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, and, Note that the times between the failures are T1, T2, …, Tn so the failures occurring at time are, Thus, 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,
2. The renewal function, m(t), satisfies the following equation: where is the distribution function of the inter-arrival time or the renewal period.