The article “Expectation Analysis of the Probability of Failure for Water Supply Pipes” (J. of Pipeline Systems Engr. and Practice, May 2012: 36–46) recommends using a Poisson process to model the number of failures in commercial water pipes. The article also gives estimates of the failure rate λ, in units of failures per 100 miles of pipe per day, for four different types of pipe and for many different years.
For PVC pipe in 2008, the authors estimate a failure rate of 0.0081 failures per 100 miles of
pipe per day. Consider a 100-mile-long segment of such pipe. What is the expected number of failures in 1 year (365 days)? Based on this expectation, what is the probability of at least one failure along such a pipe in 1 year?
For cast iron pipe in 2005, the authors’ estimate is λ 1⁄4 0.0864 failures per 100 miles per day. Suppose a town had 1500 miles of cast iron pipe underground in 2005. What is the probability of at least one failure somewhere along this pipe system on any given day?
(a)
(i)
Given:
Mean failures per day = 0.0081
So,
Expected number of failures in 1 year = 0.0081X 365 = 8.9948
(ii)
Poisson Distribution with = 8.9948 is
given by:
for x = 0, 1, 2, ...
P(At least 1 failure) = 1- P(X=0)
So,
P(At least 1failure) = 1- 0.00012 = 0.99988
So,
Answer is:
0.99988
(b)
Given:
Per 100 miles: 0.0864
Per 1500 miles: 0.0864 X 15 = 1.296
P(At least 1 failure) = 1- P(X=0)
So,
P(At least 1failure) = 1- 0.2736 = 0.7264
So,
Answer is:
0.7264
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