ABSTRACT
This chapter discusses the simplest Poisson process, which jumps upward by one unit in such a way that the times between jumps are i.i.d. exponential random variables. This is generalized to the birth-death process, which is allowed to jump downward by one unit as well as upward. Then a different generalization of the Poisson process is introduced called the renewal process, which again is constrained to jump upward by one unit, but whose inter-jump times are not necessarily exponentially distributed. The chapter covers the rudiments of one of the main applications of stochastic processes, namely queueing theory. It presents a study on a process with continuous state space called the Brownian motion, together with some of its applications. In order to derive some results easily, the chapter specializes to a particular kind of arrival process.
