Stability and Fairness of Service Networks

1. Stability and Fairness of Service Networks Jean Walrand – U.C. BerkeleyJoint work with A. Dimakis, R. Gupta, and J. Musacchio

2. Outline • Stability of Longest Queue First • Fluctuations can stabilize • Fairness through flow control • Control of long term rates • Fairness of multiple access • Impatience may help in a crowd

3. Outline • Stability of Longest Queue First • Fluctuations can stabilize Fairness through flow control Control of long term rates Fairness of multiple access Impatience may help in a crowd

4. Stability of Longest Queue Firstwith Antonis Dimakis (PhD 5/06) • Motivation • Easy Case • Subtle Effect

5. Stability of Longest Queue Firstwith Antonis Dimakis (PhD 5/06) • Motivation • Easy Case • Subtle Effect

6. LQF - Motivation • Wireless: 2 3 4 5 1 6 • Goals: Simple protocol, large throughput • Transmission priority increases with backlog

7. Q11(t) input 1 l11 output 1 l12 1 1 Q12(t) Q21(t) input 2 l21 Q22(t) output 2 l22 2 2 LQF: Motivation • Switch: • Iterated Longest Queue First (iLQF) [McKeown’95]: • Queues are considered in decreasing queue size order. • Maximum throughput?

8. Stability of Longest Queue Firstwith Antonis Dimakis (PhD 5/06) • Motivation • Easy Case • Subtle Effect

9. l1 l2 l3 i.i.d. arrivals 1 2 3 servicevectors Stability of LQF: Easy Case Example: w.p. 1/2 9 – 9 – 8 12 – 9 – 8 LQF: w.p. 1/2 9 – 9 – 8 7 – 9 – 8

10. l1 l2 l3 1 2 3 Stability of LQF: Easy Case • Necessary: l1+l2<1, l2+l3<1. • Sufficient! Under LQF, longest queues tend to decrease: • Say, Q1¼ Q2>>Q3, for some time. • Then, Q1+Q2 decreases, and so do Q1,Q2. • Key: locally in time, service from common resource pool.

11. l1 l2 l3 1 2 3 Stability of LQF: Easy Case • Local Pooling: Assume {1, 2} are longest for some time Note that {1, 2} are served at constant rate (1) (We say that {1, 2} satisfies Local Pooling.)  {1, 2} must decrease (because l1+l2<1)  longest queue must decrease

12. l1 l2 l3 1 2 3 Stability of LQF: Easy Case • Local Pooling: Assume {1, 2, 3} are longest for some time Note that {1, 2} are served at constant rate (1) (We say that {1, 2, 3} satisfies Local Pooling.)  {1, 2} must decrease (because l1+l2<1)  longest queue must decrease

13. l1 l2 l3 1 2 3 Stability of LQF: Easy Case • Local Pooling: Assume {1, 3} are longest for some time Note that {1, 3} are served at constant rate (2) (We say that {1, 3} satisfies Local Pooling.)  {1, 3} must decrease (because l1+l3<2)  longest queue must decrease

14. l1 l2 l3 1 2 3 Stability of LQF: Easy Case • Local Pooling: Set L satisfies LP if it has a subset K that LQF serves at a constant rate Theorem:If every set L satisfies LP and if the rates are feasible,then LQF makes system stable Proof: Longest queue is a Lyapunov function(Consider fluid limit ….)

15. Stability of LQF: Easy Case • Graphs that satisfy Local Pooling: 3, 4, 5 Cycles Trees Combinations

16. Stability of Longest Queue Firstwith Antonis Dimakis (PhD 5/06) • Motivation • Easy Case • Subtle Effect

17. 2 3 1 4 6 5 Stability of LQF: Subtle Effect • Graph that does not satisfy Local Pooling: Service Vectors: {1, 3, 5}, {2, 4, 6} {1, 4}, {2, 5}, {3, 6} {1, 2, 3, 4, 5, 6} has no subset served at constant rate {1, 2, 3, 4, 5, 6} does not satisfy LP Every proper subset satisfies LP E.g., {1, 2, 3, 5} longest  serve {2, 3} at rate 1

18. 2 3 1 4 6 5 Stability of LQF: Subtle Effect • Note: Deterministic inputs with rate close to 0.5  unstable (LQF serves 2/6 a positive fraction of time) Theorem: LQF stable if i.i.d. arrivals with nonzero variance Key Idea: {1, 2, 3, 4, 5, 6} cannot be set of longest queues for a positive fraction of time!  LP holds most of the time  Longest queue decreases

19. 2 3 1 4 6 5 Stability of LQF: Subtle Effect Key Idea: {1, 2, 3, 4, 5, 6} cannot be set of longest queues for a positive fraction of time! • Assume all queues are longest for a while • {2, 3} and {5, 6} served at same rate

20. Stability of LQF: Subtle Effect

21. Stability of LQF: Subtle Effect • Max – Min large at kD(n) • A subset L of queues dominates the others during interval • This subset satisfies LP  Longest queue decreases.

