Resource Allocation in Wireless Networks by nikeborome

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									           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=.0128)
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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