

MILHDBK338B: Electronic Reliability Design Handbook 
 

10.4.1.5 Model E  R&M Parameters Not Defined in Terms of Time
10.4.1.5 Model E  R&M
Parameters Not Defined in Terms of Time
A very different situation in availability modeling is
encountered when system “uptime” is not measured in hours of operation or any
time parameter but rather in terms of number of rounds fired, miles traveled,
actuations or cycles performed, etc. The reliability parameter is then no
longer expressed in terms of MTBF but rather in meanroundsbetweenfailures
(MRBF), meanmilesbetweenfailures (MMBF), meancyclesbetweenfailures
(MCBF), etc. The failure rate then also is expressed in number of failures per
round, per mile, or per cycle rather than number of failures per operating
hour.
For straightforward reliability calculations this poses no
problem since the same reliability equations apply as in the time domain,
except that the variable time, t, in hours is replaced by the variable number
of rounds, number of miles, etc. We may then calculate the reliability of such
systems for one, ten, one hundred, or any number of rounds fired or miles
traveled, as we wish. The maintainability calculations remain as before, since
downtime will always be measured in terms of time and the parameter of main
interest remains the MTTR.
However, when it comes to availability, which usually combines
two time parameters (i.e., the MTBF and the MTTR into a probability of the
system being up at some time, t), a difficult problem arises when the time, t,
is replaced by rounds or miles, since the correlation between time and rounds
or time and miles is quite variable.
An equation for the steadystate availability of machine guns is
given in Reference
[11]. This equation is based on a mission profile that at discrete times,
t_{1}, t_{2}, t_{3},
etc., requires the firing of N_{1}, N_{2},
N_{3}, etc., bursts of rounds.
When the gun fails during a firing, for example at time t, it fires only f
rounds instead of N_{3}
rounds and must undergo repair during
which time it is not available to fire; for example, it fails to fire a
required N_{4} rounds at t_{4}, and a further N_{5} rounds at
t_{5} before becoming again available (see Figure 10.45).
Its system availability, A, based on the rounds not fired during repair may be
expressed, for the described history, as:

A = (N_{1} +
N_{2} + f)/(N_{1} +
N_{2} + N_{3} +
N_{4} + N_{5}) 
(10.61) 
Each sequence of rounds fired
followed by rounds missed (not fired) constitutes a renewal process in terms
of rounds fired, as shown in Figure 10.46, where the gun fails after firing x
rounds, fails to fire g(x) rounds in the burst of rounds during which it
failed and also misses firing the required bursts of rounds while in repair
for an MTTR = M. Assume that the requirements for firing bursts of rounds
arrives at random according to a Poisson process with rate r and the average
number of rounds per burst is N, then the limiting availability of the gun may
be expressed as:

A = MRBF/(MRBF + N + gMN) 
(10.62) 
where MRBF is the mean
number of rounds between failure. The derivation of this formula, developed by
R.E. Barlow, is contained in the Appendix of Reference [11].
To calculate A from Eq. (10.62) one must know the MRBF and MTTR of the gun,
the average rounds N fired per burst, and the rate g
at which requirements for firing bursts of
rounds arrive.
FIGURE 10.45: HYPOTHETICAL
HISTORY OF MACHINE GUN USAGE
FIGURE 10.46: RENEWAL
PROCESS IN TERMS OF ROUNDS FIRED
Similar availability equations can be developed for other types
of weapons and also for vehicles where the renewal process is in terms of
miles traveled. Other approaches to calculating the availability of guns as
well as vehicles are found in Reference [12]
and are based on calculating from historical field data the maintenance ratios
and, via regression analysis, the maintenance time ratios (called the “maintenance clock hour index”)
that are in turn used in the conventional time based equation of inherent,
achieved, and operational availability.
For example, consider a machine gun
system in a tank on which historical data are available, showing that 0.014
corrective maintenance manhours are expended per round fired and that per
year 4800 rounds are fired while the vehicle travels for 240 hr per yr. The
maintenance ratio (MR) for the gun system is then computed as (Ref. [12], pp.
3638).
The dimensions for 0.28 are
gun system maintenance manhours per vehicle operating hour. According to this
example, the corrective maintenance time ratio, a
(sometimes called the maintenance clock
hour index, W),
is, given by:

a_{Gun} =
0.628(0.28)^{0.952} = 0.187 
(10.64) 
The numbers 0.628 and 0.952
are the intercept and the regression coefficients, respectively, obtained by
regression analysis as developed in Reference [12],
p. 18, Table 1. The dimension for aGun is gun system downtime per vehicle operating hour. The
inherent availability of the gun system is then, according to the conventional
time equation, Eq. (10.20).

A_{i} = (1 + aGun)^{1} =
(1.187)^{1} = 0.842 
(10.65) 
This may be interpreted as
the gun system being available for 84.2% of the vehicle operating time.
Caution is required in using this approach for weapon availability
calculations, since in the case where the vehicle would have to be stationary
and the gun would still fire rounds, MR and a
would become infinitely large and the
inherent availability of the gun system would become zero.




 
 