EE360: Lecture 11 Outline Capacity of Ad Hoc Nets Announcements HW 1 due today at 5pm Project progress reports due next Friday Introduction to network capacity Capacity definitions Shannon capacity of ad-hoc networks Time-division networks Scaling Laws Ad-Hoc Network Capacity Network capacity in general refers to how much data a network can carry Multiple definitions Shannon capacity Total network throughput (vs. delay) User capacity (bps/Hz/user or total no. of users) Data rate vs. mobility … “Google” Network Capacity Shannon Capacity of Networks The Shangri-La of Information Theory Much progress in finding the capacity limits of wireless single and multiuser channels Limited understanding about the capacity limits of wireless networks, even for simple models System assumptions such as constrained energy and delay may require new capacity definitions Is this elusive goal the right thing to pursue? Shangri-La is synonymous with any earthly paradise; a permanently happy land, isolated from the outside world Shannon Network Capacity: What is it? n(n-1)-dimensional region R34 Upper Bound Rates betweenall node pairs Upper/lower bounds Lower bounds achievable Lower Bound Upper bounds hard R12 Other axes Capacity Delay Upper Bound Energy and delay Lower Bound Energy Network Capacity Results Multiple access channel (MAC) Broadcast channel Relay channel upper/lower bounds Interference channel Scaling laws Achievable rates for small networks Relay Channel Capacity Results generally unknown or nonintuitive Gaussian relay channel (Cover,El Gamal’79) N2 Source + Destination + Broadcast or cooperative MAC N1 strategy (Markov source coding) Parallel Relay Channel (Schein,Gallager’00,Kramer’03) Upper bounds on capacity, capacity unknown. N2 + N3 Source + Destination N1 + Cooperative MAC or Distributed Source Coding Shannon Capacity of Interference Channel Unknown in general X1 Gd Y1 Gi Gi X2 Y2 Gd Known for “strong” interference Gain of interference path as strong as direct path gain Capacity is that of the MAC channel Both RXs decode both messages Clearly, information-theoretic formulas for capacity of general networks are intractable Shannon Capacity: Mutual Information Formula For n nodes, p(x(1),…,x(n)) s.t. (Cover/Thomas) Relay transmissions S Xi Sc Yk Xk Yj Rate flow across cutsets bounded by conditional MI Problems with this Formula Generally doesn’t give concrete formulas Not clear how to achieve these rates What is optimal encoding/decoding strategy? Capacity region is n(n-1) dimensional For random topologies, capacity is random Ad Hoc Network Achievable Rate Regions All achievable rate vectors between nodes Lower bounds Shannon capacity An n(n-1) dimensional convex polyhedron Each dimension defines (net) rate from one node to each of the others Time-division strategy Link rates adapt to link SINR 3 Optimal MAC via centralized scheduling Optimal routing 2 5 Yields performance bounds Evaluate existing protocols 1 4 Develop new protocols Rate Matrix Transmission scheme at time t for n users (snapshot) Rows represent original data source Negative entries represent bits to send or forward Positive entries represent bits received (data rate) Link rates dictated by link capacity given SIR (variable rate) Multihop routing and power control increase set of matrices Transmission Scheme Rate Matrix 1 Data from 1, rate 10 2 3 Data from 2, rate 20 4 Time Division Time division of two schemes is a linear combination of their rate matrices. Example: 50% of time under scheme A and 50% of time under scheme B has rate matrix: Scheme A Scheme B 50/50 Time Division User 1 sends 5 bps/Hz to User 2 User 2 sends 10 bps/Hz to User 3 and 10 bps/Hz to User 4 User 4 sends 5 bps/Hz to User 3 Achievable Rates Achievable rate Capacity region vectors achieved is convex hull of by time division all rate matrices A matrix R belongs to the capacity region if there are rate matrices R1, R2, R3 ,…, Rn such that Linear programming problem: Need clever techniques to reduce complexity Power control, fading, etc., easily incorporated Region boundary achieved with optimal routing Example: Six Node Network Capacity region is 30-dimensional Capacity Region Slice (6 Node Network) (a): Single hop, no simultaneous transmissions. (b): Multihop, no simultaneous transmissions. (c): Multihop, simultaneous transmissions. (d): Adding power control (e): Successive interference Multiple SIC cancellation, no power Spatial hops reuse control. Extensions: - Capacity vs. network size - Capacity vs. topology - Fading and mobility - Multihop cellular Optimal Routing The point is achieved by the following time division: Route Diversity Low Background Rate Single hop, one active user per timeslot Spatial Separation Multihop, one active gain user per timeslot Multihop, multiple users per timeslot Multihop Multihop, multiple gain users per timeslot 3 level power control Rij=.01, ij 23,45 Single hop with 1 active user cannot support this background rate. Other regions shrink a bit, but not by aggregate rate (.28=.0128) High Background Rate Multihop, one active user per timeslot Rij=.01, ij 23,45 Multihop, multiple users per