Information Technology Reference
In-Depth Information
The exponential distribution
When—as in the example—device failures events occur at a constant rate, the num-
ber of failure events in a fixed time period can be mathematically modelled as a Poisson
process, and the interarrival time between failure events follows an exponential distribu-
tion.
The exponential distribution is memoryless—since the rate of failure events is con-
stant across time, then the expected time to the next failure event is the same—no
matter what the current time and no matter how long it has been since the last failure.
So, if a device has an annual failure rate of 0.5 and thus a mean time to failure of 2
years, and we've been operating the device without a failure for a year, the expected
time from the current time to the next failure is still 2 years.
So, if random variable T represents the time between failures and has an exponential
distribution with representing the average number of failure events per unit of time,
then the probability density function fT
T
t is:
e
t
if t 0
f
T
(t) =
0
if t < 0
and the mean time to failure is MTTF =
1
.
Exponential distributions have a number of convenient mathematical properties. For
example, because the failure rate is constant, the mean time to failure is the inverse
of the failure rate; this is why it is easy to convert between MTTF and annual failure
rates in storage specificiations. Also, if the expected number of failures is given for one
duration (e.g., 0.1 failures per year), it can easily be converted to the expected number
for a different duration (e.g., 0.0003 failures per day). Finally, if we have k independent
failure processes with rates of
1
,
2
, . . . ,
k
, then the aggregate failure function—the
rate at which failures of any of the k kinds occurs—is
tot
=
1
+
2
+ ::: +
k
and the mean time to the next failure of any kind is MTTF
tot
=
1
tot
. For example, if we
have 100 disks, each with a MTTF
disk
= 1:5 10
6
hours or, equivalently, each failing
at a rate of 0.00585 failures per year, then the overall 100-disk system suffers failures
at a rate of 100 0:00585 = 0:585 failures per year or, equivalently, the 100-disk system
has MTTF
100disks
= 1:5 10
4
hours.
Warning. Because the exponential distribution is so mathematically convenient,
is tempting to use it even when it is not appropriate. Remember that failures in real
systems may be correlated (i.e., they are not independent) and may vary over time (i.e.,
they are not constant).
