2.1 Definitions
Let \(T\) be a subset of \(\mathbb {R}\).
A collection of random variables \(\{X(t)\, , \, t\in T\}\) is called a stochastic process.
The index \(t\) of \(X(t)\) often represents time and describe the state of the process at time \(t\) by random
variable \(X(t)\).
The set \(T\) is called the index set of the stochastic process \(\{X(t)\, , \, t\in T\}\).
If \(T\) is a countable set, the stochastic process is said to be disjoint time process.
If \(T\) is an interval of real time, the stochastic process \(\{X(t)\}\) is said to be a continuous time
process.
The set of all possible values that the random variable of the stochastic process \(\{X(t)\, , \, t\in T\}\) can assume, is
called the state space of the process.
The state space of a stochastic process can be discrete set or a continuous set.
- 1.
- Consider a computer server with jobs arriving at random points in time by customer
and queuing for service. A customer departs from the system after service completion
(all his jobs completed).
Let \(N_k\) be the number of jobs in the system at the time of departures of the \(k^{\text {th}}\) customer.
Hence we have a stochastic process \(\{N_k\, , \, k = 1\, , \, 2\, \cdots \cdots \}\) whose state space \(\, I = \{0\, , \, 1\, , 2\, \cdots \cdots \}\)
(Discrete time and discrete state space). - 2.
- Let \(X(t)\) be the number of jobs in the system at time \(t\).
Hence we have a stochastic process \(\{X(t)\, , \, t\geq 0\}\,\) where the state space \(I= \{0\, , \, 1\, , \, 2\, \cdots \cdots \}\). - 3.
- Let \(W_k\) denote the time that the \(k^{\text {th}}\) customer has to wait in the system before receiving
service.
Hence we have a stochastic process \(\{W_k\, , \, k = 1\, , \, 2\, , \cdots \cdots \, \}\) where \(I = [0\, , \, \infty )\). - 4.
- Let \(Y(t)\) denote the cumulative service time required to complete all jobs in the system
that are there at time \(t\).
Hence we have a stochastic process \(\{Y(t)\, , \, t\geq 0\}\) where state space \(I= [0\, , \, \infty )\).
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