22. Stability of LQF: Subtle Effect Theorem: Assume that whenever a set L does not satisfy LP, the corresponding service vectors have rank ≤ |L| - 2. Assume also the arrivals are i.i.d. with positive variance(and satisfy a large deviation bound). Then LQF is stable for any feasible arrival rates.

23. 2 3 1 2 1 4 8 3 6 5 7 4 5 6 Stability of LQF: Subtle Effect Examples:

24. Outline Stability of Longest Queue First Fluctuations can stabilize • Fairness through flow control • Control of long term rates Fairness of multiple access Impatience may help in a crowd

25. Motivation Analysis Fairness Through Flow Control(with John Musacchio, UCSC)

26. Motivation Analysis Fairness Through Flow Control(with John Musacchio, UCSC)

27. Motivation h = discard threshold • Example Intuitively: h large enough  max – min fair Long-term average rates  max – min for h >> 1

28. Motivation h = discard threshold

29. Motivation

30. Motivation Analysis Fairness Through Flow Control(with John Musacchio, UCSC)

31. Analysis Qn(nt): Scale thresholds and speed up

32. Analysis Qn(nt)/n: Scale space Qn(nt)/n  fluid limit Q(t) with suitable rates….

33. However, we want Analysis Roughly, x(n; t) := Qn(nt)/n  uoc fluid limit Q(t) For t ≥ t0,Q(t) = q* with suitable rates. This implies Key argument: Most of the time t ≥ 0, x(n; t) ≈ q*

34. Motivation q*:

35. Motivation q*:

36. However, we want Analysis Roughly, x(n; t) := Qn(nt)/n  uoc fluid limit Q(t) For t ≥ t0,Q(t) = q* with suitable rates. This implies Key argument: Most of the time t ≥ 0, x(n; t) ≈ q* • To show this: • Uniformly in |x(n; 0) – q*| ≤ e, E[|x(n; t) – q*|]  0 for t ≤ t0 • Uniformly in y = |x(n; 0) – q*| > e, E[|x(n; yt0) – q*|] < dy • Expected time E(t) until |x(n; t + t0) – q*| ≤ e is small for n >> 1

37. Analysis Key argument: Average throughput close to max min

38. Outline Stability of Longest Queue First Fluctuations can stabilize Fairness through flow control Control of long term rates • Fairness of multiple access • Impatience may help in a crowd

39. Motivation Protocol Analysis Simulations Fairness of Multiple Accesswith Rajarshi Gupta (PhD 5/05)

40. Motivation Protocol Analysis Simulations Fairness of Multiple Accesswith Rajarshi Gupta (PhD 5/05)

41. Motivation: Exponential Backoff is Unfair • Exponential backoff scheme (e.g. 802.11b) • Nodes pick backoff uniformly in a backoff range • If collision, double the backoff range • Multiple interference domains • Node in center sees more contention and collision • It backs off more • Gets lesser share of bandwidth • Unfair towards middle nodes in network All rates in Mbps

42. Motivation Protocol Analysis Simulations Fairness of Multiple Accesswith Rajarshi Gupta (PhD 5/05)

43. Protocol: Impatient Backoff Algorithm • Approach: Nodes that face more contention should get higher priority • Key Mechanism • Upon collision, nodes decrease their backoff • Need to worry about • Stability • Fairness • Throughput

44. Protocol: Backoff Update • If collision or quiet • Decrease the mean backoff delay • b := b/m, where m>1 • If successful transmission • Increase the mean backoff delay • b := bm • Note: Distributed reset mechanismWhen a node’s mean delay falls below threshold, node broadcasts “multiply by K” ….

45. Protocol: Simplified MAC Model • All packet lengths are same • Transmissions occur slot by slot • Local synchronization is assumed • Similar to any slotted protocol • No RTS/CTS

46. Protocol: IBA Mechanism • Backoff Contention Phase • Each node has mean backoff b • Picks backoff delay B using exponential variable with mean b • Sends out Slot Capture Message after B backoff mini-slots • If a node carrier senses another message sooner – it keeps quiet • Packet Transmission Phase • Starts after completion of Backoff Contention Phase • Nodes with successful Slot Capture Messages transmit • Constant packet length • Transmission confirmed by ack • Collision occurs if two neighbors pick same backoff • Neither hears slot capture • Both try to transmit • Packet transmission wasted

47. Motivation Protocol Analysis Simulations Fairness of Multiple Accesswith Rajarshi Gupta (PhD 5/05)

48. Two extreme topologies Star Topology (unfair) Triangle Clique Topology (symmetric) Model ratio between mean backoffs Prove stability, fairness Throughput-fairness tradeoff (in Star) Max throughput = 0 + 41 = 4 But fair throughput = 0.5 + 40.5 = 2.5 interference Markov Chain Models interference

49. Stable: Positive recurrent for m>1 Strong drifts towards stable state S0 Fair: Expected transmission rate for all nodes is 0.5 interference Star Topology: Birth-Death Chain

50. Model sleeping nodes Every 100 slots, some nodes go to sleep Fairness = 1 Average success probability Middle Node(sX)= 0.473 Outer Nodes(sZ)= 0.470 Star Topology: Varying Neighbors