Mathematics | Renewal processes in probability

A Renewal process is a general case of Poisson Process in which the inter-arrival time of the process or the time between failures does not necessarily follow the exponential distribution. A counting process N(t) that represents the total number of occurrences of an event in the time interval (0, t] is called a renewal process, if the time between failures are independent and identically distributed random variables.

The probability that there are exactly n failures occurring by time t can be written as,
$ P\{N(t) = n\} = P\{N(t)\geq n\}-P\{N(t) > n \}
and,
$T_k=W_k + W_{k-1} $

Note that the times between the failures are T1, T2, …, Tn so the failures occurring at time $W_k$ are,
$W_k=\sum_{i=1}^kT_i$
Thus,
$ P\{N(t) = n\}$
$= P\{N(t) \geq n\}-P\{N(t)>n\} $
$= P\{W_n \leq t\}-P\{W_{n+1} \leq t\} $
$= F_n(t)-F_{n+1}(t) $



Properties –

  1. The mean value function of the renewal process, denoted by m(t), is equal to the sum of the distribution function of all renewal times, that is,
    $ m(t)$
    $= E[N(t)] $
    $ = \sum_{n=1}^{\infty}F_n(t)  $

  2. The renewal function, m(t), satisfies the following equation:
    $ m(t)$
    $ = F_a(t)+\int_{0}^{t}m(t-s)dF_a(s) $
    where  F_a(t) is the distribution function of the inter-arrival time or the renewal period.


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