Upper Bounds on the Lifetime of Sensor Networks

Manish Bhardwaj, Timothy Garnett, Anantha Chandrakasan

Massachusetts Institute of TechnologyOctober 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

B

R

SensorRelayAggregatorAsleep

Functional Abstraction of DGWN Node

A/D

Sen

sor+

Ana

log

Pre

-Con

ditio

ning

SensorCore

DSP+RISC+FPGA etc.

ComputationalCore

AnalogSensor Signal

Communication &Collaboration Core

Radio+Protocol Processor

“Raw”SensorData

ProcessedSensorData

Energy Models

Etx = 11+ 2dn

d

n = Path loss index Transmit Energy Per Bit

Erx = 12Receive Energy Per Bit

Erelay = 11+2dn+12 = 1+2dn Prelay = (1+2dn)r

d

Relay Energy Per Bit

Esense = 3Sensing Energy Per Bit

1. Transceiver Electronics2. Startup Energy Power-Amp

Defining Lifetime

Three network states: Active Failure Dormant

Possible lifetime definitions: Cumulative active time Cumulative active time to first failure

The Lifetime Bound Problem

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 (1, 2, 3), path loss index n Source observability radius () Spatial distribution of the source (lsource(x,y)) Expected source rate (r bps)

Note: Bound is topology insensitive

B

R

N nodes, Initial energy E J

Preliminaries: Minimum-Energy Links and Characteristic Distance

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 –

SourceSink

D meters

K-1 nodes available

AB

K

iidPDP

1relay12link )()(

Problem: Find a solution that minimizes the measure

Claim I: Optimal Solution is Collinear w/ Non-Overlapping Link Projections

Proof: By contradiction. Suppose a non-compliant solution is optimal

Produce another solution T via the projection transformation shown

Trivial to prove that measure(T) < measure() (QED) Result holds for any radio function monotonic in d Reduces to a 1-D problem

AB

AB

T

Claim II: Optimal Solution Has Equal Hop Distances

Proof: By contradiction. Suppose a non-compliant solution is optimal

Produce solution T by taking any two unequal adjacent hops in and making them equal to half the total hop length

For any convex Prelay(d), measure(T) < measure() (recall that 2f((x1+x2)/2) < f(x1)+f(x2) for a convex function f) (QED)

AB

TAB

d1 d2

(d1+d2)/2

Optimal Solution

Measure of the optimal solution: -12+KPrelay(D/K) Prelay convex KPrelay(D/K) is convex The continuous function xPrelay(D/x) is minimized when:

ABD/K

charn

DD

n

Dx

)1(2

1

Hence, the K that minimizes Plink(D) is given by:

charcharopt D

DDDK or

charx D

Dnn

xDxP

1min 1

relay

rDD

nnDP

char

12

1link 1

)(

Corollary: Minimum Energy Relay

It is not possible to relay bits from A to B at a rate r using total link power less than:

SourceSink

D meters

AB

rDD

nnDP

char

12

1link 1

)(

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

Perfect power control

Distance

d2 behavior

d4 behavior

Overall radio

behavior

Distance

Energy/bit

Digression: Practical Radios

Results hinge only on communication energy versus distance being monotonically increasing and convex

Inflexible power-amp

Complex path loss behavior• Not a problem!• Energy/bit can be made linear• Equal hops still best strategy• But … Dchar varies with 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 Power-Control Problem

What is the best way to quantize the radio energy curve(for a given number of levels)?

Distance

Or?

Answer depends on:• Distribution of distances• Sophisticated non-linear optimization needed for best multi-hop

Maximizing Lifetime – A Simple Case

Problem: Using N nodes what is maximum sensing lifetime one can ever hope to achieve?

B

N nodes available

d A

Take I

B

d A

Take II

B

d

A

d/K

Take III

B

d1

A

d2

Need an alternative approach to bound lifetime …

Bounding Lifetime

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

B

d A

rrdd

nnP

char312

1network 1

sensinglinknetwork )( PdPP

If the network lifetime is Tnetwork, then:

networkchar

N

ii Trr

dd

nnE

312

1

1 1

rdd

nn

ENT

char

network

3121

1

.

Simulation Results

Source Moving Along A Line

B

A

dB

S0 S1dN

dW d(x)

rrdxd

nnxP

char312

1network

)(1

)(

sensinglinknetwork ))(()( PxdPxP

NB

B

ddx

dxdxxlxPPE )()()( sourcenetworknetwork

rd

ddddddddd

dnn

ENT

N

W

char

network

2

ln

)1(

.

43

2124321

1

Simulation Results

Source in a Rectangular Region

B

dN

dB

dW

A

dWx

y

NB

B

W

W

ddx

dx

dy

dydxdyyxlyxPPE ),(),()( sourcenetworknetwork

rdd

nn

ENT

char

rectnetwork

11

.

W

W

W

WWW

WNrect dd

dddddddd

dddddddddd

ddd

2

231

4

433

43

2134321 lnlnln2)(4

121

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!

Simulation Results

Source in a Semi-Circle

dWdR

dR

dB

rdd

nn

ENT

char

tornetwork

sec11

.

))((3

ln2

22

33

sectorWBWB

B

WRBWRBR

dddddddddddd

d Rdd32

circle-semi

Simulation Results

Bounding Lifetime for Sources in Arbitrary Regions: Partitioning Theorem

R1, p1

R2, p2

R3, p3

R4, p4

R5, p5

R6, p6

B

1

1 )()(

P

j j

jnetwork RT

pRT

Partitioning Relation:

R

Sub-region

Probability of residing

in a sub-region

Lifetime bound forregion 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