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# CTMCI_beamer - Introductory Engineering Stochastic...

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Introductory Engineering Stochastic Processes, ORIE 361 Instructor: Mark E. Lewis, Associate Professor School of Operations Research and Information Engineering Cornell University Introduction to Continuous-time Markov Chains 1/ 15

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Monotone Increasing DTMC Consider the following DTMC on { 0 , 1 ,... } Q = 0 1 0 0 ··· 0 0 1 0 ··· 0 0 0 1 ··· . . . . . . . . . ··· ··· Q deﬁnes what is called the monotone increasing Markov chain 2/ 15
Monotone Increasing DTMC Consider the following DTMC on { 0 , 1 ,... } Q = 0 1 0 0 ··· 0 0 1 0 ··· 0 0 0 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/ 15

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Poisson Process Facts Recall some facts about the Poisson Process It is a continuous-time process (events can occur at any time) 3/ 15
Poisson Process Facts Recall some facts about the Poisson Process It is a continuous-time process (events can occur at any time) Jumps occur one at a time 3/ 15

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Poisson Process Facts Recall some facts about the Poisson Process It is a continuous-time process (events can occur at any time) Jumps occur one at a time The time between events is exponentially distributed 3/ 15
Poisson Process Facts Recall some facts about the Poisson Process It is a continuous-time process (events can occur at any time) Jumps occur one at a time The time between events is exponentially distributed Each exponential is independent (related to independent increments ) 3/ 15

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Poisson Process Facts Recall some facts about the Poisson Process It is a continuous-time process (events can occur at any time) Jumps occur one at a time The time between events is exponentially distributed Each exponential is independent (related to independent increments ) Each exponential has the same rate (related to the stationary increments ) 3/ 15
Sample Paths of the Poisson Process time N(t) 4/ 15

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Sample Paths of the Poisson Process time N(t) t' N(t') = 2 Exp( λ29 Exp( λ29 Exp( λ29 Exp( λ29 4/ 15
Constructing Sample Paths of the Poisson Process If instead at times { 0 , 1 , 2 ,... } (as is done for DTMCs) we generate the inter-event times by independent Exp ( λ ) random variables, we have paths that look like time N(t) t' N(t') = 2 Exp( λ29 Exp( λ29 Exp( λ29 Exp( λ29 5/ 15

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Constructing Sample Paths of the Poisson Process If instead at times { 0 , 1 , 2 ,... } (as is done for DTMCs) we generate the inter-event times by independent Exp ( λ ) random variables, we have paths that look like time N(t) t' N(t') = 2 Exp( λ29 Exp( λ29 Exp( λ29 Exp( λ29 Continuous-time, jumps at exponential times, according to the DTMC, Q 5/ 15
Constructing Sample Paths of the Poisson Process If instead at times { 0 , 1 , 2 ,... } (as is done for DTMCs) we generate the inter-event times by independent Exp ( λ ) random variables, we have paths that look like time N(t) t' N(t') = 2 Exp(

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## This note was uploaded on 04/03/2008 for the course ORIE 361 taught by Professor Lewis,m. during the Spring '07 term at Cornell.

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CTMCI_beamer - Introductory Engineering Stochastic...

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