Download presentation

Presentation is loading. Please wait.

Published byDarius Davidson Modified over 3 years ago

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) ….

Similar presentations

OK

Michael J. Neely, University of Southern California CISS, Princeton University, March 2012 Asynchronous Scheduling for.

Michael J. Neely, University of Southern California CISS, Princeton University, March 2012 Asynchronous Scheduling for.

© 2019 SlidePlayer.com Inc.

All rights reserved.

To make this website work, we log user data and share it with processors. To use this website, you must agree to our Privacy Policy, including cookie policy.

Ads by Google