Stochastic Network Optimization and the Theory of Network Throughput, Energy, and Delay Michael J. Neely University of Southern California

Stochastic Network Optimization and the Theory of Network Throughput, Energy, and Delay Michael J. Neely University of Southern California http://www-rcf.usc.edu/~mjneely/ *Sponsored in part by the DARPA IT-MANET Program, NSF OCE-0520324, NSF Career CCF-0747525 General mobile network

Outline: 1.Analogy between Info Theory and Network Theory “Capacity” Definitions Canonical Models 2.*Overview talk on Stochastic Network Optimization: History Landmark Results Application to general multi-hop stochastic networks 3.*Focus talk on 1-hop multi-user wireless downlink Fundamental energy-delay tradeoff Low complexity achievability *Details can be found in the following (available on my webpage): L. Georgiadis, M. J. Neely, L. Tassiulas, “Resource Allocation and Cross-Layer Control in Wireless Networks,” Foundations & Trends in Networking, vol. 1, no. 1, pp. 1-144, 2006. M. J. Neely, “Optimal Energy and Delay Tradeoffs for Multi-User Wireless Downlinks,” IEEE Transactions on Information Theory, vol. 53, no. 9, pp. 3095-3113, Sept. 2007.

Part 1: Analogy between info theory and network theory Capacity Region Λ = Set of all end-to-end rate vectors (or matrices) achievable over a network.  Information Theory View of Capacity Optimizes over all maps of symbols into codewords Results known for point-to-point links Results known for small 1-hop systems (broadcast/MAC)  Network/Queueing Theory View of Capacity Sometimes called “transport capacity” [Gupta/Kumar] Optimizes over all routing/scheduling/resource allocation Typically “link based” (with some extensions…) Simplified PHY layer (SINR, Interference Sets, etc.) Results hold for arbitrarily large networks, with mobility 

1 Wireless Link = AWGN Channel (symbol-by-symbol transmission) 1 Wireless Link = ON/OFF Channel (slot-by-slot packet transmission) “info theory”“queueing theory” + Symbols Noise C = log(1 + SNR) Capacity maximizes time avg. bit rate. [Optimizes over all coding strategies.] [very deep math for 1 link] Packet Arrivals Pr[ON]=p C = p packets/slot Capacity = time avg packet transmission rate. [nothing to optimization here] [Basic Queue Stability theory] Capacity: Mathematical Models for a Wireless System (two meaningful perspectives) Part 1: Analogy between info theory and network theory

1 Wireless Link = AWGN Channel (symbol-by-symbol transmission) 1 Wireless Link = ON/OFF Channel (slot-by-slot packet transmission) “info theory”“queueing theory” + Symbols Noise C = log(1 + SNR) Achievability: Random Coding Converse: Sphere-Packet, Fano [very deep math for 1 link] Packet Arrivals Pr[ON]=p C = p packets/slot Achievability: Obvious Converse: Obvious [Basic Queue Stability theory] Capacity: Mathematical Models for a Wireless System (two meaningful perspectives) Part 1: Analogy between info theory and network theory

N-User Gauss. Broadcast Downlink (symbol-by-symbol transmission) N-User Downlink (Fading Channels) (opportunistic packet transmission) “info theory”“queueing theory” bits    ON/OFF Capacity REGION is set of all supportable long term bit rate vectors Optimizes over all Coding Strategies Capacity REGION is set of all supportable packet arrival rate vectors Optimizes over all scheduling strategies Example: Observe Channel states, then decide which queue to serve Mathematical Models for a Wireless System (two meaningful perspectives) Part 1: Analogy between info theory and network theory

N-User Gauss. Broadcast Downlink (symbol-by-symbol transmission) N-User Downlink (Fading Channels) (opportunistic packet transmission) “info theory”“queueing theory” bits    ON/OFF Mathematical Models for a Wireless System (two meaningful perspectives) Part 1: Analogy between info theory and network theory Capacity Region: all ( 1,…, N ) s.t. for all subsets K of users. [Tassiulas & Ephremides ‘93] Capacity Region: all ( 1,…, N ) s.t. (degraded Gauss. BC)

