Stochastic Processes
 Stochastic Processes In many practical situations the attributes of a system randomly change over time.  Examples include the number of customers in a checkout line, congestion on a highway, the number of items in a warehouse, and the price of a financial security, to name a few.  In certain instances, it is possible to describe an underlying process that explains how the variability occurs.  When aspects of the process are governed by probability theory, we have a stochastic process.  The first step in modeling a dynamic process is to define the set of states over which it can range and to characterize the mechanisms that govern its transitions.  The state is like a snapshot of the system at a point in time.  It is an abstraction of reality that describes the attributes of the system of interest.  Time is the linear measure through which the system moves, and can be thought of as a parameter.  Because of time there is a past, present, and future.  We usually know the trajectory a system has followed to arrive at its present state.  Using this information, our goal is to predict the future behavior of the system in terms of a basic a set of attributes.  As we shall see, a variety of analytic techniques are available for this purpose. From a modeling point of view, state and time can be treated as either continuous or discrete.  Both theoretical and computational considerations, however, argue in favor of the discrete state case so this will be our focus. We consider both discrete time and continuous time models. To obtain computational tractibility we assume that the stochastic process satisfies the Markov condition. That is, the path the process takes in the future depends only on the current state, and not the sequence of states visited prior to the current state. For the discrete time system this leads to the Markov Chain model.  For the continous time system the model is called a Markov Process. The model of a stochastic process describes activities that culminate in events.  The events cause a transition from one state to another.  Because activity durations are assumed to be continuous random variables, events occur in the continuum of time.  This section provides the vocabulary used in conjunction with a continuous time stochastic processes along with an example in which the model is useful. Two other sections in this Modeling part of the site will focus respectively on Markov Chains and Markov Processes.

Operations Research Models and Methods
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by Paul A. Jensen
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