Maximizing the Functional Lifetime of Sensor Networks

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					     Maximizing the Functional Lifetime of
              Sensor Networks

              Arvind Giridhar, P.R. Kumar
             Coordinated Science Laboratory




4/28/2009          CSL, University of Illinois   1
            Data Gathering Sensor Networks

 Network consisting of multiple sensors,
  one sink

 Data collection:
     – Each sensor has data to be sent to sink
       (bi bits for sensor i)
     – Periodically repeated


 Limited energy:
     – Sensor i has energy level Ei


 How to route data to maximize network
  lifetime?

4/28/2009                  CSL, University of Illinois   2
                         Related Work

 Tassiulas et al (2001)
     – Maximum lifetime routing
     – Derived equivalent linear programming formulation
     – Developed distributed algorithms to converge to optimal solution

 Sankar and Liu (2004), Madan and Lall (2004) consider similar
  problems

 Bharadwaj, Chandrakasan (2001)
     – Upper bounds on lifetime based on total energy consumption

 Our contribution: Consider simple regular networks
     – Analytically solve/provide sharp bounds, using properties of cost
       function



4/28/2009                   CSL, University of Illinois                    3
                        Network Model

 Model wireless network as complete
  graph

 Any node can transmit to any other
  node….

 ….but at a cost


 Cost function for communication
                                     Energy to transmit = bf (d(i, j))
      b bits from node i to node j  
                                     Energy to receive = bfR


4/28/2009                  CSL, University of Illinois                    4
                Model of Information bits

 Information bits modeled as fluid
     – Incompressible and infinitely divisible
     – Node can transmit arbitrary non-negative quantity of information to
       any other node


                               2        2
                    1                                    2
                          b1
                                    b1  1  2



 A routing flow is a set of non-negative reals {ij :1  i  n, 0  j  n}
 denoting quantities of information transmitted between every
ordered pair of nodes


4/28/2009                   CSL, University of Illinois                      5
                                  Flow Routing

                        Network                          Energy constraints
    Given                                          
             {(x1 , y1 ),...,(xn , yn )}           {E1 , E2 ,..., En }
                            Data Collection Task
                        +
                                 {b1 ,b2 ,...,bn }

 We seek flows which route all the bits to the sink while
  respecting the energy constraints
              n                        n

 That is,       ij   f (dij )   ki fR  Ei , for each i  Energy constraints
             j0                      k 1
              n            n

               
                   ij            ki     nbi , for each i  Flow conservation
             j0          k 1

             ij  0, for each i, j.

4/28/2009                               CSL, University of Illinois                  6
                      Functional Lifetime

 Let  be a flow for task {b1,b2 ,...,bn }

 The functional lifetime of the network is the maximum number of
  repetitions L of the flow  which is feasible given limited node
  energies {E1, E2 ,..., En }

     L repetitions of task {b1,b2 ,...,bn }  task {Lb1, Lb2 ,..., Lbn }
     L repetitions of flow   flow L
     L can be non-integral


We seek the routing flow which maximizes the functional lifetime



4/28/2009                    CSL, University of Illinois                   7
            No Direct Dependence on Time

 Functional lifetime is measured in number of “rounds,” not
  number of time units
     – Serves as metric of cost-to-lifetime for data collection task



 Actual lifetime (in time units) will thus depend on the rate of data
  generation or sampling of the environment

 Functional lifetime does not depend on time to transmit

 Thus, interference does not appear in formulation
     – Transmissions can be staggered in time so that no two transmitters
       ever interfere


4/28/2009                    CSL, University of Illinois                8
                   Linear Program Formulation

    Min z                                                    (1)
    subject to:
     n                          n

          ij   f (dij )   ki f R  Ei z, for each i
    j0                        k 1                                   z is the maximum
     n              n                                                fraction of node energy
       ij            ki     nbi , for each i                   consumed in routing
    j0            k 1

    ij  0, for each i, j.

    LEMMA: The functional lifetime is the inverse of the
    optimal value z* in the linear program (1). The corresponding
    flow {ij:1 i,j  n} scaled by 1/ z* achieves the optimum.

