renewal

Renewal theory is the branch of probability theory that generalizes the Poisson process for arbitrary holding times. Instead of exponentially distributed holding times, a renewal process may have any independent and identically distributed (IID) holding times that have finite mean. A renewal-reward process additionally has a random sequence of rewards incurred at each holding time, which are IID but need not be independent of the holding times.
A renewal process has asymptotic properties analogous to the strong law of large numbers and central limit theorem. The renewal function



m
(
t
)


{\displaystyle m(t)}

(expected number of arrivals) and reward function



g
(
t
)


{\displaystyle g(t)}

(expected reward value) are of key importance in renewal theory. The renewal function satisfies a recursive integral equation, the renewal equation. The key renewal equation gives the limiting value of the convolution of




m
′

(
t
)


{\displaystyle m'(t)}

with a suitable non-negative function. The superposition of renewal processes can be studied as a special case of Markov renewal processes.
Applications include calculating the best strategy for replacing worn-out machinery in a factory and comparing the long-term benefits of different insurance policies. The inspection paradox relates to the fact that observing a renewal interval at time t gives an interval with average value larger than that of an average renewal interval.

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