10.4.2 Mission
Reliability and Dependability Models
Although availability is a simple and appealing concept at first
glance, it is a point concept, i.e., it refers to the probability of a system
being operable at a random point in time. However, the ability of the system
to continue to perform reliably for the duration of the desired operating
(mission) period is often more significant. Operation over the desired period
of time depends, then, on clearly defining system operating profiles. If the
system has a number of operating modes, then the operating profile for each
mode can be considered.
The term mission reliability has been used to denote the system
reliability requirement for a particular interval of time. Thus, if the system
has a constant failure rate region so that its reliability R can be expressed
as:
where:
l = failure rate =
1/MTBF
t = time for
mission
then mission reliability RM for a mission duration of T is
expressed as:
This reliability assessment, however, is conditional upon the
system being operable at the beginning of its mission or its (point)
availability.
In order to combine these two concepts, a simplified system
effectiveness model may be used where the system effectiveness may be
construed simply as the product of the probabilities that the system is
operationally ready and that it is mission reliable.
If A is the mean availability of a system at any point in time
t_{o} when we want to use the system and if R_{M} is the
system reliability during mission time T, then system effectiveness E, not
including performance, may be defined as:
Thus, A is a weighting factor, and E represents an assessment of
system ability to operate without failure during a randomly chosen mission
period.
One concept of dependability used by the Navy (Ref. [13]) takes
into account the fact that for some systems a failure which occurs during an
operating period t_{1} may be acceptable if the failure can be
corrected in a time t_{2} and the system continues to complete its
mission. According to this concept, dependability may be represented
by:

D = R_{M} + (1
 R_{M})M_{o} 
(10.69) 
where:
D 
= 
system dependability  or the probability that the
mission will be successfully completed within the mission time
t_{1}, providing a downtime per
failure not exceeding a given time t_{2} will not adversely affect the overall
mission

RM 
= 
mission reliability  or the probability
that the system will operate without failure for the mission time
t_{1}

Mo 
= 
operational maintainability  or the
probability that when a failure occurs, it will be repaired in a time
not exceeding the allowable downtime t_{2}

t2 
= 
specified period of time within which the
system must be returned to
operation 
For this model, the exponential approximation of the lognormal
maintainability function is used,
or

M_{o} = ( 1  e^{mt2} ) 
(10.70) 
Then, the system effectiveness is:

E = AD = A [ R_{M} + (1  R_{M}) M_{o}
] 
(10.71) 
In the case where no maintenance is allowed during the mission
(t_{2} = 0 or M_{o} = 0), as in the case of a missile, then
this reduces to Eq. (10.68).
E = AD = AR_{M}
This concept of dependability is compatible with the WSEIAC
model and indeed can be taken into account in the dependability state
transition matrices.
Let us examine an airborne system with the following parameters
and requirements:
l = 0.028 failures/hr
m = 1
repair/hr
Mission time (T) = 8
hours
t_{a} = 30 minutes to
repair a failure during a mission
Thus,
\ E
= A [ R_{M} + (1
 R_{M}) M_{o} ]
= 0.973
[ 0.8 + (1  0.8)
(0.4) ]
= 0.973 [ 0.8 + 0.08 ] = 0.86