timeslot Multihop, multiple users per timeslot 3 level power control Rij=.1, ij 23,45 Fading and Mobility increase Capacity Gain matrix alternates between N fading states (a): No routing, no simultaneous transmissions. (b): Routing, no simultaneous transmissions. (c): Routing, simultaneous transmissions. (d): Adding power control. (e): Successive interference cancellation, no power control. In a similar way, mobility also increases capacity General Trends Relaying greatly increases capacity Multiple users per slot increases capacity, but rate adaptation limits these gains Rate adaptation mostly eliminates the need for power control Throughput trends for large networks still unknown Special case: Multihop Cellular System All nodes communicate with a common node (BS) BS Impact of resource allocation on throughput Multihop routing Time slot allocation Power control Numerical Results Users transmit to BS at maximum common rate R BS is centrally located, other nodes uniformly distributed Single hop, one active user per slot R=0.023. Multihop, one active user per slot R=0.68. Multihop routing, many active users per slot R=0.75. Multihop routing, many active users per slot, heuristic power control algorithm R=0.77. Similar trends as for the general network structure Other Open Questions • Capacity of time-varying links (with/without feedback) Yi-1 Xi Yi Tx p(yi,si|x i,si-1) Rx Si-1 Si D • Capacity of basic network building blocks • Capacity of large dynamic networks Capacity for Large Networks (Gupta/Kumar’00) Make some simplifications and ask for less Each node has only a single destination All nodes create traffic for their desired destination at a uniform rate l Capacity (throughput) is maximum l that can be supported by the network (1 dimensional) Throughput of random networks Network topology/packet destinations random. Throughput l is random: characterized by its distribution as a function of network size n. Find scaling laws for C(n)=l as n . Extensions Fixed network topologies (Gupta/Kumar’01) Similar throughput bounds as random networks Mobility in the network (Grossglauser/Tse’01) Mobiles pass message to neighboring nodes, eventually neighbor gets close to destination and forwards message Per-node throughput constant, aggregate throughput of order n, delay of order n. S D Throughput/delay tradeoffs Piecewise linear model for throughput-delay tradeoff (ElGamal et. al’04, Toumpis/Goldsmith’04) Finite delay requires throughput penalty. Achievable rates with multiuser coding/decoding (GK’03) Per-node throughput (bit-meters/sec) constant, aggregate infinite. Rajiv will provide more details Some capacity questions How to parameterize the region Power/bandwidth Channel models and CSI Outage probability Security/robustness Defining capacity in terms of asymptotically small error and infinite delay has been highly enabling Has also been limiting Cause of unconsummated union in networks and IT What is the alternative? Limitations in theory of MANETs today Wireless Wireless Information Network Theory Theory B. Hajek and A. Ephremides, “Information theory and communications networks: An unconsummated union,” IEEE Trans. Inf. Theory, Oct. 1998. Optimization Theory Shannon capacity pessimistic for wireless channels and intractable for large networks – Large body of wireless (and wired) network theory that is ad-hoc, lacks a basis in fundamentals, and lacks an objective success criteria. – Little cross-disciplinary work spanning these fields – Optimization techniques applied to given network models, which rarely take into account fundamental network capacity or dynamics Consummating Unions Wireless Wireless Information Network Theory Theory Menage a Trois Optimization Theory When capacity is not the only metric, a new theory is needed to deal with nonasymptopia (i.e. delay, random traffic) and application requirements Shannon theory generally breaks down when delay, error, or user/traffic dynamics must be considered Fundamental limits are needed outside asymptotic regimes Optimization provides the missing link to address these issues MANET Metrics New Paradigms Constraints for Upper Capacity and Fundamental Limits Bounds Capacity Delay Upper Models and Bound Dynamics Layerless Lower Dynamic Bound Networks Degrees of Energy Freedom Generalized Network Inferface Source Coding and Network Utility Capacity Delay (C*,D*,E*) Utility=U(C,D,E) Energy/SNR Metrics Problem Definition Fundamental Limits New MANET Theory of Wireless Systems Application Metrics Is a capacity region all we need to design networks? Yes, if the application and network design can be decoupled Application metric: f(C,D,E): (C*,D*,E*)=arg max f(C,D,E) Capacity Delay (C*,D*,E*) Energy Summary Ad-hoc networks provide a flexible network infrastructure for many emerging applications Recent advances in communication techniques should be incorporated into ad-hoc network design Design issues traverse all layers of the protocol stack, and cross layer designs are needed Protocol design in one layer can have unexpected interactions with protocols at other layers.
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