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chap6f - Probability and Statistics with Reliability...

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Copyright © 2005 by K.S. Trivedi 1 Probability and Statistics with Reliability, Queuing and Computer Science Applications Second edition by K.S. Trivedi Publisher-John Wiley & Sons Chapter 6 : Stochastic Processes Dept. of Electrical & Computer Engineering Duke University Email: [email protected] URL: www.ee.duke.edu/~kst
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Copyright © 2005 by K.S. Trivedi 2 What is a Stochastic Process? So far we dealt with either a single random variable or a few random variables together. Now we will consider a whole family of random variables. Stochastic Process: is a family of random variables { X(t) | t ε T } defined on a given probability space. T is an index set; it may be discrete or continuous. Values assumed by X ( t ) are called states. State space (I) : set of all possible states A stochastic process is also known as a random process or a chance process.
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Copyright © 2005 by K.S. Trivedi 3 Stochastic Process Characterization At a fixed time t=t 1 , we have a random variable X(t 1 ) . Similarly, we can have X(t 2 ), .., X(t k ) . X(t 1 ) can be characterized by its distribution function, We can also consider the joint distribution function, Discrete and continuous cases: Index set T may be discrete/continuous State space I may be discrete/continuous
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Copyright © 2005 by K.S. Trivedi 4 Classification of Stochastic Processes Four classes of stochastic processes: Discrete-state process Æ chain Discrete-time process Æ stochastic sequence (e.g., probing a system every 10 minutes.) } | { T n X n
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Copyright © 2005 by K.S. Trivedi 5 Example: a Queuing System Interarrival times Y 1 , Y 2 , … (common dist. Fn. F Y ) Service times: S 1 , S 2 , … ( iid with a common CDF F S ) Notation for a queuing system: F Y / F S / m Some examples: M/M/1, M/G/1, M/M/k, GI/M/1, M/D/1
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