ISIT - Nice Nice ISIT - June 27, 2007 June 27, 2007 Competition and Competition and Collaboration in Wireless Collaboration in Wireless Networks Networks Vince Poor Vince Poor (poor@princeton poor@princeton. .edu edu) ) ( Competition & Collaboration in Wireless Networks
Research Trends in Wireless Nets Nets Research Trends in Wireless • The Past Two Decades: Key Developments at the Link Level – MIMO – MUD – Turbo • Today: An Increased Focus on Interactions Among Nodes – Competition • Cognitive radio • Information theoretic security • Game theoretic modeling, analysis & design ← – Collaboration • Network coding • Cooperative transmission & relaying • Multi-hop transmission & coalition games • Collaborative beam-forming • Collaborative inference ← Competition & Collaboration in Wireless Networks
Today’ ’s Talk - Two Parts s Talk - Two Parts Today • Energy Games: Competition in Multiple Access Communication Networks • Distributed Inference: Collaboration in Wireless Sensor Networks (WSNs) Competition & Collaboration in Wireless Networks
ENERGY GAMES: ENERGY GAMES: COMPETITION IN COMPETITION IN THE MAC THE MAC [Joint work with Farhad Meshkati Farhad Meshkati, , Stuart Schwartz Stuart Schwartz, et al.] , et al.] [Joint work with Competition & Collaboration in Wireless Networks
Energy Games Energy Games • Terminals transmit to an access T point via a multiple-access T channel. • Users are like players in a game, competing for resources to AP T transmit their data to the AP. T • The action of each users affects the others. • We can model this as a competitive game, with payoff measured in bits-per-joule. Competition in MA Communication Networks
Game Theoretic Framework Game Theoretic Framework [Goodman, Mandayam Mandayam,Yates, ,Yates, et et al.] al.] [Goodman, Game: G = [{1 ,…,K }, { A k }, { u k }] K : total number of terminals A k : set of strategies for terminal k u k : utility function for terminal k transmit power = T k throughput bits � � u k = utility = � � p k Joule � � T k = R k f( γ k ) , where f( γ k ) is the frame success rate, and γ k is the received SINR of user k . Competition in MA Communication Networks
An Uplink Game An Uplink Game • For a fixed linear MUD at the uplink receiver, each user selects its transmit power to maximize its own utility. Th’m [w/ Mandayam, T-COM 05] : f sigmoidal ⇒ Nash equilibrium • (i.e., no user can unilaterally improve its utility) is reached when each user chooses a transmit power that achieves γ *: f( γ *) = γ * f ′ ( γ *) • I.e., Nash equilibrium (NE) requires SINR balancing. Competition in MA Communication Networks
Remarks Remarks • The NE is unique, and can be reached iteratively as the unique fixed point of a nonlinear map. • The NE as an Analytical Tool: – We can use the NE to examine the effects on energy efficiency of various network design choices. – E.g., we can compare receiver choices: the matched filter, (zero-forcing) decorrelator, and MMSE detector. Competition in MA Communication Networks
Flat SIMO Model Flat SIMO Model { : { h h k } ,p } Channel Gains: Channel Gains k,p h 1 h 1 , , 1 1 User 1: 010… r 1 ( t ) User 2: 110… r 2 ( t ) h 1 h 1 ,P ,P h 2 h 2 ,P ,P ... ... ... ... h K h K, , 1 1 h K h K,P ,P User K: 011… r P ( t ) Competition in MA Communication Networks
Nash Equilibrium Nash Equilibrium Utility vs vs. Load (Large-System Limit) . Load (Large-System Limit) Utility m = # receive antennas • Random CDMA: K terminals; spreading gain N • Load: α = K/N (i.e., the number of users per dimension) • Large-system limit: K, N →∞ , with α fixed. Competition in MA Communication Networks
Social Optimality Social Optimality • The Pareto (or socially) optimal solution, chooses the transmit power so that no user’s utility can be improved without decreasing that of another. • The Pareto solution is generally hard to find. • The Nash equilibrium solution not generally Pareto optimal. • But, it’s close. Competition in MA Communication Networks
Example: Nash & Pareto Optima Example: Nash & Pareto Optima Utility vs. Load Competition in MA Communication Networks
