Statistics for Maintenance Modelling: Renewal Processes

# Calculating the Renewal Function Analytically: Exponential Distribution¤

The simplest case is that of the exponential distribution for the time between failures. Here the result is trivial. We have f(x) = (from The Weibull Distribution screen in the Continuous Lifetime Distributions 1 session), where the mean is E(X) = = 1/. The calculation of the renewal function M(t) relies on the generalism that if at least events k occur by time t, this is the same as saying that the time to the kth event must be no later than t. Mathematically, this becomes:
N(t) k iff Wk t.

The distribution of Wk is Erlang, because Wk is the sum of k exponential random variables, so the pdf. is:

fk(x) = (x)k - 1/(k - 1)!

Now the probability that the kth item fails before time t is the cdf (see The Cumulative Density Function screen in the Basic Probability Concepts session), so:

Prob(N(t) k) = Prob(Wk t) = Fk(t) = fk(x) dx

= {1/(k - 1)!} (substituting the pdf. above)

Integrating progressively by parts we obtain: Prob(N(t) k) = 1 - | Another way of looking at this is that the number of renewals in the interval from 0 to t is a Poisson random variate with mean t where is the rate of occurrence (see The Poisson Distribution screen in the Continuous Lifetime Distributions 1 session).