Upper Bounds on the Lifetime of Sensor
Networks
Manish Bhardwaj, Timothy Garnett,
Anantha Chandrakasan
Massachusetts Institute of Technology
October 2001
Outline

Wireless Sensor Networks

Energy Models

The Lifetime Problem

Bounding Lifetime

Extensions

Summary
Wireless Sensor Networks

Sensor Types: Low Rate
(e.g., acoustic and seismic)

Bandwidth: bits/sec to kbits/sec

Transmission Distance: 5-10m
(< 100m)

Spatial Density
 0.1

nodes/m2 to 20 nodes/m2
Node Requirements

Small Form Factor

Required Lifetime: > year
Key Challenge: Maximizing Lifetime
Data Gathering Wireless Networks:
A Primer
Sensor
Relay
Aggregator
Asleep
r
B
R
Analog
Sensor Signal
Sensor+
Analog Pre-Conditioning
Functional Abstraction of DGWN Node
“Raw”
Sensor
Data
Processed
Sensor
Data
A/D
DSP+RISC
+FPGA etc.
Sensor
Core
Computational
Core
Radio+
Protocol Processor
Communication &
Collaboration Core
Energy Models
d
Etx = a11+ a2dn
Transmit Energy Per Bit
1. Transceiver Electronics
2. Startup Energy
n = Path loss index
Power-Amp
Erx = a12
Receive Energy Per Bit
d
Relay Energy Per Bit
Sensing Energy Per Bit
Erelay = a11+a2dn+a12 = a1+a2dn
Prelay = (a1+a2dn)r
Esense = a3
Defining Lifetime

Three network states:
 Active
 Failure
 Dormant

Possible lifetime definitions:
 Cumulative
active time
 Cumulative active time to first failure
The Lifetime Bound Problem
N nodes, Initial energy E J
r
B
R

Bound the lifetime of a network given:
 A description of R and the relative position of the base-station
 The number of nodes (N) and initial energy in each node (E)
 Node energy parameters (a1, a2, a3), path loss index n
 Source observability radius (r)
 Spatial distribution of the source (lsource(x,y))
 Expected source rate (r bps)

Note: Bound is topology insensitive
Preliminaries: Minimum-Energy
Links and Characteristic Distance
D meters
B
Sink
A
Source
 
K-1 nodes available

Given: A source and sink node D m apart and K-1
available nodes that act as relays and can be placed at will
(a relay is qualified by its source and destination)

Solution: Position, qualification of the K-1 relays

Measure of the solution: Energy needed to transport a bit
or equivalently, the total power of the link –
K
Plink ( D)  a12   Prelay (d i )
i 1

Problem: Find a solution that minimizes the measure
Claim I: Optimal Solution is Collinear
w/ Non-Overlapping Link Projections
S
B
A
ST
B
A

Proof: By contradiction. Suppose a non-compliant
solution S is optimal

Produce another solution ST via the projection
transformation shown

Trivial to prove that measure(ST) < measure(S) (QED)

Result holds for any radio function monotonic in d

Reduces to a 1-D problem
Claim II: Optimal Solution Has Equal
Hop Distances
d1
S
d2
B
A
(d1+d2)/2
ST
B
A

Proof: By contradiction. Suppose a non-compliant
solution S is optimal

Produce solution ST by taking any two unequal adjacent
hops in S and making them equal to half the total hop
length

For any convex Prelay(d), measure(ST) < measure(S) (recall
that 2f((x1+x2)/2) < f(x1)+f(x2) for a convex function f) (QED)
Optimal Solution
D/K
B
A

Measure of the optimal solution: -a12+KPrelay(D/K)

Prelay convex  KPrelay(D/K) is convex

The continuous function xPrelay(D/x) is minimized when:
D
x
n

a1
D

Dchar

a 2 (n  1)
 D  na1 D
min xPrelay   
x
 x  n  1 Dchar
Hence, the K that minimizes Plink(D) is given by:
K opt
 D   D 

 or 

D
D
 char   char 

 na1 D

Plink ( D)  
 a12 r
 n  1 Dchar

Corollary: Minimum Energy Relay
D meters
B
Sink

A
Source
It is not possible to relay bits from A to B at a rate r
using total link power less than:
 na1 D

Plink ( D)  
 a12 r
 n  1 Dchar

with equality  D is an integral multiple of Dchar

Key points:
 It
is possible to relay bits with an energy cost linear in distance,
regardless of the path loss index, n
 The most energy efficient multi-hop links result when nodes
are placed Dchar apart
Digression: Practical Radios

