MURI ADCN Workshop

1. MURI ADCN Workshop John Doyle, Steven Low EAS, Caltech OSU, Columbus October 14, 2010 Post-docs Lijun Chen KristerJacobsson Nader Motee Chee-Wei Tan Grad students MasoudFariva JavadLavaei JK Nair SomayehSojoudi

2. Outline • Overview of Caltech projects (40mins, Low) • Optimal wireless protocols and devices (40 mins, Lavaei)

3. Overview • Heavy-tailed traffic • File fragmentation to mitigate heavy-tailed delay • Tail-robust scheduling algorithms • Wireless • Random access game • Smart antenna design • Power control • Congestion control • Effect of ack-clocking • Reverse-engineering transients

4. Overview • Heavy-tailed traffic • File fragmentation to mitigate heavy-tailed delay • Tail-robust scheduling algorithms • Wireless • Random access game • Smart antenna design (JavadLavaei) • Power control • Congestion control • Effect of ack-clocking • Reverse-engineering transients

5. File fragmentation File fragmentation over an unreliable channel J. Nair, M. Andreasson, L. Andrew, S. Low and J. Doyle. IEEE Infocom, San Diego, CA, March 2010

6. File fragmentation: summary • Motivation: how to mitigate heavy tail? • Recent work showed file transfer time can be heavy-tailed even if file size is light-tailed (Fiorini, Sheahan, & Lipsky, 2005; Jelenkovic & Tan 2007; etc.) • Results • Independent or bounded fragmentation preserves light-tailedness • Constant fragmentation min expected delay • Asymptotically optimal design: blind fragmentation • Optimal or blind fragmentation preserves tail index

7. Model • Given file of random size L • L is fragmented into K packets for transmission at unit rate • n-th transmission of size • n-th transmission is successful if where are iid with distribution F constant overhead file fragment

8. Model fragment size at n remaining file size at time n+1 per-packet overhead iid random var of distrF

9. Model per-stage cost: total cost:

10. Prior work Theorem[Fiorini, Sheahan, & Lipsky, 2005; Jelenkovic & Tan 2007] Without fragmentation T(L) has heavy tail even when L is light-tailed, provided F has unbounded support

11. Result: LT-preserving frag independent fragmentation: bounded fragmentation: Theorem With independent frag or bounded frag: T(L) is light-tailed provided L is light-tailed Then, heavy-tailed delay originates only from heavy-tailed files

12. Result: optimal fragmentation per-bit cost: • Theorem • Constant fragmentation is uniquely optimal • Optimal #fragments: K*(L) = • Optimal fragment size: x*(L) = L/K*(L)

13. Result: blind fragmentation blind fragmentation: expected total cost: • Theorem • for allL • Blind fragmentation is asymptotically optimal

14. Result: tail distribution of T(L) optimal frag: blind frag: • Theorem • If L light-tailed, so is T(L) • If L RV(a) (heavy-tailed), so isT(L)

15. Overview • Heavy-tailed traffic • File fragmentation to mitigate heavy-tailed delay • Tail-robust scheduling algorithms • Wireless • Random access game • Smart antenna design (JavadLavaei) • Power control • Congestion control • Effect of ack-clocking • Reverse-engineering transients

16. Tail-robust scheduling Tail-robust scheduling via Limited Processor Sharing J. Nair, A. Wierman, and B. Zwart. Proc. IFIP Performance, 2010; to appear in Performance Evaluation

17. The “simplest” scheduling model Q: What policy minimizes mean response time? A: Shortest Remaining Processing Time (SRPT) Optimal regardless of interarrival times, job sizes, etc. Robust A Wierman

18. Q: Can a policy be optimal & robust for the tail? Lot’s of analysis over the last 20+ years… Power-law job sizes Light-tailed job sizes We’ll study the tail index: We’ll study the decay rate: A Wierman

19. Q: Can a policy be optimal & robust for the tail? Lot’s of analysis over the last 20+ years… Power-law sizes Light-tailed sizes SRPT Optimal [NWZ 08] Worst possible [NZ 06] Optimal [BBQZ 06] Worst possible [MZ 06] Worst possible [B76] Optimal [RS 01] Optimal [MT 80] Worst possible [NWZ 08] Worst possible [A99] Worst possible [N 07] PS FCFS PLCFS LCFS A Wierman

20. (non-learning) Q: Can a policy be optimal & robust for the tail? ^ A:NO! A Wierman

21. Theorem: There does not exist a work-conserving, online, non-learning scheduling policy ν that has: for all ε>0 and work-conserving, online policies πunder both light-tailed and power-law job sizes. Corollary: Optimal under power-laws  worst-case under light-tails,and vice-versa A Wierman

22. (non-learning) (non-learning) Q: Can a policy be optimal & robust for the tail? ^ ^ A: NO! Q: Can a policy be weakly robust for the tail? better-than-worst-case under bothlight-tailed and power-law workloads A: No known policies are. A Wierman

23. Our candidate:Limited Processor Sharing, LPS(c) at most c jobs PS …but it uses ρ FCFS queue A Wierman is weakly robust and optimal for large classes of power-law and light-tailed distributions.