N-Node Static Multi-Hop Network (multiple sources and destinations) “info theory”“queueing theory” N-Node Static Multi-Hop Network (multiple sources and destinations) Infinite Traffic Symbol-by-Symbol Transmissions Interference Channels Optimize the coding Capacity = ??? Pr[Channel k = ON] = p k Random Packet Arrivals Optimize Scheduling/Routing p1p1 p2p2 p3p3 p4p4 p5p5 p6p6 Capacity = Known Exactly (Multi- Commodity Flow Subject to “Graph Family” Link Constraints) Mathematical Models for a Wireless System (two meaningful perspectives) Part 1: Analogy between info theory and network theory

N-Node Static Multi-Hop Network (multiple sources and destinations) “info theory”“queueing theory” N-Node Static Multi-Hop Network (multiple sources and destinations) Coding Tools (inner bounds): Net Coding, Cooperative Trans., etc. Converse Tools (outer bounds): cut-sets, multi-terminal info theory Capacity = ??? Scheduling Tools: Max Weight Matching (MWM), Backpressure Converse Tools: Queue Stability, Flow Conservation, Optimization p1p1 p2p2 p3p3 p4p4 p5p5 p6p6 Capacity = Known Exactly Mathematical Models for a Wireless System (two meaningful perspectives) Part 1: Analogy between info theory and network theory

N-Node MANET Scheduling Tools: Max Weight Matching (MWM), Backpressure Routing Converse Tools: Queue Stability, Flow Conservation, Optimization Capacity = ??? N-Node MANET Capacity = Known Exactly Mathematical Models for a Wireless System (two meaningful perspectives) Part 1: Analogy between info theory and network theory “info theory”“queueing theory” T/R

N-Node MANET Capacity = ??? N-Node MANET Mathematical Models for a Wireless System (two meaningful perspectives) Part 1: Analogy between info theory and network theory “info theory”“queueing theory” T/R Stochastic Network Theory extends to: Bursty Traffic Arbitrary Mobility Performance Optimization [Neely thesis 2003, Infocom 2005] [Neely, Modiano, Rohrs JSAC 2005] [Georgiadis, Neely, Tassiulas F&T 2006]

Part 2: Overview of Stochastic Network Optimization What Problems Can be Solved Today? Slotted time system t = {0, 1, 2, …} Q(t) = Queue State Vector (possibly multi-hop) S(t) = “Topology State” (random, chosen by environment) I(t) = “Control Action” or “Tranmission mode”, I(t) in I General “Link Transmission Rate Vector Function: Rate vector(t) = C(I(t), S(t)) General “Penalty/Reward vector”: Penalty vector(t) = x(I(t), S(t)) Queue Evolution: Q(t) Q(t+1) I(t), S(t) Q(t+1) = max[Q(t) – out(t), 0] + in(t) f() convex h i (x) convex

Part 2: Overview of Stochastic Network Optimization How is this solved? We have a general and extensive theory: Lyapunov Drift and Stability for Networks [Tassiulas & Ephremides TAC 1992, IT 1993] Drift for Joint Network Stability and Performance Optimization [Neely thesis 2003, Infocom 2005], [Georgiadis, Neely, Tassiulas F&T 2006] Virtual Queues [Neely Infocom 2005, IT 2006], [Georgiadis, Neely, Tassiulas F&T 2006] Auxiliary Variables and “Flow State” Queues [Neely, Modiano, Li Infocom 2005], [Georgiadis, Neely, Tassiulas F&T 06] Alternative Approaches: Downlink, Linear Utilities [Tsibonis, Georgiadis, Tassiulas Infocom 03] Flow Based, Static Channels [Cruz & Santhanam, Infocom 03], [Lin, Shroff CDC 04, Infocom 05] Fluid Model Analysis for Multi-Hop and General Utilities [Stolyar, Queueing Systems 05] --- “Primal-Dual” Alg. Infinite Backlog Assumption, 1-hop downlink [“Prop. Fair” Alg] [Agrawal, Subramanian, Allerton 02], [Kushner, Whiting Allert. 02] [Eryilmaz, Srikant Infocom 2005], [Liu, Chong, Shroff 03]

