1. MURI ADCN Workshop John Doyle, Steven Low EAS, Caltech OSU, Columbus September 9, 2009 Post-docs Lijun Chen Krister Jacobsson Chee-Wei Tan Grad students Javad Lavaei JK Nair Somayeh Sojoudi Undergrad/Staff Martin Andreasson Tom Quetchenbach

2. Outline  File fragmentation to mitigate heavy- tailed delay (Low)  Network arch theory (Doyle)  Nonconvex power control in ad hoc wireless networks (Tan)

3. 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.)  Model  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

4. 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 file fragment constant overhead

5. Model remaining file size at time n+1 fragment size at n per-packet overhead iid random var of distr F

6. Model per-stage cost: total cost:

7. 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

8. 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 Implication: Heavy tail can originate from protocol interaction alone! Motivation: How to mitigate?

9. Prior work Theorem [ Jelenkovic & Tan 2008] If fragment size = largest of k previous T(L) still has (lighter) heavy tail with first k moments

10. Prior work Intuition: HT is created by repeated comparison of a sequence of iid rv’s with the same rv L with unbounded support Avoid such fragmentation policies

11. Two fragmentation policies independent fragmentation: bounded fragmentation:

12. 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

13. What is optimal fragmentation? Dynamic programming formulation optimal fragmentation:

14. Per-bit cost per-bit cost: x g(x) a Key assumption : is non-decreasing Intuition: a minimizes per-bit cost; Optimal fragment size close to a ?

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

16. Simpler fragmentation Optimal fragmentation requires knowledge of L, in addition to failure distr F Want: blind fragmentation that only requires F optimal frag:

17. Result: blind fragmentation Theorem for all L Blind fragmentation is asymptotically optimal blind fragmentation: expected total cost:

18. Result: robustness Theorem What happen if the optimal or blind policy is designed wrt failure distribution G when the actual distribution is F ? Optimal cost under F : Optimal cost under G : Blind cost under G :

19. Result: tail distribution of T(L) Definition G is regularly varying(RV) with index >0 if where is a slowly varying function

20. Result: tail distribution of T(L) Theorem If L light-tailed, so is T(L) If L RV() (heavy-tailed), so is T(L) optimal frag: blind frag:

21. Result: tail distribution of T(L) Theorem If L light-tailed, so is T(L) If L RV() (heavy-tailed), so is T(L) Optimal or blind policy preserves the index of tail distribution

22. Summary  Independent or bounded fragmentation preserves light-tailedness  Under IFR, optimal fragmentation is unique and constant  Blind fragmentation is asymptotically optimal  Optimal or blind fragmentation preserves tail index

23. Outline  File fragmentation to mitigate heavy- tailed delay (Low)  Network arch theory (Doyle)  Nonconvex power control in ad hoc wireless networks (Tan)

24. Network arch theory  Key elements of network architecture Robust yet fragile Layering as optimization decomposition/distributed IPC Constraints that deconstrain (Gerhart & Kirschner) Resources Deconstrained Applications Deconstrained Constraints that deconstrain

25. Status: very early stage  To better understand layering From familiar: congestion control optimization To: optimal dynamics, wireless, network coding Layering as recursive control: physical layer antenna design  To better understand constraints Energy constraint Constraints from optimal tradeoffs  Still working on component problems, but optimistic they will point to a general theory

26. Each layer is abstracted as an optimization problem Operation of a layer is a distributed solution Results of one problem (layer) are parameters of others Operate at different timescales Layering as optimization decomposition Application: utility IP: routing Link: scheduling Phy: power IP TCP/AQM Physical Application Link/MAC

27. Layering as optimization decomposition r Networkgeneralized NUM r Layerssub-problems r Interfacefunctions of primal/dual variables r Layeringdecomposition methods Vertical decomposition: into functional modules of different layers Horizontal decomposition: into distributed computation and control IP TCP/AQM Physical Application Link/MAC

