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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

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

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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

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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

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

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Model per-stage cost: total cost:

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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

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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?

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

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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

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Two fragmentation policies independent fragmentation: bounded fragmentation:

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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

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What is optimal fragmentation? Dynamic programming formulation optimal fragmentation:

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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 ?

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

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

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

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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 :

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

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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:

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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

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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

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

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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

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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

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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

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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

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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

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

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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

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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 Chen, Low and Doyle, ToN, submitted r Network coding Chen, Ho, Chiang, Low and Doyle. T-IT, submitted

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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 Chen, Low and Doyle, ToN, submitted r Network coding Chen, Ho, Chiang, Low and Doyle. T-IT, submitted

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Resources Deconstrained Applications Deconstrained From optimization to optimal control

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

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

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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?

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

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

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Where we are going

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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

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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

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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

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

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

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

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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

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

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Wireless Network Model Wireless Ad-hoc Network Model : 48

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

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Geometrical Illustration 50 University of Illinois at Urbana-Champaign

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Two Related Problems 51 Constraints: Individual or total power

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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

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Max-min SIR 53 Interpretation: SIR threshold

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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.

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Weighted Sum MSE 55

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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

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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

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Weighted Sum MSE: Algorithm 58

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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

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Weighted Sum Rate 60

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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

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Weighted Sum Rate: Algorithm 62

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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)

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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

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

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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

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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

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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) ….

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