Effects of Delay QoS QoS Effects of Delay [w/ FM & SC [w/ FM & SC, submitted to , submitted to Trans. IT Trans. IT ] ] • For some traffic, delay is a key element of service quality. • Delay model (ARQ): – X represents the number of transmissions needed for a given packet to be received without error, so that: P(X=m) = f( γ ) [1 - f( γ ) ] m-1 , m = 0, 1, … – We can represent a delay requirement as a pair (D, β ) : P(X ≤ D) ≥ β ⇔ γ ≥ γ ’ – Thus, we have a constrained game , with γ k ≥ γ k ’. Competition in MA Communication Networks
NE for Multiple Delay Classes NE for Multiple Delay Classes • Traffic is typically heterogeneous ---------------------- ---------------------- with multiple delay classes. ------------------- ------------------- Utility, u • A given delay class c will have its own SINR constraint: γ c ’ • At NE all users in class c will SINR-balance to max{ γ * , γ c ’ }. γ ’ γ * SINR, γ • Tight delay constraints on one class can affect the energy efficiencies of all traffic due to increased interference levels. Competition in MA Communication Networks
2-Class Example: Utility Loss 2-Class Example: Utility Loss α = 0.1 α = 0.9 • RCDMA in the large-system limit: K, N →∞ , with α = K/N fixed. • Class A: (D A , β A ) = (1, 0.99) • Class B: (D B , β B ) = (3, 0.90) Competition in MA Communication Networks
Finite Backlog Case Finite Backlog Case [w/ R. Balan, T-COM , to appear.] Poisson packet arrivals • FIFO’ed packets transmitted via ARQ • QoS: (source rate, ave. delay) • Translates into a lower bound on SINR • Constrained Nash game (on transmit power & rate) • Leads to “size” of a user quantifying the resources required to deliver QoS. • NE exists only when the sum of the users “sizes” is < 1. • Competition in MA Communication Networks
Utility vs vs. Delay . Delay Utility Utility is normalized by B × SNR, and the normalized delay is D × B . The combined “size” of the other users is 0.2. Competition in MA Communication Networks
Enhancements Enhancements • Nonlinear MUD (ML, MAP, PIC, etc.): Results apply to nonlinear MUD for RCDMA in the large system limit. [w/ D. Guo, FM & SC; T-WC, to appear] • Multicarrier CDMA: Actions include choice of a carrier . [w/ M. Chiang, FM & SC; JSAC’ 06] • UWB: Rich scatter. [w/ G. Bacci, M. Luise & A. Tulino: JSTSP , to appear] • Adaptive Modulation: Actions include choice of a modulation index [w/ A. Goldsmith, FM & SC: JSAC’07 ] or waveform [w/ S. Buzzi; EWC’07 ]. Competition in MA Communication Networks
COLLABORATIVE COLLABORATIVE INFERENCE IN INFERENCE IN WSNs WSNs [Joint work with Sanj Kulkarni Sanj Kulkarni, , Joel Joel Predd Predd, et al.] , et al.] [Joint work with Competition & Collaboration in Wireless Networks
Sensor Field Sensor Field Collaborative Inference
Motivation Motivation • Salient features of WSNs: – The primary application is inference – Information at different terminals is often correlated – Energy is often severely limited • Collaborative inference: – Sensors work together to make inferences, while conserving resources (i.e., “bandwidth & batteries”) – Here, we’ll examine collaborative learning Collaborative Inference
Classical (Supervised) Learning Classical (Supervised) Learning • Input space X X = R d ; output space Y Y = R • (X,Y) is an X X × Y Y -valued r.v. with (X,Y) ~P XY • Design f: X X → Y Y to predict outputs from inputs and minimize expected loss; e.g., E {| f(X)-Y | 2 } • P XY is unknown • So, construct f from examples: Collaborative Inference
A Model for Dist Dist’ ’d d Learning in Learning in WSNs WSNs A Model for “A distributed sampling device A distributed sampling device with a wireless interface with a wireless interface” ” “ • Sensor i measures . S 10 S 2 S 6 S 8 S 11 S 4 S 1 S 9 S 7 S 3 S 5 • This division defines a • This division defines a topology topology, , which in turn which in turn shapes the shapes the nature of collaboration. nature of collaboration . Collaborative Inference
A Centralized Approach A Centralized Approach • Sensor i sends to a centralized processor • “Learn” using (reproducing) kernel methods: – For a positive semi-definite kernel K(,) : • Assumption: energy and bandwidth constraints preclude the sensors from sending for centralized processing. Collaborative Inference
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