In this chapter, we concentrate on a generalization of the Poisson processes. Recall the memoryless property of the exponential distribution. Essentially, this property makes the Poisson process restart every time we observe it. Consider a light bulb that has been in operation for two years. Suppose we assume the light bulb is replaced according to a Poisson process. This implies that the probability this light bulb lasts for one more day given that it has been operating for two years is exactly the same as the probability of a totally new light bulb operating one day. Obviously, this is not realistic, but using anything else than the exponential for the inter-arrival time distribution will lose the simplicity of the Poisson process.
If we consider some other type of inter-arrival time distribution, we obtain a renewal process. This type of process is very general. Due to their general nature, they provide some of the most powerful (and general theorems) in stochastic processes. Here, “powerful” means applicable to a wide class of processes. However, since these results are very general, in order to obtain specific results one needs to hypothesize specific distribution, and oftentimes this leads to more specialized processes. We cover renewal processes in this chapter since studying them provide strong results which will be applied later to Markov chains and processes that are widely used in practice.