4/28/2009                              CSL, University of Illinois                       9
                            Cost Function

 Consider following cost function:

            f (x)  Et x e x
     2  path loss exponent
     0  dispersion coefficient

 Inverse of distance-attenuation function of signal power over
wireless medium


                   Assumption: fR  f (dmin )(1  0.5 1 )

                        For =3, require fR  0.75 f1


4/28/2009                        CSL, University of Illinois      10
                      Network Topology


 Linear regular topology

      – Regularly spaced sensors on                         E1 E2 E3   
                                                                          En
        a line
                                                    0 1 2 3            
                                                                          n
      – Sink node at one corner                       b1 b2 b3         
                                                                          bn

      – Inter-node distance d

      – Define fi : f (id)




4/28/2009                     CSL, University of Illinois                       11
                            “Near-Far Flow”

 Near-far flow is a flow for which:

 1. Each node transmits only to
    nearest node towards sink, and
    to the sink
     • For each node i, ij  0 for 0<j  i
                         *




 2. Fraction of energy consumed
       same for all nodes
        1 *                     1 *                                  1 *
           (10 f1  21 fR ) 
                      *
                                   (21 f1  20 f2  32 fR )  L 
                                              *        *
                                                                        (nn1 f1  n0 fn )
                                                                                     *

        E1                      E2                                   En
                      
       Tx En. Rx. En.
4/28/2009                        CSL, University of Illinois                        12
                   Uniqueness of Near-far Flow

     The near-far flow is solution to following linear equations

         10 f1  21 fR  E1z*
          *        *


20 f2  21 f1  32 fR  E2 z*
 *        *        *


                             L
          n0 fn  nn 1 f1  En z*
           *        *
                                                   Equations have unique
                                                    solution….
                      10  21  b1
                       *     *


                20  21  32  b2
                 *     *     *
                                                   …but solutions may not be non-
                             L                      negative

                  n0  nn 1  bn
                    *     *
                                                   Thus, near-far flow exists only if
                                                                 ii 1, i*0  0,1  i  n
                                                                  *




    4/28/2009                      CSL, University of Illinois                           13
                              Main Result

THEOREM: Consider the linear regular network.
1. The functional lifetime is upper bounded as follows

                     n
                         Ei n
                                                 n     f1 
                      f gi                   1  f 
                                             j i 1 
                     i 1 i                               j
              Tf                      gin 
                                           &
                                           & n1
                      n
                                                      fR 
                     b g     n
                            i i               1  f 
                     i 1                      j i     j 

              
2. If a near-far flow exists, then the above upper bound is achievable


LEMMA: If bifi/Ei and Ei are non-decreasing in i, then a near-far flow
  exists
      – Lemma provides sufficient conditions

4/28/2009                         CSL, University of Illinois            14
                               Proof Outline

 Construct feasible solution to dual program with same objective
  function value

 Weak duality proves upper bound

 Proof of feasibility of dual solution depends on particular
  structure of the cost function f(.)

     – Through the following inequality

   LEMMA: Given a  0,b  c  1,   2,   0,fR  f (1)(1  0.5 1 )

        ( f (a  b)  f (a))( f (b  c)  f (b))  ( f (a  b  c)  f (b  c))( f (b)  fR )

   where f (x) : Et e x x

4/28/2009                         CSL, University of Illinois                             15
 Crossing Paths in a Flow are Suboptimal

       Equivalently…


                                                           1  2
                 1


                                         
            
                                                            2
                                  1  2
                                                      1
                                            

                       
                                       
4/28/2009               CSL, University of Illinois                   16
                                Example 1


 b1=b2=…=bn-1=bn=b,
  E1/f1=E2/f2=…=En/fn=E

 Optimal strategy consists of each node
  transmitting directly to sink
     – Lifetime is E/b
     – Does not depend on fR or n

                        n
                            Ei n         n n
                         f gi            g
                                      E  i 1 i  E
                        i 1 i
                 Tf     n             n         =
                                      b              b
                        b g     n
                               i i         gin 
                        i 1             i 1 

4/28/2009                      CSL, University of Illinois   17
                                Example 2
 b1=b2=…=bn-1=0, bn=b,
  E1=E2=…=En=E

