ABSTRACT
To study those real-world phenomena in which systems evolve randomly, we need probabilistic models rather than deterministic ones. Such systems are usually studied as a function of time, and their mathematical models are called stochastic models. The building blocks of stochastic models are stochastic processes, defined as sets of random variables { X n : n ∈ I } $ \{X_{n}:n \in I \} $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429457951/63b50632-c513-448c-ba5f-4cb005c69347/content/inline-math12_1.tif"/> for a finite or countable index set I, or { X ( t ) : t ∈ T } $ \big \{X(t) :t \in T\big \} $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429457951/63b50632-c513-448c-ba5f-4cb005c69347/content/inline-math12_2.tif"/> for an uncountable index set T. For example, let X n $ X_{n} $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429457951/63b50632-c513-448c-ba5f-4cb005c69347/content/inline-math12_3.tif"/> be the number of customers served in a bank at the end of the nth working day. Then { X n : n = 1 , 2 , … } $ \{X_{n} :n = 1,2, \ldots \} $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429457951/63b50632-c513-448c-ba5f-4cb005c69347/content/inline-math12_4.tif"/> is a stochastic process. It is called a discrete-time stochastic process since its index set, I = { 1 , 2 , … } , $ I = \{1,2, \ldots \}, $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429457951/63b50632-c513-448c-ba5f-4cb005c69347/content/inline-math12_5.tif"/> is countable. As another example, let X(t) be the sum of the remaining service times of all customers being served in a bank at time t. Then { X ( t ) : t ≥ 0 } $ \big \{X(t) :t \ge 0\big \} $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429457951/63b50632-c513-448c-ba5f-4cb005c69347/content/inline-math12_6.tif"/> is a stochastic process, and since its index set, T = [ 0 , ∞ ) , $ T = [0, \infty ), $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429457951/63b50632-c513-448c-ba5f-4cb005c69347/content/inline-math12_7.tif"/> is uncountable, it is called a continuous-time stochastic process. The set of all possible values of X n $ X_{n} $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429457951/63b50632-c513-448c-ba5f-4cb005c69347/content/inline-math12_8.tif"/> ’s in the discrete-time case and X(t)’s in the continuous-time case is called the state space of the stochastic process, and it is usually denoted by S $ \mathcal S $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429457951/63b50632-c513-448c-ba5f-4cb005c69347/content/inline-math12_9.tif"/> . The state space for the number of customers served in a bank at the end of the nth working day is S = { 0 , 1 , 2 , … } . $ \mathcal S = \{0, 1, 2, \ldots \}. $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429457951/63b50632-c513-448c-ba5f-4cb005c69347/content/inline-math12_10.tif"/> The state space for the sum of the remaining service times of all customers being served in a bank at time t is S = [ 0 , ∞ ) $ \mathcal S = [0, \infty ) $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429457951/63b50632-c513-448c-ba5f-4cb005c69347/content/inline-math12_11.tif"/> . Other examples of stochastic processes follow.
