What is a renewal process?

It is a process that counts events whose interarrival times are independent and identically distributed; the Poisson process is the special case where those times are exponential.

What is the inspection paradox?

If you observe a renewal process at a fixed time, the interval containing that time tends to be longer than a randomly chosen interval, because longer intervals are more likely to be the one you land in.

Renewal Theory

Renewal theory studies processes that restart afresh at each occurrence of an event, where the times between successive events are independent and identically distributed.

Definition

A renewal process is a counting process whose interarrival times are independent and identically distributed positive random variables, and renewal theory analyses the resulting renewal function, its asymptotics, and the limiting behaviour of associated quantities such as age and residual lifetime.

Scope

This topic covers the renewal process and renewal function, the renewal equation and its solution, the elementary renewal theorem giving the long-run renewal rate, the key renewal theorem and Blackwell's theorem on asymptotic increments, the inspection paradox and the distribution of the age and residual life, and the delayed and stationary renewal processes.

Core questions

How is the renewal function defined and computed from the interarrival distribution?
What is the long-run rate of renewals?
What do the key renewal and Blackwell theorems describe asymptotically?
Why does the inspection paradox make a typical observed interval longer than average?

Key theories

Elementary and key renewal theorems: The long-run renewal rate equals the reciprocal of the mean interarrival time, and the key renewal theorem gives the limiting value of the convolution of a function with the renewal density, supplying the asymptotics of a wide class of renewal equations.
Age, residual life, and the inspection paradox: The interval covering a fixed observation time is stochastically longer than a typical interarrival interval because longer intervals are more likely to be sampled, and the limiting age and residual-life distributions follow from the equilibrium renewal density.

Clinical relevance

Renewal theory models the replacement of components in reliability engineering, the recurrence of events in maintenance and warranty analysis, and the long-run cost of operating systems with random failure times, and it provides the asymptotic backbone for queueing and inventory models.

History

Renewal theory grew from problems of population replacement and industrial replacement in the 1930s and 1940s, with Feller and Smith establishing the renewal equation and its asymptotics, Blackwell proving his renewal theorem in 1948, and Cox's 1962 monograph giving the subject its standard exposition.

Key figures

William Feller
David Blackwell
David Cox
Walter Smith

Seminal works

coxRenewal1962

Frequently asked questions

What is a renewal process?: It is a process that counts events whose interarrival times are independent and identically distributed; the Poisson process is the special case where those times are exponential.
What is the inspection paradox?: If you observe a renewal process at a fixed time, the interval containing that time tends to be longer than a randomly chosen interval, because longer intervals are more likely to be the one you land in.