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Unformatted text preview: Copyright © 2005 by K.S. Trivedi 1 Probability and Statistics with Reliability, Queuing and Computer Science Applications Second edition by K.S. Trivedi PublisherJohn Wiley & Sons Chapter 6 : Stochastic Processes Dept. of Electrical & Computer Engineering Duke University Email: kst@ee.duke.edu URL: www.ee.duke.edu/~kst 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. 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 Copyright © 2005 by K.S. Trivedi 4 Classification of Stochastic Processes ¡ Four classes of stochastic processes: ¡ Discretestate process Æ chain ¡ Discretetime process Æ stochastic sequence (e.g., probing a system every 10 minutes.) }  { T n X n ∈ 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 Copyright © 2005 by K.S. Trivedi 6 Example: a Queuing System ¡ Some interarrival/service time distributions types are: ¡ M : Memoryless (i.e., EXP( )) ¡ D : Deterministic ¡ E k : kstage Erlang etc. ¡ H k : kstage Hyperexponential distribution ¡ G : Generally distributed ¡ GI : General Independent interarrival times ¡ PH : PhaseType ¡ MMPP: Markov modulated Poisson process ¡ Thus M/M/1 denotes Memoryless interarrival/service time distributions with a single server. Copyright © 2005 by K.S. Trivedi 7 Discrete time, Discrete state Stochastic Process ¡ N k : Number of jobs in the system at the time of k th job’s departure Æ Stochastic process {N k  k=1,2,…}: ¡ Discrete time, discrete state N k D i s c r e t e k Discrete Copyright © 2005 by K.S. Trivedi 8 Continuous Time, Discrete State ¡ X(t): Number of jobs in the system at time t. {X(t)  t є T} forms a continuoustime, discretestate stochastic process, with, X(t) D i s c r e t e Continuous Copyright © 2005 by K.S. Trivedi 9 Discrete Time, Continuous State ¡ W k : waiting time for the k th job. Then {W k  k є T} forms¡a¡ Discretetime, Continuousstate stochastic process, where, W k Discrete C o n t i n u o u s k Copyright © 2005 by K.S. TrivediCopyright © 2005 by K....
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 Spring '10
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