Adaptive Routing with Stale Information

1. Adaptive Routing with Stale Information S. Fischer and B. Vöcking ACM PODC 2005 CS591IG, Spring 2006 UIUC

2. Outline • Motivation • System Model • Results & Proofs • Stability • Efficiency • Conclusion

3. Motivation • Routing metrics currently used are static (i.e. hop count) • Inefficient in terms of packet delay, bandwidth • Common adaptive routing that use dynamics metrics (i.e. delay) can introduce instability to networks • Stale routing information can cause periodic route changes and oscillations • Is it possible to achieve both stability and efficiency using adaptive routing in the presence of stale routing information?

4. Contributions • Theoretically, this paper shows that • Common adaptive routing policies can stabilize the network, given that information is up-to-date • However, common adaptive routing policies can cause network to be unstable, given that information is old • Based on some assumptions, certain adaptive routing policies can guarantee system stability, even with stale information

5. System Model • Consider a network graph G=(V,E) with set I of commodities. • Commodity iI needs to send normalized trafficdi [0,1] from a sourcesiV to a sinkti V • Let Pi = set of all possible paths connecting si and ti • LetP = iIPi • Let a Flow vector (x)pPdenotes a traffic allocation on G • Let le : [0,1] → R0+, e  Ebe the latency function of normalized traffic at edge e • Assume that each leis continuous, non-decreasing and has finite first derivative • Hence, the latency at edge e is le(xe) where xeis the total normalized traffic at edge e

6. 2 commodities 1. A wants to send traffic 0.7 to B 2.A wants to send traffic 0.3 to C P1 = {AB,ACB} P2 = {AC,ABC} P = P1P2= {AB,AC,ABC,ACB} Example B lAB(x) = x lBC(x) = x1/2 A lAC(x) = x1/4 C

7. P = P1P2= {AB,AC,ABC,ACB} One feasible traffic allocation is (x)pP = [0.4,0.1,0.2,0.3] Example B lAB(x) = x lBC(x) = x1/2 A lAC(x) = x1/4 C lAB = lAB(0.4+0.2) = 0.6 lAC = lAC(0.1+0.3) = (0.4)1/4 = 0.795 lABC = lAB(0.4+0.2)+ lBC(0.2+0.3) = 0.6 + (0.5)1/2= 1.307 lACB = lAC(0.1+0.3) + lBC(0.2+0.3) = (0.4)1/4 + (0.5)1/2 = 1.502

8. Adaptive routing • Assume each flow consists of an infinite number of agents carrying an infinitesimal load • Each agent will try to change her path to minimize her own latency • Each agent revises her own routing policy independently at a fixed Poisson rate • At revision point, the agent using path p samples a pathq in he same commodity (p,q  Pi for some i  I) with sampling probability σpq • After choosing path q, the agent switches from path p to path q with migration probabilityμ(lp,lq) • The flow allocation either keeps changing, or reaches a state where no agent can improve her latency individually (a.k.a. Wardrop Equilibrium)

9. Adaptive routing(cont.) Sample? (Poisson) yes no Work (current path = p) Pick a path q with prob = σpq 1 - μ(lp,lq) Migrate to the new path? μ(lp,lq) Change route Flow chart of an agent’s activity

10. Sampling probability σpq • There are 2 sampling schemes in the paper • Uniform sampling : σpq = 1/|Pi| for all iI, p,q Pi • Proportional sampling : σpq = xq/di , where xq is the normalized traffic on path q

11. Migrate probability μ(lp,lq) • Intuitively, most adaptive routing protocols use Better Response Migrate Policy μ(lp,lq) = 1 if lp > lq 0 otherwise • A migration policy is smooth if there exists a value α such that μ(lp,lq) ≤ α|lp-lq| • Obviously, the better response migrate policy is not smooth • An example of smooth policies is Linear Migration Policy μ(lp,lq) = max {(lp-lq)/lmax , 0}

12. System solution • Based on sample-and-migrate model, and assume that every agent revises her route policy with Poisson rate = 1, the migration rate from path p to path q (rpq) can be calculated as follows rpq = xp . σpq . μ(lp,lq) • Hence, the fraction of load using path p (xp) can be solved by the following differential equation d(xp)/dt = qP(rqp - rpq)

13. Potential function • In order to proof the stability of the system, we don’t have to find the exact solution of (x)pP • In stead, the potential function Φ(X) can be used to represent the state of the system • The lowest possible value of the potential function indicates the equilibrium of the system

