Introduction
First let us consider few examples to initiate the reader for a better understanding of the concept of rrenewal process, renewal theory and ensuing theorems based on the concept of renewal process and renewal theory.

Example 4.1
Assume you are playing a game in which you roll a dice and you are interested to find the pattern of occurrence of the number 2. Let us define the event which corresponds to the pattern that the number 2 occurs if the rolling of the dice is repeated continuously. Then this event can be defined as a simple example of renewal process.

Example 4.2
In the second example consider you as the marketing manager of a refrigerator manufacturing firm and you are conducting a survey to find the need of giving a free service contract to all the customers who have purchased your company's refrigerator in the past. In case the working life of the refrigerator which is being used is more than 4 years then you would give the customer the new free service scheme, otherwise no such scheme or benefit is given to that customer. Suppose now you are interested to find the event which corresponds to the pattern of finding refrigerators which are more than 4 years old, then, you can term this study as an illustration of renewal process.

Example 4.3
Let us continue our discussion about renewal process with another example. You are the maintenance-in-charge of a sophisticated CNC machine. The machine has n number of similar components which are critical for the working of the CNC machine. Each of the n components has an operational life which is distributed exponentially with a mean of  years. You as the maintenance-in-charge may either replace the component or let it fail till you change it. Thus if you define the event as the number of component failures in some specific time, say T, then one can study this phenomena using the concepts of renewal theory.

Example 4.4
Consider you have a light socket in your room and you put a CFL light in it. The moment the first light bulb fails you replace that with a similar type of second bulb. In case the second bulb fails you replace that with the third CFL bulb and so on. Then the event which denotes the successive lifetimes of bulbs which are being replaced can be explained using the concept of renewal process.