chap6f - Copyright © 2005 by K.S. Trivedi 1 Probability...

Info iconThis preview shows pages 1–11. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 Publisher-John 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: ¡ Discrete-state process Æ chain ¡ Discrete-time 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 : k-stage Erlang etc. ¡ H k : k-stage Hyperexponential distribution ¡ G : Generally distributed ¡ GI : General Independent interarrival times ¡ PH : Phase-Type ¡ 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 continuous-time, discrete-state 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¡ Discrete-time, Continuous-state 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....
View Full Document

Page1 / 54

chap6f - Copyright © 2005 by K.S. Trivedi 1 Probability...

This preview shows document pages 1 - 11. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online