if the renewal process is modelling the numbers of breakdown of different machines, then the holding time represents the time between one machine breaking down before another one does. The Poisson process is the unique renewal process with the Markov property, as the exponential distribution is the unique continuous random variable with the property of memorylessness. Renewal-reward processes Let be a sequence of IID random variables (rewards) satisfying Then the random variable is called a renewal-reward process. Note that unlike the , each may take negative values as well as positive values. The random variable depends on two sequences: the holding times and the rewards These two sequences need not be independent. In particular, may be a function of . Interpretation In the context of the above interpretation of the holding times as the time between successive malfunctions of a machine, the "rewards" (which in this case happen to be negative) may be viewed as the successive repair costs incurred as a result of the successive malfunctions. An alternative analogy is that we have a magic goose which lays eggs at intervals (holding times) distributed as . Sometimes it lays golden eggs of random weight, and sometimes it lays toxic eggs (also of random weight) which require responsible (and costly) disposal. The "rewards" are the successive (random) financial losses/gains resulting from successive eggs (i = 1,2,3,...) and records the total financial "reward" at time t. Renewal function We define the renewal function as the expected value of the number of jumps observed up to some time : Elementary renewal theorem The renewal function satisfies {| class="toccolours collapsible collapsed" width="80%" style="text-align:left" !Proof |- |The strong law of large numbers for renewal processes implies To prove the elementary renewal theorem, it is sufficient to show that is uniformly integrable. To do this, consider some truncated renewal process where the holding times are defined by where is a point such that which exists for all non-deterministic renewal processes. This new renewal process is an upper bound on and its renewals can only occur on the lattice . Furthermore, the number of renewals at each time is geometric with parameter . So we have |} Elementary renewal theorem for renewal reward processes We define the reward function: The reward function satisfies Renewal equation The renewal function satisfies where is the cumulative distribution function of and is the corresponding probability density function. {| class="toccolours collapsible