# 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

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

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

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

– 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  

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 }
+
{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

 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

 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
– 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

 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

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