Renewal theory is the branch of
probability theory
Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set o ...
that generalizes the
Poisson process
In probability, statistics and related fields, a Poisson point process is a type of random mathematical object that consists of points randomly located on a mathematical space with the essential feature that the points occur independently of one ...
for arbitrary holding times. Instead of
exponentially distributed
In probability theory and statistics, the exponential distribution is the probability distribution of the time between events in a Poisson point process, i.e., a process in which events occur continuously and independently at a constant averag ...
holding times, a renewal process may have any
independent and identically distributed
In probability theory and statistics, a collection of random variables is independent and identically distributed if each random variable has the same probability distribution as the others and all are mutually independent. This property is usua ...
(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
In probability theory, the law of large numbers (LLN) is a theorem that describes the result of performing the same experiment a large number of times. According to the law, the average of the results obtained from a large number of trials shou ...
and
central limit theorem
In probability theory, the central limit theorem (CLT) establishes that, in many situations, when independent random variables are summed up, their properly normalized sum tends toward a normal distribution even if the original variables themselv ...
. The renewal function
(expected number of arrivals) and reward function
(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
In mathematics (in particular, functional analysis), convolution is a operation (mathematics), mathematical operation on two function (mathematics), functions ( and ) that produces a third function (f*g) that expresses how the shape of one is ...
of
with a suitable non-negative function. The superposition of renewal processes can be studied as a special case of
Markov renewal process
In probability and statistics, a Markov renewal process (MRP) is a random process that generalizes the notion of Markov jump processes. Other random processes like Markov chains, Poisson processes and renewal processes can be derived as special ...
es.
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.
Renewal processes
Introduction
The renewal process is a generalization of the
Poisson process
In probability, statistics and related fields, a Poisson point process is a type of random mathematical object that consists of points randomly located on a mathematical space with the essential feature that the points occur independently of one ...
. In essence, the Poisson process is a
continuous-time Markov process
A continuous-time Markov chain (CTMC) is a continuous stochastic process in which, for each state, the process will change state according to an exponential random variable and then move to a different state as specified by the probabilities of ...
on the positive integers (usually starting at zero) which has independent
exponentially distributed
In probability theory and statistics, the exponential distribution is the probability distribution of the time between events in a Poisson point process, i.e., a process in which events occur continuously and independently at a constant averag ...
holding times at each integer
before advancing to the next integer,
. In a renewal process, the holding times need not have an exponential distribution; rather, the holding times may have any distribution on the positive numbers, so long as the holding times are independent and identically distributed (
IID
In probability theory and statistics, a collection of random variables is independent and identically distributed if each random variable has the same probability distribution as the others and all are mutually independent. This property is us ...
) and have finite mean.
Formal definition
![Renewal process](https://upload.wikimedia.org/wikipedia/commons/5/55/Renewal_process.reetep.png)
Let
be a sequence of positive
independent identically distributed
In probability theory and statistics, a collection of random variables is independent and identically distributed if each random variable has the same probability distribution as the others and all are mutually independent. This property is usual ...
random variable
A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the po ...
s such that
:
We refer to the random variable
as the "
-th holding time".