Everything about Renewal Process totally explained
Renewal theory is the branch of
probability theory that generalizes Poisson processes for arbitrary
holding times. Applications include calculating the expected time for a
monkey who is randomly tapping at a keyboard to type the word
Macbeth and comparing the long-term benefits of different insurance policies.
Renewal processes
Introduction
A
renewal process is a generalisation of the
Poisson process. In essence, the Poisson process is a
continuous-time Markov process on the positive integers (usually starting at zero) which has
independent identically distributed holding times at each integer
(
exponentially distributed) before advancing (with probability 1) to the next integer:
. In the same informal spirit, we may define a renewal process to be the same thing, except that the holding times take on a more general distribution. (Note however that the
IID property of the holding times is retained).
Formal definition
Let
be a sequence of
independent identically distributed random variables such that
»
this implies that the turning points satisfy:
»
» :
and thus
»
We take the only solution
t in [0, 2]:
t = 2/3. This is indeed a minimum (and not a maximum) since the cost per unit time tends to infinity as
t tends to zero, meaning that the cost is decreasing as
t increases, until the point 2/3 where it starts to increase.
Further Information
Get more info on 'Renewal Process'.
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