Part 2: Overview of Stochastic Network Optimization How is this solved? We have a general and extensive theory: Lyapunov Drift and Stability for Networks [Tassiulas & Ephremides TAC 1992, IT 1993] Drift for Joint Network Stability and Performance Optimization [Neely thesis 2003, Infocom 2005], [Georgiadis, Neely, Tassiulas F&T 2006] Virtual Queues [Neely Infocom 2005, IT 2006], [Georgiadis, Neely, Tassiulas F&T 2006] Auxiliary Variables and “Flow State” Queues [Neely, Modiano, Li Infocom 2005], [Georgiadis, Neely, Tassiulas F&T 06] Note: Our work is unique in that it: -Solves full problem on the actual queueing network of interest -Links very nicely to the previous Tassiulas drift framework -Gets Strongest Results, and Explicit performance-delay Tradeoffs [O(1/V) ; O(V)] peformance-delay for any network, any utility Question: Is this the optimal tradeoff?

The Basic Stability Theory for Networks:  Tassiulas & Ephremides [Trans. Autom. Control 1992] Multi-hop network with Random Packet Arrival Processes Link Scheduling according to “Feasible Activation Sets” Lyapunov drift for stability “Backpressure” Routing and Max-Weight Scheduling Gives Stability for any rate vector inside capacity region Does not require knowledge of traffic rates  Tassiulas & Ephremides [Trans. Inform. Theory 1993] Single-Hop Dynamic Channels “Opportunistic” scheduling (ie, “channel-aware”) Lyapunov drift for stability Max-Weight Algorithm does not require channel statistics or traffic arrival rates, gets stability whenever possible

Quick Description of “Backpressure Routing” and “Max-Weight Scheduling”: n n [Red data is destined for red node, Yellow data is destined for yellow node] No pre-defined routes! Optimal Link Activation Set determined by a max-weight rule “Which packet to send over a link” is determined by a differential backlog index (backpressure) The max-weight link activation can be NP hard for networks with interference, but is trivial (and distributed) for orthogonal links

An Abbreviated History of Lyapunov Drift for Network Stability: (for various computer networks and switching systems) Tassiulas & Ephremides (Backpressure, Max-Weight) [1992, 1993] Kumar & Meyn [1995] McKeown, Anantharam, Walrand [1996, 1999] Kahale & Wright [1997] Andrews, Kumaran, Ramanan, Stolyar, Whiting [2001] Leonardi, Mellia, Neri, Marsan [2001] Neely, Modiano, Rohrs [2003, 2005]  Extends to MANETs Performance Optimization (time varying channels) but with no queueing or stability constraints (infinite backlog assumption): R. Agrawal, V. Subramanian [2002] (“Proportionally Fair Alg”) Kushner, Whiting [2002] (“Proportionally Fair Alg”) Liu, Chong, Shroff [2003]

Corresponding Results for Static Networks (non-stochastic): These use static convex optimization theory to maximize network utility. Lagrange multipliers are “shadow prices.” There is either no queueing analysis, or approximate queueing analysis.  Wireline Networks, Fixed Route Selection, Flow Based  Kelly [1997]  Kelly, Maullou, Tan [1998]  Low, Lapsley [1999]  Wireless Networks, Fixed Route Selection, Flow Based Xiao, Johansson, Boyd [2001] Lee, Mazumdar, Shroff [2002] Julian, Chiang, O’Neill Boyd [2002] Chiang [2004, 2005]  Scheduling for Utility Optimization (static networks) Cruz & Santhanam [2003] (scheduling decisions are chosen over time) Lin & Shroff [2004, 2005] (scheduling decisions are chosen over time)