24. Response time tail gets lighter Power-law c Response time tail gets lighter Light-tailed c c=1FCFS c=∞PS A Wierman

25. better-than-worst-case Power-law kc ≥ 2 better-than-worst-case Light-tailed c < ∞ c=1FCFS c=∞PS A Wierman

26. better-than-worst-case Power-law optimal (if sizes have a finite variance) better-than-worst-case Light-tailed optimal (if sizes are more “variable” thanan Exponential dist.) c=1FCFS c=∞PS A Wierman

27. Overview • Heavy-tailed traffic • File fragmentation to mitigate heavy-tailed delay • Tail-robust scheduling algorithms • Wireless • Random access game • Smart antenna design (JavadLavaei) • Power control • Congestion control • Effect of ack-clocking • Reverse-engineering transients

28. Random access game Random Access Game and MAC Design, L. Chen, S. H. Low and J. C. Doyle, IEEE/ACM Transactions on Networking, 2010

29. Contention-based MAC (contention control) • Two components • A contention resolution algorithm: adjusts channel access probability in response to the contention • A feedback mechanism: updates a contention measureand sends it back to wireless nodes L. Chen

30. Dynamical model • The exact form of and are determined by or can be designed for the specific MAC protocol • Present a game-theoretic model to understand the dynamical system (1) and use it to design new protocols (1) L. Chen

31. Random access game fixed point • Only determined by the contention resolution algorithm • Usually continuous, increasing and concave

32. Definition: A random access game is defined as a quadruple • is a set of players (wireless nodes) • Strategy with • Payoff function with given contention measure • MAC (i.e., system (1)) as strategy update algorithm achieving the equilibrium of random access game • The equilibrium properties can be understood and designed through the specification of and • The adaptation of channel access probability can be specified through , corresponding to different strategies to approach the equilibrium.

33. Conditional collision probability as contention measure • Assumptions (single cell wireless LANs): • A0: is continuously differentiable, strictly concave, and with bounded curvature away from zero, i.e., • A1: let and denote the smallest eigenvalue of by . Then, . • A2: functions are all strictly increasing or all strictly decreasing

34. Equilibrium • Theorem: Under assumption A0, there exists a Nash equilibrium for random access game. Suppose additionally A1 holds. Then random access game has a unique Nash equilibrium. • A channel access probability is a Nash equilibrium of random access game, if • Proof: By showing the equilibrium condition is the optimality condition for a strictly convex optimization problem.

35. Nontrivial Nash Equilibrium • A Nash equilibrium is a nontrivial equilibrium if for all nodes , the equilibrium strategy satisfies and trivial equilibrium otherwise. • Theorem: Suppose A2 holds. If the random access game has a nontrivial Nash equilibrium, it must be unique. • Proof by contradiction: Note that a nontrivial Nash equilibrium

36. Definition: A Nash equilibrium is said to be symmetric if for all , and an asymmetric equilibrium otherwise. • By symmetry, there must have multiple asymmetric equilibria if there exists any. • Theorem: For a system with several classes of users, suppose A1 and A2 hold. If random access game has a nontrivial equilibrium, it must be unique and symmetric. • Guarantees fair sharing of wireless channel among the same class of wireless nodes • Provides service differentiation among different classes of wireless nodes

37. Gradient play • Have a nice economic interpretation • Theorem: Suppose A0 and A1 hold. The gradient play converges to the unique Nash equilibrium of the random access game if for any , the stepsize • Proof by Lyapunov method. • Also studied its robust verification to the estimation error.

38. A concrete MAC design • Consider a single-cell network with classes of users • Each class associated with a weight • Assume • Want to achieve maximal throughput under the weighted fairness constraint

39. Utility design • Let . Under the assumption of Poisson arrival, the throughput achieves maximum at that satisfies • the duration of idle slot, the duration of a collision • Under the decoupling approximation, to achieve weighted fairness requires

40. Requires • A convenient choice • Utility function

41. Equilibrium and dynamics • Theorem: Suppose • The random access game has a unique and nontrivial equilibrium • The gradient play converges if the stepsize

42. Performance: throughput

43. Performance: collision

44. Performance: short-term fairness

45. Performance: dynamic scenario

46. Performance: service differentiation

47. A natural progression Centralized optimization Optimization Distributed but cooperative actions with rich information and signaling allowed Less cooperation (economic perspective) Less information or signaling available (engineering perspective) Game theory

48. Overview • Heavy-tailed traffic • File fragmentation to mitigate heavy-tailed delay • Tail-robust scheduling algorithms • Wireless • Random access game • Smart antenna design (JavadLavaei) • Power control • Congestion control • Effect of ack-clocking • Reverse-engineering transients