Statistics for Maintenance Modelling: Renewal Processes

# Calculating the Renewal Function Analytically: Exponential Distribution (3)¤

This method of finding M(t) can be modified to make the calculations easier. In general, the following identity is useful:

Since Prob(N(t) = k) = Fk(t) - Fk+1(t) (because Fk is the probability that k or more events have occurred),

M(t) = k(Fk(t) - Fk+1(t)) = F1 - F2 + 2( F2 - F3) + ...

= F1 + F2 + F3 + ... = Fk(t)

So in general, the renewal density v(t) = dM(t)/dt, or

v(t) = fk(t) is sometimes a good way in to calculate M(t).

Click here for an intuitive proof of M(t).

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