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As an example of a discrete index set, consider the net number count of delivery and returns of a particular item in an online store at time t within 24 hours of operation

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As an example of a discrete index set, consider the net number count of delivery and returns of a particular item in an online store at time t within 24 hours of operation. Hence, the process of delivery and returning is a stochastic process with the index set T as the set {0,1,2,#,24} and with the state space as S = ± {0, 1, 2± ,#}. As an example of a continuous index set, let us consider arrival of customers at a bank, which is assumed to be random in nature. The number of customers arrived during the interval [0,t] is a stochastic process in which the index set is T = ∞ [0, ) and the state space S = {0,1,2,...}. This is also an example of a pure birth process that we will consider later.