Page 151 - Applied Statistics And Probability For Engineers

c04.qxd 5/13/02 11:19 M Page 129 RK UL 6 RK UL 6:Desktop Folder:TEMP WORK:MONTGOMERY:REVISES UPLO D CH114 FIN L:Quark Files:

4-10 ERLANG AND GAMMA DISTRIBUTIONS 129

occur in a Poisson process. The random variable that equals the interval length until r counts
occur in a Poisson process has an Erlang random variable.

EXAMPLE 4-23 The failures of the central processor units of large computer systems are often modeled as a
Poisson process. Typically, failures are not caused by components wearing out, but by more
random failures of the large number of semiconductor circuits in the units. Assume that the
units that fail are immediately repaired, and assume that the mean number of failures per hour
is 0.0001. Let X denote the time until four failures occur in a system. Determine the probabil-
ity that X exceeds 40,000 hours.
Let the random variable N denote the number of failures in 40,000 hours of operation.
The time until four failures occur exceeds 40,000 hours if and only if the number of failures
in 40,000 hours is three or less. Therefore,

P1X 40,0002 P1N 32

The assumption that the failures follow a Poisson process implies that N has a Poisson distri-
bution with

E1N2 40,00010.00012 4 failures per 40,000 hours

Therefore,

4 k
3 e 4
P1X 40,0002 P1N 32 a 0.433
k 0 k!
The cumulative distribution function of a general Erlang random variable X can be obtained
from P1X x2 1 P1X x2, and P1X x2 can be determined as in the previous exam-
ple. Then, the probability density function of X can be obtained by differentiating the cumula-
tive distribution function and using a great deal of algebraic simplification. The details are left
as an exercise. In general, we can obtain the following result.

Definition
The random variable X that equals the interval length until r counts occur in a
Poisson process with mean 0 has an Erlang random variable with parameters
and r. The probability density function of X is

r r 1 x
x e
f 1x2 , for x 0 and r 1, 2, p (4-17)
1r 12!

Sketches of the Erlang probability density function for several values of r and are
shown in Fig. 4-25. Clearly, an Erlang random variable with r 1 is an exponential
random variable. Probabilities involving Erlang random variables are often determined by
computing a summation of Poisson random variables as in Example 4-23. The probability
density function of an Erlang random variable can be used to determine probabilities;
however, integrating by parts is often necessary. As was the case for the exponential
distribution, one must be careful to define the random variable and the parameter in
consistent units.