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.