The Poisson process is a prominent stochastic process, mainly because it frequently appears in a wealth of physical phenomena and because it is relatively simple to analyze. Therefore, we will first treat the Poisson process before considering the more general Markov processes.

A stochastic process

Introduction and definitions

A stochastic process, formally denoted as {X(t)t є T}, is a sequence of random variables X(t), where the parameter t – most often the time – runs over an index set T. The state space of the stochastic process is the set of all possible values for the random variables X(t) and each of these possible values is called the state of the process. If the index set T is a countable set, X[k] is a discrete stochastic process. Often k is the discrete time or a time slot in computer systems. If T is a continuum, X(t) is a continuous stochastic process. For example, the outcome of n tosses of a coin is a discrete stochastic process with state space {heads, tails} and the index set T = {0, 1, 2, …, n}. The number of arrivals of packets in a router during a certain time interval [a, b] is a continuous stochastic process because t ξ [a, b]. Any realization of a stochastic process is called a sample path.