Work on Utility Optimization for Stochastic Networks: Wireless Downlink, Time Varying Channels, Infinite Data, No Queueing R. Agrawal, V. Subramanian [2002] (“Proportionally Fair Alg”) Kushner, Whiting [2002] (“Proportionally Fair Alg”) Liu, Chong, Shroff [2003] Joint Stability and Performance Optimization (Time Varying Channels): Tsibonis, Georgiadis, Tassiulas [2003]  Solves downlink with special structure, and with linear utilities Neely [2003, 2005] “Dual Method”  Solves the general problem! (multi-hop, stochastic, concave utilities)  Obtains explicit [O(1/V), O(V)] utility-delay tradeoff! Stolyar [2005] “Primal-Dual Method”  A different approach to the general problem, solves on a fluid model Eryilmaz & Srikant [2005] “Dual Method”  Downlink, infinite backlog, fluid model analysis Lee, Mazumdar, Shroff [2006]  Stochastic gradients, flow based, infinite backlog

λ1λ1 λ2λ2 λ1λ1 λ2λ2 Input Rate Output Rate 1 2 S 1 (t) {ON, OFF} S 2 (t) {ON, OFF} “Proportionally Fair” algorithm (designed for infinite backlog) [Example from Neely, Modiano, Li Infocom 2005, TON 2008]

λ1λ1 λ2λ2 λ1λ1 λ2λ2 Input Rate Output Rate 1 2 S 1 (t) {ON, OFF} S 2 (t) {ON, OFF} “Max-Weight” algorithm (designed only to achieve stability, Tass. & Eph. 93) [Example from Neely, Modiano, Li Infocom 2005, TON 2008]

λ1λ1 λ2λ2 λ1λ1 λ2λ2 Input Rate Output Rate 1 2 S 1 (t) {ON, OFF} S 2 (t) {ON, OFF} “An Ideal” algorithm [Neely 2003, 2005] [Example from Neely, Modiano, Li Infocom 2005, TON 2008]

Work on Utility Optimization for Stochastic Networks: Wireless Downlink, Time Varying Channels, Infinite Data, No Queueing R. Agrawal, V. Subramanian [2002] (“Proportionally Fair Alg”) Kushner, Whiting [2002] (“Proportionally Fair Alg”) Liu, Chong, Shroff [2003] Joint Stability and Performance Optimization (Time Varying Channels): Tsibonis, Georgiadis, Tassiulas [2003]  Solves downlink with special structure, and with linear utilities Neely [2003, 2005] “Dual Method”  Solves the general problem! (multi-hop, stochastic, concave utilities)  Obtains explicit [O(1/V), O(V)] utility-delay tradeoff! Stolyar [2005] “Primal-Dual Method”  A different approach to the general problem, solves on a fluid model Eryilmaz & Srikant [2005] “Dual Method”  Downlink, infinite backlog, fluid model analysis Lee, Mazumdar, Shroff [2006]  Stochastic gradients, flow based, infinite backlog

General Theory in F&T book: (Georgiadis, Neely, Tassiulas F&T 2006) Unifies Lyapunov drift and Performance Optimization with a very simple modification: Every slot, observe queues and channels, and take a greedy action to minimize “Drift + Penalty”: Δ(Q(t)) + VE{Φ(t)|Q(t)} General mobile network DriftDrift Penalty

General Theory in F&T book: (Georgiadis, Neely, Tassiulas F&T 2006) Unifies Lyapunov drift and Performance Optimization with a very simple modification: Every slot, observe queues and channels, and take a greedy action to minimize “Drift + Penalty”: Δ(Q(t)) + VE{Φ(t)|Q(t)} General mobile network Theorem (Neely 2003, 2005): | E{Φ} - E{Φ*} | < O(1/V) E{Queue Size} < O(V)

