Models
 Terminology
 Terminology
 Stochastic Process A random variable, {X(t)}, where t is a time index that takes values from a given set T. T may be discrete or continuous. X(t) is a scalar that may take discrete or continuous values. We consider here only finite-discrete stochastic processes. Time The parameter of a stochastic process. State A vector that describes attributes of a system at any point in time. The state vector has m components. X(t) describes some feature of the state. State Space Collection of all possible states. Activity An activity begins at some point in time, has a duration and culminates in an event. Generally the duration of the activity is a random variable with a known probability distribution. Event The culmination of an activity. The event has the potential to change the state of the process. Calendar The set of events that can occur in a specified state, Y(s). Next Event While in some state when one or more events can occur, the one that occurs next is called the next event. Measured from the current time, the time of the next event is: The next event is the value of x that obtains the minimum. When the durations of events are random variables both the next event and the time of the next event are random variables. Transition A function that determines the next state, s', based on the current state, s, and the event, x. The number of elements of the transition function is the same as the number of elements in the state vector. s' = T(s,x). State-transition network A graphical representation of the states, represented by nodes, and events, represented by arcs. A transition is shown as a directed arc going from one node to another. Markovian Property Given that the current state is known, the conditional probability of the next state is independent of the states prior to the current state. Discrete-Time Markov Chain A stochastic process that satisfies the Markovian property and has a discrete time parameter. Sometimes such a process is call simply Markov Chain. Continuous-Time Markov Chain A stochastic process that satisfies the Markovian property and has a continuous time parameter. Sometimes such a process is call a Markov Process.

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by Paul A. Jensen