Renewal theory
（重定向自Renewal equation）
![Sample evolution of a renewal process with holding times Si and jump times Jn.](/Images/godic/202502/06/Renewal_process.reetep5236.png")
![Sample evolution of a renewal-reward process with holding times Si, jump times Jn and rewards Wi](/Images/godic/202502/06/Renewal-reward_process.reetep5236.png")
![The renewal interval determined by the random point t (shown in red) is stochastically larger than the first renewal interval.](/Images/godic/202502/06/Inspection_paradox.reetep5236.png")
Renewal theory is the branch of probability theory that generalizes Poisson processes for arbitrary holding times. Applications include calculating the best strategy for replacing worn-out machinery in a factory and comparing the long-term benefits of different insurance policies.