This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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] =&gt; 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...
View
Full
Document
This document was uploaded on 02/27/2010.
 Spring '09

Click to edit the document details