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
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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 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
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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 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
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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 ) 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
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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
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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 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
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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 ...can we generalize??
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