Renewal theory is the branch of probability theory that generalizes the Poisson process for arbitrary holding times. Instead of exponentially distributed holding times, a renewal process may have any independent and identically distributed (IID) holding times that have finite mean. A renewal-reward process additionally has a random sequence of rewards incurred at each holding time, which are IID but need not be independent of the holding times. A renewal process has asymptotic properties analogous to the strong law of large numbers and central limit theorem. The renewal function (expected number of arrivals) and reward function (expected reward value) are of key importance in renewal theory. The renewal function satisfies a recursive integral equation, the renewal equation. The key renewal equation gives the limiting value of the convolution of with a suitable non-negative function. The superposition of renewal processes can be studied as a special case of Markov renewal processes. Applications include calculating the best strategy for replacing worn-out machinery in a factory and comparing the long-term benefits of different insurance policies. The inspection paradox relates to the fact that observing a renewal interval at time t gives an interval with average value larger than that of an average renewal interval. Renewal processes Introduction The renewal process is a generalization 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 exponentially distributed holding times at each integer before advancing to the next integer, . In a renewal process, the holding times need not have an exponential distribution; rather, the holding times may have any distribution on the positive numbers, so long as the holding times are independent and identically distributed (IID) and have finite mean. Formal definition Let be a sequence of positive independent identically distributed random variables such that We refer to the random variable as the "-th holding time". is the expectation of . Define for each n > 0 : each is referred to as the "-th jump time" and the intervals are called "renewal intervals". Then is given by random variable where is the indicator function represents the number of jumps that have occurred by time t, and is called a renewal process. Interpretation If one considers events occurring at random times, one may choose to think of the holding times as the random time elapsed between two consecutive events. For example,