Lecture7 - Burke's Theorem

Lecture7 - Burke's Theorem - Lecture 7 Burkes Theorem and...

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Unformatted text preview: Lecture 7 Burkes Theorem and Networks of Queues Eytan Modiano Massachusetts Institute of Technology Eytan Modiano Slide 1 Burkes Theorem An interesting property of an M/M/1 queue, which greatly simplifies combining these queues into a network, is the surprising fact that the output of an M/M/1 queue with arrival rate is a Poisson process of rate This is part of Burke's theorem, which follows from reversibility A Markov chain has the property that P[future | present, past] = P[future | present] Conditional on the present state, future states and past states are independent P[past | present, future] = P[past | present] => P[X n =j |X n+1 =i, X n+2 =i 2 ,...] = P[X n =j | X n+1 =i] = P* ij Eytan Modiano Slide 2 Burkes Theorem (continued) The state sequence, run backward in time, in steady state, is a Markov chain again and it can be easily shown that p i P* ij = p j P ji (e.g., M/M/1 (p n ) =(p n+1 ) ) A Markov chain is reversible if P*ij = Pij Forward transition probabilities are the same as the backward probabilities If reversible, a sequence of states run backwards in time is statistically indistinguishable from a sequence run forward A chain is reversible iff p i P ij =p j P ji All birth/death processes are reversible Detailed balance equations must be satisfied Eytan Modiano Slide 3 Implications of Burkes Theorem...
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Lecture7 - Burke's Theorem - Lecture 7 Burkes Theorem and...

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