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CTMCII_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 CTMCs II 1/ 1
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Time Homogeneity Definition A continuous-time stochastic process { X ( t ) , t 0 } is said to be time homogeneous if it satisfies P ( X ( t + s ) = j | X ( s ) = i ) = P ( X ( t ) = j | X (0) = i ) for any t , s 0 and any states i and j. the transition probabilities depend only on the given states and the exact time between transitions and not on the initial time 2/ 1
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Time Homogeneity Definition A continuous-time stochastic process { X ( t ) , t 0 } is said to be time homogeneous if it satisfies P ( X ( t + s ) = j | X ( s ) = i ) = P ( X ( t ) = j | X (0) = i ) for any t , s 0 and any states i and j. the transition probabilities depend only on the given states and the exact time between transitions and not on the initial time The CTMCs we constructed last class have this property 2/ 1
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Explosiveness For a CTMC, it is possible that there are an infinite number of transitions in a finite amount of time 3/ 1
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Explosiveness For a CTMC, it is possible that there are an infinite number of transitions in a finite amount of time Generalize the Poisson process slightly to let the rate at which we go from i to i + 1 to be λ i = 2 i (called a pure birth process ). 3/ 1
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Explosiveness For a CTMC, it is possible that there are an infinite number of transitions in a finite amount of time Generalize the Poisson process slightly to let the rate at which we go from i to i + 1 to be λ i = 2 i (called a pure birth process ). The time of the i th jump in S i = i - 1 n =0 T n (where T n Exp ( λ i )). 3/ 1
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Explosiveness For a CTMC, it is possible that there are an infinite number of transitions in a finite amount of time Generalize the Poisson process slightly to let the rate at which we go from i to i + 1 to be λ i = 2 i (called a pure birth process ). The time of the i th jump in S i = i - 1 n =0 T n (where T n Exp ( λ i )). All jumps will be complete by S and E S = X n =0 E T n = X n =0 1 λ n = X n =0 1 2 n = 1 1 - 1 2 = 2 . 3/ 1
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Explosiveness For a CTMC, it is possible that there are an infinite number of transitions in a finite amount of time Generalize the Poisson process slightly to let the rate at which we go from i to i + 1 to be λ i = 2 i (called a pure birth process ). The time of the i th jump in S i = i - 1 n =0 T n (where T n Exp ( λ i )). All jumps will be complete by S and E S = X n =0 E T n = X n =0 1 λ n = X n =0 1 2 n = 1 1 - 1 2 = 2 . This is called explosiveness . 3/ 1
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Explosiveness For a CTMC, it is possible that there are an infinite number of transitions in a finite amount of time Generalize the Poisson process slightly to let the rate at which we go from i to i + 1 to be λ i = 2 i (called a pure birth process ).
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