A Brief Introduction to Point Process, Counting Process, Renewal Process, Regenerative Process, Poisson Process

Abstract

This chapter started with the definition of point process as a basic stochastic process, continued with the definitions of the arrival time and the interarrival time; and based on the definition of point process, counting process has been defined. The definition of renewal process as a special counting process, and the definition of Poisson process as a special renewal process have been provided. Poisson process has been classified as homogeneous and nonhomogeneous Poisson process, and the basic properties of a homogeneous Poisson process including the stationary and independent increments, exponential random variable as the interarrival time distribution, and gamma random variable as the arrival time distribution have been explained. In addition to the typical example of the arrival of the customers; other illustrative examples including the breakdown of the machines, time of the earthquakes, and the two-server systems have also been presented, and solved by using the basic formulae and the schematic representations.