Upload
serge
View
76
Download
0
Tags:
Embed Size (px)
DESCRIPTION
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. - PowerPoint PPT Presentation
Citation preview
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