)
= 0.768896. The reliability
of the standby elements for a fifteen year lifetime is e(-0.2*0.1314)
= 0.974062.
P0
= (0.768896)4
= 0.349518. P1
= P0
* (4 * 2) * ((1-0.974062) / 0.2) = 0.362628.
P2
= (P1
/ 2) * ((1 * 0.2) + (4 * 2)) * ((1-0.974062) / 0.2) = 0.192817.
The K-of-N reliability
is P0
+ P1
+ P2
= 0.349518 + 0.362628 + 0.192817 = 0.9050.
For the common case of
a 1-of-2 standby configuration, a simplified equation PP
+ ((λ
*
PP)
/
λs)
* (1-PS)
can be used. As an example,
a system with 0.5 per million hours primary failure rate and 0.05 per
million hours standby failure rate has a desired lifetime of eight
years. The PP
is 0.9656 and the PS
is 0.9965. The combined
reliability is 0.9656 + ((0.5 * 0.9656) / 0.05) * (1 - 0.9965) = 0.9993.
Failure Rate | Serial Reliability | Parallel Reliability | K of N Reliability | Standby Reliability | Duty Cycle Reliability