Can Solve Problems of the Type [F&T 2006]: where x is a vector of general “network penalties/rewards” (throughput, energy, reliability, etc., and penalties are arbitrary functions of control decisions) and f() and h() are convex functions. Joint Lyap. Drift and Performance Opt. Virtual Queues Auxiliary Variables Important and New Techniques of Stochastic Network Optimization [F&T 2006]:

Flow control reservoir raw data -flow state queues -aux. vars. Structure of the Algorithms: Flow Control (separable at each source) Modified Max-Weight/Backpressure Scheduling and Routing Flow Control

Application to Mobile Networks with Unreliable Channels = Stationary Node = Locally Mobile Node = Fully Mobile Node = Sink

= Stationary Node = Locally Mobile Node = Fully Mobile Node = Sink Application to Mobile Networks with Unreliable Channels

Avg. Power Avg. Delay Performance-Delay Tradeoff: [O(1/V), O(V)] Average Power Versus Delay (Fix a set of transmission rates for each node) [Neely, Urgaonkar 2006, 2009]

Avg. Power Avg. Delay Average Power Versus Delay (Fix a set of transmission rates for each node) [Neely, Urgaonkar 2006, 2009] Performance-Delay Tradeoff: [O(1/V), O(V)]

Avg. Power Avg. Delay Performance-Delay Tradeoff: [O(1/V), O(V)] Average Power Versus Delay (Fix a set of transmission rates for each node) [Neely, Urgaonkar 2006, 2009]

Delay Theory For Networks: 1. Network Calculus for Deterministic Networks [Rene Cruz, Trans. Inf. Theory 1991] (no performance-delay tradeoff for deterministic networks) 2. Achievable [O(1/V); O(V)] performance-delay tradeoff for stochastic networks (random traffic and/or channels) [Neely 2003, 2005][Georgiadis, Tassiulas, Neely F&T 2006] 3. Energy-Delay square root Tradeoff for a single queue [Berry-Gallager Trans. Information Theory 2002] 4.For general network, optimal tradeoff is either a square root law or a logarithm law! [Neely IEEE JSAC 2006, IEEE Transactions on Information Thoery 2007]

Optimal Energy and Delay Tradeoffs for Multi-User Wireless Downlinks Michael J. Neely University of Southern California http://www-rcf.usc.edu/~mjneely/ Infocom 2006, Barcelona, Spain *Sponsored by NSF OCE Grant 0520324 1 2 N Avg. Delay Avg. Power

Assumptions: 1)Random Arrivals A(t) i.i.d. over slots. Rate vector  (bits/slot) 2) Random Channel states S(t) i.i.d. over slots. 3) Transmission Rate Function P(t) --- Power allocation during slot t (P(t)  ) S(t) --- Channel state during slot t t 0 1 2 3 … Time slotted system (t {0, 1, 2, …}) rate  i power P  (P(t), S(t)) Good Med Bad 1 2 N

rate  i power P Good Med Bad 1 2 N Control: Allocate Power (P(t)  ) in Reaction to Current Channel State And Current Queue Backlogs. Goal: Stabilize with Minimum Average Power while also Maintaining Low Average Delay.

rate  i power P Good Med Bad 1 2 N Control: Allocate Power (P(t)  ) in Reaction to Current Channel State And Current Queue Backlogs. Goal: Stabilize with Minimum Average Power while also Maintaining Low Average Delay. [ Avg. Power and Avg. Delay are Competing Objectives! ] What is the Fundamental Energy-Delay Tradeoff?

