Reliability Characteristics for Two Subsystems in Series or Parallel or n Subsystems in m_out_of_n Arrangement
(by Don L. Lin)
1Introduction
This is for reliability of two subsystems in series or in parallel. Also discussed is the system consists of n identical subsystems in parallel, the system is declared as failed if m or more subsystems fail (the m_out_n case).
The system characteristics such as “system failure rate”, “system Mean Time Between Failure”, “system availability and unavailability”, and “system mean down time” are derived.
2Connection in series
This section describes a system consists of two (non-identical) subsystems in series.
2.1System Failure Rate
For just one subsystem, the failure rate is λ1. The probability of failure in dt is λ1dt. For two subsystems in series, the probability of failure in dt is (λ1dt + λ2dt ). The system failure rate is thus (λ1 + λ2 ).
λseries = λ1 + λ2
The reliability function is R(t)=exp[-(λ1 + λ2)t].
2.2System MTBF
From the exponential form of the reliability function, it is obvious that
MTBFseries=1/(λ1 + λ2 )=
2.3System Availability and unavailability
For the system to be available, each subsystem should be available. Thus,
Conversely, the unavailability is
2.4System Mean Down Time for Repairable subsystems
If two subsystems are both repairable, one with mean down time MDT1 and the other MDT2, what is the mean down time for the two subsystems in series?
At any instance in time, the system is in one of the 4 states:
- both subsystems functional,
- only subsystem #1 is non-functional,
- only subsystem #2 is non-functional,
- both subsystems are non-functional.
The last 3 cases are responsible for the system being non-functional. It is assumed that the 4th case has negligible probability. Given the system is down, what is the probability that it is because the subsystem #1 is non-functional? It is obviously . Since subsystem #1 needs MDT1 to repair, the repair time associated with repairing subsystem #1 is then
A similar expression is true for subsystem #2. Summing them up, one gets
3Connection is Parallel
Here the two subsystems are repairable. The mean down times are MDT1 and MDT2.
3.1System Failure Rate
If the system just consists of subsystem #1, then the system failure rate is λ1. The probability of failure in dt is λ1dt. Adding subsystem #2 in parallel, the probability for system failure in dt is λ1dt scaled down by the probability that the subsystem #2 is in the failure state. The probability to find the subsystem #2 in the failure state is given by . Assuming and using , the scaled down failure rate for subsystem #1 is then given by. Likewise, the scaled down failure rate for subsystem #2 is . Consequently,
3.2System MTBF
Taking the approach that the inverse of the failure rate is MTBF (true for exponential distribution), one gets
MTBFparallel=1/λparallel=
It is noted that if the two subsystems are not repairable, then the MTBF for the parallel case is the sum of the individual MTBF’s.
3.3System Availability and unavailability
For the system to be available, either subsystem should be available. Thus,
Conversely, the unavailability is
(1)
3.4System Mean Down Time for Repairable subsystems
From the definition of
one can get the MDT for the parallel case by using Eq.(1) above.
Consequently,
4M out of N Parallel Subsystems
If a system consists of n parallel, identical subsystems and the system is down if there are m or more subsystems down, what are the formulas for system failure rate, system MTBF, system availability, and system mean down time?
4.1System Failure Rate
If the system just consists of subsystem #1, then the system failure rate is λ. The probability of failure in dt is λdt. To have a system failure, we need to have other (m-1) subsystems in the failure state. The chance that any one subsystems is in the failure state is given by MDT/(MTBF+MDT), or (MDT/MTBF), if we assume MDT< MTBF. To find (m-1) subsystems in the failure state, the probability is . There are ways to group (m-1) subsystems out of (n-1) subsystems. Also, we can choose any subsystem to be the #1 subsystem in the analysis. Putting all together, one has
(2)
This is the failure rate for exactly m subsystem failures. The failure rate for more than m subsystem failures is going to be smaller by a factor of (.
For a consistency check, we consider n=m=2. This is a system consisting of two parallel, identical subsystems. When m=2 subsystems fail, the system fails. This is what discussed in Section 2. And Eq.(2) for this case is which agrees with the formula in Section 2.
4.2System MTBF
Taking the approach that the inverse of the failure rate is MTBF (true for exponential distribution), one gets
MTBFm_out_of_n=1/λm_out_of_n=
.
4.3System Availability and unavailability
For the system to be available, at least (n-m+1) subsystems should be available. Thus,
Using the following equality,
we can rewrite the availability as
And the unavailability is given (again, for MDT<MTBF), by
4.4System Mean Down Time for Repairable subsystems
From the definition of
one can get the MDT for the m_out_of_n case by using in Sections 4.2 and 4.3 for UAm_out_of_n and MTBFm_out_of_n.
Consequently,
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5All Formulas in One Table
Two subsystems In Series( is failure rate ) / Two subsystems In Parallel / n identical subsystems in parallel; system fails if m or more subsystems fail. (m_out_of_n)
System
Failure
Rate / λseries = λ1 + λ2 / /
System
MTBF / MTBFseries= / MTBFparallel= /
System
Avail-
ability
(A) / / /
System
Unavail-
ability
(UA) / / /
System
Mean
Down
Time
(MDT) / MDTseries=
/ /
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