2.1 Definitions
Definition 2.1.1. Let \(T\subset \mathbb {R}\). A collection of random variables \(\{X(t) : t\in T\}\) defined on a common probability space is a stochastic process. The set \(T\) is the index set, and the set \(I\) of all values the \(X(t)\) can take is the state space.
The index \(t\) usually represents time, and \(X(t)\) is the state of the process at time \(t\). If \(T\) is countable, for example \(T = \{0, 1, 2, \dots \}\), the process is a discrete-time process and is usually written \(\{X_n : n = 0, 1, 2, \dots \}\); if \(T\) is an interval, such as \([0, \infty )\), it is a continuous-time process. The state space may likewise be discrete (finite or countable) or continuous. All four combinations occur.
Example 2.1.2. Jobs arrive at random times at a computer server, queue, and leave when served.
- 1.
- Let \(N_k\) be the number of jobs in the system just after the \(k\)th job leaves. Then \(\{N_k : k = 1, 2, \dots \}\) has discrete time and discrete state space \(I = \{0, 1, 2, \dots \}\).
- 2.
- Let \(X(t)\) be the number of jobs in the system at time \(t\). Then \(\{X(t) : t\geq 0\}\) has continuous time and discrete state space \(I = \{0, 1, 2, \dots \}\).
- 3.
- Let \(W_k\) be the time the \(k\)th job waits before its service starts. Then \(\{W_k : k = 1, 2, \dots \}\) has discrete time and continuous state space \(I = [0, \infty )\).
- 4.
- Let \(Y(t)\) be the total remaining service time of all jobs present at time \(t\). Then \(\{Y(t) : t\geq 0\}\) has continuous time and continuous state space \(I = [0, \infty )\).
A process is described by its finite-dimensional distributions, the joint distributions of \((X(t_1), \dots , X(t_n))\) for every choice of \(n\) and of times \(t_1<\cdots <t_n\). In general these can be very complicated. The processes studied in this course are tractable because of one simplifying property.
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