28. Examples application transport network link physical Optimal web layer: Zhu, Yu, Doyle ’01 HTTP/TCP: Chang, Liu ’04 TCP: Kelly, Maulloo, Tan ’98, …… TCP/IP: Wang et al ’05, …… TCP/power control: Xiao et al ’01, Chiang ’04, …… TCP/MAC: Chen et al ’05, …… Rate control/routing/scheduling: Eryilmax et al ’05, Lin et al ’05, Neely, et al ’05, Stolyar ’05, Chen et al ‘06 detailed survey in Proc. of IEEE, 2006

29. Example: Cross-layer congestion/routing/scheduling design Rate controlScheduling Routing Rate constraint Schedulability constraint

30. Cross-layer implementation  Rate control:  Routing: solved with rate control or scheduling  Scheduling: Network Transport Physical Application Link/MAC A Wi-Fi implementation by Warrier, Le and Rhee shows significantly better performance than the current system. Rate controlScheduling Routing

31. Recent generalizations r Optimal control  Lavaei, Doyle and Low, CDC, 2009 r Robust control  Jacobsson, Andrew and Tang, CDC, 2009  Jacobsson, Andrew, Tang, Low and Hjalmarsson, TAC, March 2009 r Game theory  Chen, Cui and Low. JSAC, September 2008.  Chen, Low and Doyle, ToN, submitted r Network coding  Chen, Ho, Chiang, Low and Doyle. T-IT, submitted

32. Recent generalizations r Optimal control  Lavaei, Doyle and Low, CDC, 2009 r Robust control  Jacobsson, Andrew and Tang, CDC, 2009  Jacobsson, Andrew, Tang, Low and Hjalmarsson, TAC, March 2009 r Game theory  Chen, Cui and Low. JSAC, September 2008.  Chen, Low and Doyle, ToN, submitted r Network coding  Chen, Ho, Chiang, Low and Doyle. T-IT, submitted

33. Resources Deconstrained Applications Deconstrained From optimization to optimal control

34. router TCPAQM my PC source algorithm (TCP) iterates on rates link algorithm (AQM) iterates on prices Primal: Dual horizontal decomposition Static optimization: dual algorithm

35. Controller is fully decentralized Globally stable to optimal equilibrium Generalizations to delays, other controllers

36. Implications Views TCP as solving an optimization problem Clarifies tradeoff at equilibrium Generalizes to other strategies, other layers Framework for cross layering But are the dynamics optimal?

37. State weight Control weight dynamics IQ penalty on deviation from equilibrium Balance state versus control penalty What is this controller optimal for?

38. Other implications Elegant proofs of stability Clarifies the tradeoff in dynamics Insights about joint congestion control and routing Can derive more general control laws

39. Where we are going

40. Layering: Rethinking fundamentals Distributed IPC (Inter-process comms/controls) –Book: John Day, Patterns in network architecture –Generalizes OS as IPC to networks –Natural fit with optimization framework –Layering/Control recurses, with changes in scope Compatible with “platform-based design” (A. S-V) –Recursive design from applications to silicon? –Optimization/decomposition –Illustrate with wireless circuit design Emphasis continues on central challenges –Wireless –Mobility –Real time

41. application Physical From layering as DIPC to platform-based design Recursive design process From applications to silicon Optimization/decomposition Illustrate with antennae design Recursion Scope Physical Circuit Logical Instructions Next steps

42. Transistors operating at wavelengths << chip dimensions Forcing (facilitating) integrated E&M, circuits, and systems. Design difficult but also truly novel systems/capabilities New and elegant solution for the large-scale radiating circuit problems where the conventional circuit assumptions are no longer valid (Lavaie, Babakhani, Hajimiri, Doyle) Application to diverse wireless communication problems Unifying theme: Layering as optimization Duality and convexity

43. Heterogeneous applications ubiquitous at every scale mobility/wireless real-time/sense/control exploding complexity and diversity Unifying theme: Layering as optimization Duality and convexity

44. Outline  File fragmentation to mitigate heavy- tailed delay (Low)  Network arch theory (Doyle)  Nonconvex power control in ad hoc wireless networks (Tan)

