# 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 –**

- 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,
- The renewal function, m(t), satisfies the following equation:

where is the distribution function of the inter-arrival time or the renewal period.

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