1 2 N Our Previous Work on Minimum Power Scheduling with Delay Tradeoffs [Neely Infocom 2005]: U N (t)U 1 (t) U 2 (t) V P V Av. Delay O(1/V) O(V) P* ( P* = Min Av. Power for Stability )

1 2 N Our Previous Work on Minimum Power Scheduling with Delay Tradeoffs [Neely Infocom 2005]: U N (t)U 1 (t) U 2 (t) ( P* = Min Av. Power for Stability ) V P V Av. Delay O(1/V) O(V) P*

1 2 N Our Previous Work on Minimum Power Scheduling with Delay Tradeoffs [Neely Infocom 2005]: U N (t)U 1 (t) U 2 (t) Analysis: Use theory of Performance Optimal Lyapunov Scheduling: -Neely, Modiano 2003, 2005 -Georgiadis, Neely, Tassiulas [F&T 2006, NOW Publishers] Achieves: [O(1/V), O(V)] energy-delay tradeoff V P V Av. Delay O(1/V) O(V) P*

1 2 N Our Previous Work on Minimum Power Scheduling with Delay Tradeoffs [Neely Infocom 2005]: V P V Av. Delay O(1/V) O(V) U N (t)U 1 (t) U 2 (t) Analysis: Use theory of Performance Optimal Lyapunov Scheduling: -Neely, Modiano 2003, 2005 -Georgiadis, Neely, Tassiulas [F&T 2006, NOW Publishers] Achieves: [O(1/V), O(V)] energy-delay tradeoff P*

V P V Av. Delay O(1/V)  ( V ) P* The Fundamental Berry-Gallager Bound for Energy-Delay Tradeoffs in a Single Wireless Downlink: A(t)  (t) =  (P(t), S(t)) Av. Delay >=  ( V ) [Berry, Gallager 2002] Approach Achievability via Technique of Buffer Partitioning.

Precedents for Energy and Delay Optimization for Single Wireless Links: -Berry and Gallager [2002] ( Fundamental Square Root Law ) -Uysal-Biyikoglu, Prabhakar, El Gamal [2002] -Khojastepour and Sabharwal [2004] ( “Lazy Scheduling” and Filter Theory for Static Links ) -Fu, Modiano, Tsitsiklis [2003] -Goyal, Kumar, Sharma [2003] -Zafer and Modiano [2005] ( Dynamic Programming, Markov Decision Theory )

Challenging to extend optimal delay results for stochastic systems beyond a single queue because… 1)Parameter Explosion: (cannot practically measure) Number of channel state vectors S grows geometrically with number of links N. Markov Decision Theory and Dynamic Programming requires knowledge of:  S = Pr[ S(t) = S] (for each channel state S ). 2) State Space Explosion: (cannot practically implement) Number of Queueing State Vectors U grows geometrically.

Idea: Combine Techniques of Buffer Partitioning and Performance Optimal Lyapunov Scheduling. 1 2 N V P V Av. Delay P* Goals: 1)Establish the fundamental energy-delay curve for multi-user systems (extend Berry-Gallager to this case). 2)Design a dynamic algorithm to achieve optimal energy-delay tradeoffs. (Must overcome the complexity explosion problem).

Specifically: Define a general power cost metric h( P ) : 1 2 N Define average power cost: Define: h* = Min. avg. power cost for network stability (Push h arbitrarily close to h*, with optimal delay tradeoff…)

Theorem 1: (Characterize h*) Assume . The min average power cost h* is given as the solution to: Define  ( ) = min. avg. power cost h* above. Corollary: For each, there is a stationary randomized alg. such that: 

The Fundamental Energy-Delay Tradeoff:    mild admissibility assumptions Theorem 2 (Multi-User Berry-Gallager Bound): If Then if avg. cost satisfies: We necessarily have: 1 2 N V h V Av. Delay h*  ( V ) O(1/V)

Achieving Optimal Tradeoffs via Buffer Partitioning… Recall the Berry-Gallager threshold algorithm for single queues:  (t) =  (P(t), S(t)) U(t)  max Q U Q drift    L R [Requires full knowledge of channel probs  S ]

