Lecture7 - Burke's Theorem

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

This preview shows pages 1–4. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the 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] =&amp;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.

### Page1 / 12

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

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online