# CTMCII-beamer - Dealing with Some Assumptions Computation...

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Unformatted text preview: Dealing with Some Assumptions Computation of Transition Probabilities Solving the Backward or Forward Equations 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/ 19 Dealing with Some Assumptions Computation of Transition Probabilities Solving the Backward or Forward Equations Embedded DTMC has No Self-Loops Explosiveness (no need for finite state space) Embedded DTMC has No Self-Loops We assumed that the embedded DTMC has Q ii = 0. Is this required? Recall we are trying to model a natural phenomenon by CTMCs...and so we would not actually view self loops (it would just stay in the same state at time t ). Just for fun, suppose we want a CTMC from an embedded DTMC with Q ii > 0 for some i and some rates v i for the exponential times between jumps (including self-loops) So the rate at which we see jumps from i to j (for i 6 = j ) should be Q ij v i (as before) But the rate at which actually leave state i is not v i 2/ 19 Dealing with Some Assumptions Computation of Transition Probabilities Solving the Backward or Forward Equations Embedded DTMC has No Self-Loops Explosiveness (no need for finite state space) Embedded DTMC has No Self-Loops We assumed that the embedded DTMC has Q ii = 0. Is this required? Recall we are trying to model a natural phenomenon by CTMCs...and so we would not actually view self loops (it would just stay in the same state at time t ). Just for fun, suppose we want a CTMC from an embedded DTMC with Q ii > 0 for some i and some rates v i for the exponential times between jumps (including self-loops) So the rate at which we see jumps from i to j (for i 6 = j ) should be Q ij v i (as before) But the rate at which actually leave state i is not v i 2/ 19 Dealing with Some Assumptions Computation of Transition Probabilities Solving the Backward or Forward Equations Embedded DTMC has No Self-Loops Explosiveness (no need for finite state space) Embedded DTMC has No Self-Loops We assumed that the embedded DTMC has Q ii = 0. Is this required? Recall we are trying to model a natural phenomenon by CTMCs...and so we would not actually view self loops (it would just stay in the same state at time t ). Just for fun, suppose we want a CTMC from an embedded DTMC with Q ii > 0 for some i and some rates v i for the exponential times between jumps (including self-loops) So the rate at which we see jumps from i to j (for i 6 = j ) should be Q ij v i (as before) But the rate at which actually leave state i is not v i 2/ 19 Dealing with Some Assumptions Computation of Transition Probabilities Solving the Backward or Forward Equations Embedded DTMC has No Self-Loops Explosiveness (no need for finite state space) Embedded DTMC has No Self-Loops We assumed that the embedded DTMC has Q ii = 0. Is this required?...
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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 University (Engineering School).

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CTMCII-beamer - Dealing with Some Assumptions Computation...

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