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Unformatted text preview: The Poisson Process as a CTMC CTMC – Motivation Continuoustime Markov Chains (CTMCs) Introductory Engineering Stochastic Processes, ORIE 3510 Instructor: Mark E. Lewis, Associate Professor School of Operations Research and Information Engineering Cornell University Disclaimer : Notes are only meant as a lecture supplement not substitute! 1/ 17 The Poisson Process as a CTMC CTMC – Motivation Continuoustime Markov Chains (CTMCs) The Monotone Increasing DTMC Constructing Sample Paths of the Poisson Process Monotone Increasing DTMC Consider the following DTMC on { , 1 ,... } Q = 1 ··· 1 ··· 1 ··· . . . . . . . . . ··· ··· Q defines what is called the monotone increasing Markov chain If we start at zero, then at each step we increase by 1 time Xn 2/ 17 The Poisson Process as a CTMC CTMC – Motivation Continuoustime Markov Chains (CTMCs) The Monotone Increasing DTMC Constructing Sample Paths of the Poisson Process Monotone Increasing DTMC Consider the following DTMC on { , 1 ,... } Q = 1 ··· 1 ··· 1 ··· . . . . . . . . . ··· ··· Q defines what is called the monotone increasing Markov chain If we start at zero, then at each step we increase by 1 time Xn 2/ 17 The Poisson Process as a CTMC CTMC – Motivation Continuoustime Markov Chains (CTMCs) The Monotone Increasing DTMC Constructing Sample Paths of the Poisson Process Monotone Increasing DTMC Consider the following DTMC on { , 1 ,... } Q = 1 ··· 1 ··· 1 ··· . . . . . . . . . ··· ··· Q defines what is called the monotone increasing Markov chain If we start at zero, then at each step we increase by 1 time Xn 2/ 17 The Poisson Process as a CTMC CTMC – Motivation Continuoustime Markov Chains (CTMCs) The Monotone Increasing DTMC Constructing Sample Paths of the Poisson Process Sample Paths of the Poisson Process Recall the paths of the monotone increasing DTMC look like time Xn 3/ 17 The Poisson Process as a CTMC CTMC – Motivation Continuoustime Markov Chains (CTMCs) The Monotone Increasing DTMC Constructing Sample Paths of the Poisson Process Sample Paths of the Poisson Process Recall the paths of the monotone increasing DTMC look like time Xn Compare this to the paths of the Poisson process time N(t) 3/ 17 The Poisson Process as a CTMC CTMC – Motivation Continuoustime Markov Chains (CTMCs) The Monotone Increasing DTMC Constructing Sample Paths of the Poisson Process Sample Paths of the Poisson Process Recall the paths of the monotone increasing DTMC look like time Xn Compare this to the paths of the Poisson process time N(t) t' N(t') = 2 Exp( λ29 Exp( λ29 Exp( λ29 Exp( λ29 3/ 17 The Poisson Process as a CTMC CTMC – Motivation Continuoustime Markov Chains (CTMCs) The Monotone Increasing DTMC Constructing Sample Paths of the Poisson Process Poisson Process Facts Recall some facts about the Poisson Process It is a continuoustime process (events can occur at any time)...
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This note was uploaded on 10/07/2010 for the course ORIE 3510 taught by Professor Resnik during the Spring '09 term at Cornell.
 Spring '09
 RESNIK

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