14. Result 1: Convergence under up-to-date information • Assume that the functions le(x) are strictly increasing for all eE. Also assume that σpqassign positive probability to any path and let σpq, μ(lp,lq) be Lipschitz continuous, then the system converges towards a Wardrop equilibrium Proof • We can see that the derivative of potential function is always negative (except at the equilibrium). However, the potential function itself is always positive. Since le(x), σpqandμ(lp,lq) are continuous, the potential function and its derivative are also continuous. Using Liapunov’s second method, we can conclude that all solutions converge towards the Wardrop equilibrium

15. Stale information • The paper uses bulletin board model (introduced by Mitzenmacher) • Every agent receives system information from a bulletin board • Every period length T, all system information (flow allocation, link delay, path delay) will be posted into the board. The information will be the same throughout each period

16. lBD(x) = c lAB(x) = c B x2 = 1- x1 D A x1 C lCD(x) = xd lAC(x) = xd Result 2 : instability from stale information • Better response dynamics can cause instability in the system in bulletin board model Proof • Consider the following graph • The flow at Wardrop equilibrium is x1*= c1/d • Using better response policy and bulletin board model, x1(t) = x1(0).e-t if x1(0) >x1* 1-(1-x1(0)).e-t if x1(0) <x1* • Given that x1*  (α,β) with α = (1/(eT+1)) and β= (eT/(eT+1)), let x1(0) = α, we’ll see that x1(2nT) = βandx1((2n+1)T) = α

17. Result 3 : Convergence under stale information • Given the following properties • The slope of le(x) is bounded by βfor all eE • The migration policy μ(l1,l2) is smooth with smooth value α(i.e. μ(l1,l2) ≤ α|l1-l2|) • The length of all paths pPis bounded by L • The functions le(x) are strictly increasing for all eE. σpqassigns positive probability to any path. σpq, μ(lp,lq) are Lipschitz continuous • Then updating the bulletin board every T = 1/(4 L β α) is sufficient for the system to converge to Wardrop equilibrium (Rough) Proof • Show that for every phase beginning at time t with an update of the bulletin board and ending at time t + ,  ≤ T, the change of potential function is always non-increasing • Use Liapunov’s second method for differential equation with time delay to prove convergence of the system

18. Convergence Speed • When the system reaches the equilibrium, all agents from the same commodity achieve the same delay • The paper describes the speed of convergence by the number of periods that the system does not loosely converge • some agents spend more delay than the other agents from the same commodity

19. (,)-Approximate Equilibrium • Strong definition An agent is -unsatistfied when it uses a path p  Pi with lp > lmin,i+ , where lmin,I := minq Pi{lq}. A flow allocation xis said to be at a (,)-approximate equilibrium if at most  agents are -unsatisfied. • Weak definition An agent is -unsatistfied when it uses a path p  Pi with lp > (1+ )lav,i, where lav,i := q Pi{(xq/di)lq} is the average latency of commodity i. A flow allocation xis said to be at a (,)-approximate equilibrium if at most  agents are -unsatisfied.

20. Result 4 : How quick convergence can be • Assume the linear migration policy is used. Assume the bulletin board model is used with update interval length T = 1/(4L.α.β). • For the uniform sampling policy, the number of update periods not starting in a strong(,)-approximate equilibrium is bounded by ,where m = maxiI{|Pi|} • For the proportional sampling policy, the number of update periods not starting in a weak(,)-approximate equilibrium is bounded by (Rough) Proof • Showing that for each period not starting in (,)-approximate equilibrium, the decrease of potential function will be at least some values k. However, the potential function is bounded by lmax. Hence, the number of period not starting in (,)-approximate equilibrium can be no more than lmax/k

21. Conclusion • The paper • Proved the instability from non-smooth adaptive routing with stale information • Showed a way to achieve routing stability by using smooth adaptive routing with stale information that is periodically update with period T = 1/(4 L β α) • Quantified how quickly the system can converge to stability in the form of the number of periods the system does not loosely converge

22. Discussion • How practical is the proposed model? • The paper considers only delay as the dynamic metric • How about throughput? • Bulletin board model is not scalable • Practically, routing information is distributed • Distributed, multiple bulletin boards model may be good to try • The model assume lossless channel • Packet retransmission?

23. Discussion (Cont.) • How can we benefit from this model? • Internet routing? • Obviously, bulletin board model is not practicable • The model is too static • Overlay network? • Dedicated network infrastructure? • Might work (since there is no churn)