Results hinge only on communication energy versus
distance being monotonically increasing and convex
Overall
radio
behavior
Inflexible
power-amp
d2 behavior
Energy/bit
Perfect
power
control
d4 behavior
Distance
Complex path loss behavior
• Not a problem!
• Energy/bit can be made linear
• Equal hops still best strategy
• But … Dchar varies with distance
Distance
Finite Power-Control Resolution
• “Too Coarse” quanta a problem
• Energy/bit no longer linear
• Equal hops NOT best for energy
• No concept of Dchar
Digression: The Optimum PowerControl Problem

What is the best way to quantize the radio energy
curve(for a given number of levels)?
Or?
Distance
Answer depends on:
• Distribution of distances
• Sophisticated non-linear optimization needed for best multi-hop
Maximizing Lifetime – A Simple Case
r
d
B

A
N nodes available
 
Problem: Using N nodes what is maximum sensing
lifetime one can ever hope to achieve?
Take I
r
d
B
A
Take II
r
d
A
d/K
B
Take III
r
A
d2
B
d1
Need an alternative approach to bound lifetime …
Bounding Lifetime
r
A
d
B

Claim: At any instant in an active network:
 There
is a node that is sensing
 There is a link of length d relaying bits at r bps
Pnetwork  Plink (d )  Psensing


 na1 d

Pnetwork  
 a12 r  a 3r
 n  1 d char

If the network lifetime is Tnetwork, then:
 na1 d


Ei  
 a12 r  a 3 r Tnetwork

i 1

 n  1 d char

N
Tnetwork 
N .E
 na1 d


 a12  a 3 r
 n  1 d char

Simulation Results
Source Moving Along A Line
S0
r
dN
S1
A
dW
d(x)
dB
B
 na1 d ( x)

Pnetwork ( x)  Plink (d ( x))  Psensing  Pnetwork ( x)  
 a12 r  a 3r
 n  1 d char

xd B  d N
E ( Pnetwork )  
Pnetwork ( x)lsource ( x)dx
xd B
Tnetwork 
N .E


 d1  d 2 
2
 d1d 2  d 3d 4  dW ln 




na1
 d 3  d 4   r r

(n  1)d char 
2d N






Simulation Results
Source in a Rectangular Region
r
dW
A
B
y
dB
dW
x
E ( Pnetwork )  
xd B  d N
xd B

y  dW
y   dW
Tnetwork
d rect
dN
Pnetwork ( x, y )lsource ( x, y )dxdy
N .E

d rect
n
a1
r
n  1 d char
1000 node network,
2 J on a node has the
potential to report finite
velocity tank intrusions in a
sq. km, a km away for more
than 7 years!

 d1  d 2 
 d 4  dW
1
3
3



 d 3 ln 
4dW (d1d 2  d 3d 4 )  2dW ln 

12d N dW 
 d3  d 4 
 d 4  dW

 d  dW
  d13 ln  2

 d 2  dW

  r

Simulation Results
Source in a Semi-Circle
dR
dR
Tnetwork 
dW
dB
N .E
d sector
n
a1
r
n  1 d char
 d R  dW
2d  d B d R dW  d ln 
dB


3( (d B2  dW2 )  d B dW )
3
R
d sector
r
3
B


r

2
d semi -circle  d R  r
3
Simulation Results
Bounding Lifetime for Sources in
Arbitrary Regions: Partitioning Theorem
R 1 , p1
Probability of
residing
in a sub-region
Sub-region
R 5 , p5
R 2, p2
R 4 , p4
R 6 , p6
R 3 , p3
B
R
Partitioning Relation:
 P pj 

Tnetwork ( R )   
 j 1 T ( R ) 
j 

1
Lifetime
bound for
region Rj
Work completed subsequently …

Factoring in topology

Factoring in source movement

Factoring in aggregation:
 Flat
aggregation
 2-step hierarchical

Non-constructive approaches don’t seem to work here

Bounds derived by actual construction of the optimal
strategy

Strategy (and hence bound) can be derived in
polynomial time
Summary

Maximizing network lifetime is a key challenge in
wireless sensor networks

Using simple arguments based on minimum-energy
relays and energy conservation, it is possible to derive
tight or near-tight bounds on the lifetime of sensor
networks

It is possible to derive extremely sophisticated bounds
that factor in the exact graph topology, source
movement and aggregation