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Reliability Characteristics for Two Subsystems in Series
or Parallel or n Subsystems in m_out_of_n Arrangement
(by Don L. Lin)
1 Introduction
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.
2 Connection in series
This section describes a system consists of two (non-identical) subsystems in series.
2.1 System 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].
2.2 System MTBF
From the exponential form of the reliability function, it is obvious that
MTBFseries=1/(λ1 + λ2 )=
MTBF1  MTBF2
MTBF1  MTBF2
2.3 System Availability and unavailability
For the system to be available, each subsystem should be available. Thus,
Aseries  A1  A2
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Conversely, the unavailability is
UAseries  1  Aseries  1  (1  UA1 )  (1  UA2 )  UA1  UA2  UA1  UA2
2.4 System 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
1
1  2
. Since subsystem #1 needs MDT1 to repair, the repair time associated with
repairing subsystem #1 is then
1
1  2
* MDT1
A similar expression is true for subsystem #2. Summing them up, one gets
MDT series 
MTBF1  MDT2  MTBF2  MDT1
MTBF1  MTBF2
3 Connection is Parallel
Here the two subsystems are repairable. The mean down times are MDT1 and MDT2.
3.1 System 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
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is in the failure state. The probability to find the subsystem #2 in the failure state is
MDT2
1
given by
. Assuming MDT2  MTBF2 and using MTBF2 
,
2
MTBF2  MDT2
the scaled down failure rate for subsystem #1 is then given by 1  2  MDT2 .
Likewise, the scaled down failure rate for subsystem #2 is 1  2  MDT1 .
Consequently,
 parallel  1  2  ( MDT1  MDT 2 )
3.2 System MTBF
Taking the approach that the inverse of the failure rate is MTBF (true for exponential
distribution), one gets
MTBF1  MTBF2
MTBFparallel=1/λparallel=
MDT1  MDT2
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.
3.3 System Availability and unavailability
For the system to be available, either subsystem should be available. Thus,
Aparallel  A1  A2  A1  A2
Conversely, the unavailability is
UAparallel  1  A parallel  1  ( A1  A2  A1  A2 )  (1  A1 )  (1  A2 )  UA1  UA2 (1)
3.4 System Mean Down Time for Repairable subsystems
From the definition of
Unavailability 
MDT
MDT

MTBF  MDT MTBF
one can get the MDT for the parallel case by using Eq.(1) above.
UAparallel 
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MDT parallel
MTBF parallel

MDT parallel
MTBF1  MTBF 2
MDT1  MDT 2
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UA1  UA2 
MDT1 MDT2

MTBF1 MTBF2
Consequently,
MDT parallel 
MDT1  MDT2
MDT1  MDT2
4 M 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?
4.1 System 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 (m1) 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
MDT m 1
(
) . There are n 1 C m1 ways to group (m-1) subsystems out of (n-1)
MTBF
subsystems. Also, we can choose any subsystem to be the #1 subsystem in the
analysis. Putting all together, one has
m _ out _ of _ n    (
MDT m1
n!
) n1 C m1  n 
m  MDT m1 (2)
MTBF
(n  m)!(m  1)!
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 (   MDT ) .
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 system  2  (2  MDT ) which
agrees with the formula in Section 2.
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4.2 System MTBF
Taking the approach that the inverse of the failure rate is MTBF (true for exponential
distribution), one gets
MTBF m
MTBFm_out_of_n=1/λm_out_of_n=
n!
 MDT m 1
(n  m)!(m  1)!
.
4.3 System Availability and unavailability
For the system to be available, at least (n-m+1) subsystems should be available. Thus,
Am _ out _ of _ n 
n
n!
A i (1  A) n i
i  n  m 1 ( n  i )!i!

Using the following equality,
n
1  [ A  (1  A)] n  
i 0
n!
Ai (1  A) n i
(n  i )!i!
we can rewrite the availability as
nm
Am _ out _ of _ n  1  
i 0
n!
n!
A i (1  A) n i  1 
(1  A) m
(n  i )!i!
m!(n  m)!
And the unavailability is given (again, for MDT<<MTBF), by
UAm _ out _ of _ n 
n!
UAm
m!(n  m)!
4.4 System Mean Down Time for Repairable subsystems
From the definition of
UAm _ out _ of _ n 
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MDT m _ out _ of _ n
MTBFm _ out _ of _ n
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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,
MDT m _ out _ of _ n 
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MDT
m
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5 All Formulas in One Table
Two subsystems In Series
(  is failure rate )
Two subsystems In Parallel
System
Failure
Rate
λseries = λ1 + λ2
 parallel  1  2  ( MDT1  MDT 2 )
System
MTBF
MTBFseries=
System
Availability
(A)
MTBF1  MTBF2
MTBF1  MTBF2
Aseries  A1  A2
System
Unavail- UAseries  UA1  UA2  UA1  UA2
ability
(UA)
System
Mean
MDTseries=
Down
MTBF1  MDT2  MTBF2  MDT1
Time
MTBF1  MTBF2
(MDT)
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MTBFparallel=
MTBF1  MTBF2
MDT1  MDT2
Aparallel  A1  A2  A1  A2
m _ out _ of _ n 
n!
m  MDT m1
(n  m)!(m  1)!
MTBFm _ out _ of _ n 
Am _ out _ of _ n  1 
UAparallel  UA1  UA2
MDT parallel 
n identical subsystems in parallel; system
fails if m or more subsystems fail.
(m_out_of_n)
UAm _ out _ of _ n 
MDT1  MDT2
MDT1  MDT2
n!
(1  A) m
m!(n  m)!
n!
UAm
m!(n  m)!
MDT m _ out _ of _ n 
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MTBF m
n!
 MDT m 1
(n  m)!(m  1)!
MDT
m