Lecture 7 Burkes Theorem and Networks of Queues Eytan Modiano Massachusetts Institute of Technolog
Lecture 7 Burke’s Theorem and Networks of Queues Eytan Modiano Massachusetts Institute of Technology Eytan Modiano Slide 1
口 Burkes theoren 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 n is a Poisson process of rate n 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 ndependent PLpast present, future]= P[past present =>PDX可a+1=,Xn+22]=Px到x-=P
�Burke’s 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 λ is a Poisson process of rate λ – 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[Xn=j |Xn+1 =i, Xn+2=i2,...] = P[Xn=j | Xn+1=i] = P*ij Eytan Modiano Slide 2
Burke's Theorem(continued) The state sequence, run backward in time, in steady state, is a Markov chain again and it can be easily shown that (e.g, M/M/1(Pn)=(Pn+)u A Markov chain is reversible if P*j=Pi 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 Pi=p Pji All birth/death processes are reversible Detailed balance equations must be satisfied
Burke’s Theorem (continued) • The state sequence, run backward in time, in steady state, is a Markov chain again and it can be easily shown that piP*ij = pj Pji (e.g., M/M/1 (p n)λ=(pn+1)µ) • 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 pi Pij=pj Pji • All birth/death processes are reversible – Detailed balance equations must be satisfied Eytan Modiano Slide 3
Implications of Burke's Theorem Arrivals time Departures Since the arrivals in forward time form a poisson process, the departures in backward time form a Poisson process Since the backward process is statistically the same as the forward process, the(forward)departure process is Poisson By the same ty pe of argument, the state( packets in system) left by a (forward) departure is independent of the past departures In backward process the state is independent of future arrivals
Implications of Burke’s Theorem time Arrivals Departures time • Since the arrivals in forward time form a Poisson process, the departures in backward time form a Poisson process • Since the backward process is statistically the same as the forward process, the (forward) departure process is Poisson • By the same type of argument, the state (packets in system) left by a (forward) departure is independent of the past departures – In backward process t he state is independent of future arrivals Eytan Modiano Slide 4
NETWORKS OF QUEUES Exponential Exponential Poisson Poisson Poisson M/M/1 M/M/ ? The output process from an M/M/1 queue is a Poisson process of the same rate n as the input Is the second queue M/M/1?
NETWORKS OF QUEUES Exponential Exponential M/M/1 M/M/1 ? Poisson λ λ Poisson Poisson λ • The output process from an M/M/1 queue is a Poisson process of the same rate λ as the input • Is the second queue M/M/1? Eytan Modiano Slide 5