Achieving Optimal Tradeoffs via Buffer Partitioning… Recall the Berry-Gallager threshold algorithm for single queues:  (t) =  (P(t), S(t)) U(t)  max Q U Q drift    L R

Let’s Consider Multi-Dimensional Buffer Partitioning:         Q U1U1 U2U2 Case N=2 1 2 N Q

Let’s Consider Multi-Dimensional Buffer Partitioning:         Q U1U1 U2U2 Case N=2 Q

Let’s Consider Multi-Dimensional Buffer Partitioning:         Q U1U1 U2U2 Case N=2 (not implementable) Q

Analysis of the Threshold Algorithm: (exchanging sums over the 2 N regions yields…)  i L (t) = Pr[U i (t) <Q]  i R (t) = 1 -  i L (t)

An Online Algorithm for Optimal Energy-Delay Tradeoffs: 1 2 N Define the bi-modal Lyapunov Function: UiUi Q Designing “gravity” into the system:

An Online Algorithm for Optimal Energy-Delay Tradeoffs: 1 2 N Define the bi-modal Lyapunov Function: UiUi Q Designing “gravity” into the system: “Usually” creates proper drift direction…

1 2 N *Key inequality that holds with equality for the stationary threshold algorithm. Need to strengthen the drift guarantees… Want to also ensure for all i {1, 2, …, N}

Need to strengthen the drift guarantees… Want to also ensure for all i {1, 2, …, N} Use Virtual Queue Concept from [Neely Infocom 2005]: X i (t) A i (t) +  1 i R (t)  i (t) +  1 i L (t) indicator functions X i (t+1) = max[X i (t) -  i (t) -  1 i L (t), 0] + A i (t) +  1 i R (t)

Need to strengthen the drift guarantees… Want to also ensure for all i {1, 2, …, N} Use Virtual Queue Concept from [Neely Infocom 2005]: X i (t) A i (t) +  1 i R (t)  i (t) +  1 i L (t) indicator functions X i (t) Stable  i +  1 i L > i +  1 i R

To Stabilize Virtual Queues X i (t) and Actual Queues U i (t):

The Tradeoff Optimal Control Algorithm (TOCA): 1) Every slot t, observe channel state S(t) and queue backlogs U(t), X(t). Allocate power P(t) = P, where P solves: 2) Transmit with rate  i (t) =  i (P(t), S(t)). 3) Update the Virtual Queues X i (t): X i (t+1) = max[X i (t) -  i (t) -  1 i L (t), 0] + A i (t) +  1 i R (t)

Theorem 3 (TOCA Performance): For suitable , Q: 1 2 N V hAv. Delay  ( ) V O(1/V)  ( V )

Comparing TOCA (statistics unaware) to Berry-Gallager (statistics aware) for a single queue

Beyond the Berry-Gallager Bound: Logarithmic delay! If the Minimum Energy function  ( ) is peicewise linear (not strictly concave), then under suitable , Q, TOCA yields:  ( ) (shown in 1 dimension)

Further, logarithmic delay in this scenario is optimal! Simple One Queue Example: P(t)  ={0, 1} Watt. Two Equally Likely Channel States (GOOD, BAD): U(t)  (t)=  (P(t),S(t))  ( ) Can show that logarithmic delay is necessary and achievable!

Conclusions: 1 2 N V hAv. Delay  ( ) V O(1/V)  ( V ) -Extend Berry-Gallager Square Root Law to Multi-User Systems. -Novel Lyapunov Technique for Achieving Optimal Energy-Delay Tradeoffs. -Overcome the Complexity Explosion Problem. -Channel Statistics, Traffic Rates not Required. -Superior Tradeoff via a Logarithmic Delay Law in exceptional (piecewise linear) cases.

Stochastic Network Optimization and the Theory of Network Throughput, Energy, and Delay Michael J. Neely University of Southern California

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Stochastic Network Optimization and the Theory of Network Throughput, Energy, and Delay Michael J. Neely University of Southern California

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