Bathtub Curve and System Reliability
Initial Failure Phase
During the initial failure phase, the failure rate decreases over time. The failure rate function λ(t)λ(t) is generally a decreasing function.
Normal Operation Phase
In the normal operation phase, the failure rate is constant, i.e., λ(t)=λλ(t)=λ. The reliability function R(t)R(t) can be defined as follows:
R(t)=e−λt
R(t)=e−λt
Wearout Phase
In the wearout phase, the failure rate increases over time. The Weibull distribution is often used to model this, given by:
f(t)=λmtm−1e−λtm
f(t)=λmtm−1e−λtm
F(t)=1−e−λtm
F(t)=1−e−λtm
R(t)=e−λtm
R(t)=e−λtm
λ(t)=λmtm−1
λ(t)=λmtm−1
Mean Time between Failures (MTBF) and Mean Time to Failures (MTTF)
Defined as the average operational time until a failure occurs. For a constant failure rate λλ, it is defined as 1λλ1.
Mean Time to Repair (MTTR)
Defined as the average time required for repair.
Availability
Defined as the ratio of the system being operational over a given period.
Maintainability
Defined as the ratio of maintenance work to prevent failures.
Length of Operating Period
The period during which the system can be used, considering factors like wear and tear or obsolescence.
System Reliability in Serial and Parallel Configurations
For nn subsystems with reliabilities Ri(t)Ri(t) and failure rates λi(t)λi(t):
In a serial system:
R(t)=R1(t)⋅R2(t)⋅…⋅Rn(t)
R(t)=R1(t)⋅R2(t)⋅…⋅Rn(t)
λ(t)=λ1(t)+λ2(t)+…+λn(t)
λ(t)=λ1(t)+λ2(t)+…+λn(t)
In a parallel system:
R(t)=1−(1−R1(t))⋅(1−R2(t))⋅…⋅(1−Rn(t))
R(t)=1−(1−R1(t))⋅(1−R2(t))⋅…⋅(1−Rn(t))
Fault Tolerance Strategies
Various strategies like Fail-Safe, Fail-Soft, and Fool-Proof are implemented to improve system reliability.
Discussion