 Under optimal strategy:
     – Node n transmits to sink and node
       n-1
     – Every other node i transmits only to
       node i-1
     – Amounts transmitted are determined
       by equalizing energy consumed

               nn 1 f1  (b  nn 1 ) fn  ( fR  f1 )nn 1
                *                *                        *



                             f 
                      E 1  R 
                             fn 
                Tf 
                      ( fR  f1 )b
4/28/2009                       CSL, University of Illinois       18
                 Regular Planar Network

   Nodes located on concentric rings

      N rings

      Ring i has Mi nodes and radius iR

      Total of N(N+1)M/2 nodes

      Sink node at center of circle

      Other regular networks, such as square
       lattice, can also be considered
        – Similar results hold, but more analysis is more
          cumbersome


4/28/2009                     CSL, University of Illinois   19
            Mapping to Linear Network

                                    3M               2M              M

                                    b      3j       b     2j       b    1j
                                     j1             j1             j1

                                    3M               2M
                                                                 R   M

                                    E      3j       E     2j       E    1j

                                   j1
                                                    
                                                     j1             j1




                                                
                                                       
                                 Map each ring to a “super-node”
                                 Super-nodes located on a line at
                                  intervals of length R
                                 Energy and information bits of each
                                  super-node are sums of respective
                                  quantities over corresponding ring
                                 Lifetime of linear network upper
                                  bounds the lifetime of original network

4/28/2009          CSL, University of Illinois                                  20
                              Upper bound

     Therefore…
                    n
                          1 Mi  n              n         f1 
                     f  E j gi
                                            1           
                                                              f j 
                     i1 i j1              j i1          
             Tf  n Mi                     gi  n
                                            n
                                              Ý
                                              Ý
                                                        f R 
                        b j gin
                                            1           
                                                              f j 
                        i1 j1              j i           
             f i  f (iR)
                 Ý
                 Ý


THEOREM: Under “circular symmetry” of node energies and bits, and
existence of a near-far flow for the linear network of super-nodes, a
      
lifetime of K times the above upper bound is achievable, where
                                      f (R)
                         K
                                f (R 1  8 / M )

4/28/2009                       CSL, University of Illinois            21
                     Scaling of Lifetime



             Take   2,   0, fR  0, d  1
                      E1  E2  ...  En  E
                      b1  b2  ...  bn  b

             Choice of parameters gives best case lifetime

             For these parameters, we compare optimal flow
              to simple nearest neighbor scheme




4/28/2009                   CSL, University of Illinois       22
                          Linear Network Case

  Simple nearest neighbor scheme
      – Transmit all to nearest neighbor towards sink

                                  E                E 1
            Tf1                                 
                                 n             b nf1
                                 
                    ( f1  f R ) bi  f R b1
                                i1 

                         
  Optimal flow
 Summation in denominator can be computed

                      E            E      1        1
       Tf*                    
                     n
                         i         bf1 (n  1) 1 log n               Ratio goes to 1 as
               bf1 
                   i1 i  1
                                                     n                network size grows large


4/28/2009                              CSL, University of Illinois                      23
                Planar Network Case

 Consider circular planar network

 Similar calculation gives

                      2E     log N
             Tf* 
                     bf (R) N(N  1)


 While the nearest neighbor scheme has a lifetime lower bounded
   
  as follows
                           2E                             Optimal is better by
             Tf1 
                     Kbf (R)N(N  1)                     factor of log N


       
4/28/2009                  CSL, University of Illinois                      24
                         Implications


 Thus, simple nearest neighbor communication scheme is nearly
  optimal



 Such a scheme is preferable from an operational point of view
      – Lower interference and medium access contention
      – Need not use multiple power levels



 Broader implication: Optimal lifetime scales poorly with network
  size



4/28/2009                 CSL, University of Illinois             25
                          Future Work

 Problem needs to be solved for all values of per-node energies
  and information quantities

      – Lifetime may be determined by the energy/number of bits of a single
        node - degenerate case

      – Need to determine conditions when this happens, and what
        happens otherwise



 May need to consider more general cost functions

 Tradeoff between lifetime and delay?


4/28/2009                  CSL, University of Illinois                  26

				
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