45. Nonconvex Power Control in Ad Hoc Wireless Networks Chee Wei Tan Caltech Joint Work with Mung Chiang (Princeton) & R. Srikant (UIUC) 45

46. Motivation Objective: Performance Optimization in Multi-hop Ad-hoc Wireless Networks Questions: –What are the important performance objectives in wireless network? –Are there fast algorithms that optimize the performance objectives? –How to extend the solution to optimize power and beamformer jointly? 46

47. Ad Hoc Wireless Networks 47 Data communication, low power constraint, low complexity signal sets, multiuser interference

48. Wireless Network Model Wireless Ad-hoc Network Model : 48

49. Throughput Maximization Total power constraint Individual power constraint Vector w as queue length 49

50. Geometrical Illustration 50 University of Illinois at Urbana-Champaign

51. Two Related Problems 51 Constraints: Individual or total power

52. Power Control Algorithms 52 Goal: Fast algorithms under –Weighted Sum Rate maximization –Weighted Max-min SIR –Weighted Sum MSE minimization –Why? Time-varying network conditions, i.e., optimization problem parameters change –Users come and go –Queues of each user change continuously –Due to mobility of users in network –Time-varying fading channel condition

53. Max-min SIR 53 Interpretation: SIR threshold

54. Max-min SIR Why? - Can express our iterative algorithm as Result follows from Blondel, Nivone, Van Dooren (2005), a special case of nonlinear Perron- Frobenius theoy 54 Main result: converges geometrically fast to right eigenvector of where where is a nonnegative matrix and is a nonnegative vector.

55. Weighted Sum MSE 55

56. Weighted Sum MSE The problem can be written as For a nonnegative matrix where Condition: (either low-medium SNR regime or low interference regime) Derive using Friedland-Karlin inequalities in nonnegative matrix theory 56

57. Key Ideas Previous approaches: Using geometric programming technique and subgradient technique –Parameter tuning (step-size) –Slow convergence Our approach: –Geometric programming change-of-variable –Show that KKT optimality conditions can be obtained using a fixed-point approach 57

58. Weighted Sum MSE: Algorithm 58

59. Weighted Sum MSE Why use this algorithm? –Geometrically fast convergence –No step-size tuning required 59 Proof outline: – z = I (z) –Under conditions on I(.), convergence is geometric, results followed from Yates (1995) –Our MSE algorithm can be shown to satisfy these conditions

60. Weighted Sum Rate 60

61. Weighted Sum Rate The problem can be written as For a nonnegative matrix where Same idea as Weighted Sum MSE problem KKT optimality conditions can be obtained using a fixed-point approach 61

62. Weighted Sum Rate: Algorithm 62

63. In general, Max-min SIR not the same as Weighted Sum Rate 63 Connection between Weighted Sum Rate & Weighted Max-min SIR User 1 rate User 2 rate Vector w (queue size)

64. In general, Max-min SIR not the same as Weighted Sum Rate 64 Connection between Weighted Sum Rate & Weighted Max-min SIR Vector w (queue size) User 1 rate User 2 rate

65. Connection between Weighted Sum Rate & Weighted Max-min SIR 65

66. Extensions So far work for ad hoc networks or single-antenna power controlled networks For MIMO networks, need to optimize beamformers Initial work: Access-point controlled network 66

67. Downlink Transmit Beamformer Optimize power and transmit beamformer for all users Goal: Max-min SIR over power and beamformers 67 Transmit beamformer u1u1 u2u2 User 1 User 2 User 1 Receiver User 2 Receiver Power control

68. Uplink Receive Beamformer 68 Receive beamformer u1u1 u2u2 User 1 User 2 User 1 Transmitter User 2 Transmitter Power control Virtual uplink as auxiliary mechanism Our approach: Iterative solution is easier, reuses existing CDMA power control module and converges geometrically fast time Slot 1 (Downlink) Slot 2 (Uplink) Slot 3 (Downlink) ….