bounding the lifetime of sensor networksweb.mit.edu/6.454/www/ · 2001-11-06 · power-amp complex...
TRANSCRIPT
Bounding the Lifetime of Sensor Networks
Manish Bhardwaj
Massachusetts Institute of Technology
November 2001
Acknowledgments: Timothy Garnett, Anantha Chandrakasan
?
Data Gathering Wireless Networks: A Primer
B
R
SensorRelayAggregatorAsleep
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
Step I
B
?
Single SourceNo topology information (only N)Degenerate R (Fixed Source)
Step II
B
?
Single SourceNo topology information (only N)Resides over R with a certain PDF
R
Step III
B
?
Single SourceTopology information Degenerate R
Step IV
B
?
Single SourceTopology information Degenerate R Aggregation
Step V
B
?
Multiple Fixed SourcesTopology information Degenerate R
Step VI
B
?
Single SourceTopology information Resides over R with a certain PDF
R
Step VII
B
?
Single Moving SourceTopology information Specified Trajectory
R
?
Step VIII
B
?
Multiple Moving SourcesTopology information Specified Trajectories
R
Preview of Tools
? Energy Conservation Arguments
? Simple properties of convex functions
? LLN
? Linear Programming
? Transformation of Programs
? Network Flow Formulations
? Miscellaneous tricks …
Step I
B
?
Single SourceNo topology information (only N)Degenerate R (Fixed Source)
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
Eagg = ? 4Aggregation Energy Per Bit
1. Transceiver Electronics2. Startup Energy Power-Amp
Step I
B
?
? Bound the lifetime of a network given:?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 (? )?Source rate (r bps)
? Note: Bound is topology insensitive
Preliminaries: Minimum-Energy Links and Characteristic Distance
? Given: A source and sink node D m apart and K-1available 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
DD
DK or
charx D
Dnn
xD
xP1
min 1relay ?
????
??? ??
rD
Dnn
DPchar
???
????
??
?? 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
rD
Dnn
DPchar
???
????
??
?? 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?
Maximizing Lifetime
? 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
rrd
dnn
Pchar
3121
network 1??
????
?
????
??
??sensinglinknetwork )( PdPP ??
? If the network lifetime is Tnetwork, then:
networkchar
N
ii Trr
dd
nn
E???
???
????
????
??
???
?312
1
1 1??
?
rd
dnn
ENT
char
network
???
????
???
?
?
3121
1
.
???
?
1000 node network,2 J on a node has the potential to listen to human conversations 1 km away for 128 hours
Simulation Results
Sources Residing in Regions
? Source locations X1, X2, … assumed IID drawn from a “source location pdf”, fX(x)
? Each sustained for time T
? Lifetime: kT
…x2x1 x3 xkxk-1 xk+1 …
? Assumption: E, T chosen such that k >> 1
Step II
B
?
Single SourceNo topology information (only N)Resides over R with a certain PDF
R
Bounding Strategy
B R
?
sensinglinknetwork ))(()( PxdPxP ??
rd
xdnn
xPchar
???
????
???
?? 312
1network
)(1
)( ???
d(x) A
rd
xdnn
xPchar
???
????
???
?? 312
1network
)]([1
)]([ ??? E
E
Bounding Strategy
? ?TPPPE k
N
ii ?????
?
?211
networkK T
KPPP
EN ???
??? ???
???21
rrdd
nn
Pchar
ii 312
1
1??
????
?
????
??
??
avnetwork P
ENT
??
? ?? ? 2
2
network )]([Pr?
??
KxPPav ??? E
2
2
network )]([Pr
??
? KxPEN
Tnetwork ????
???
???
????
??
??
E
Bounding Strategy
? Bound depends on region only via E[d(x)]
? For brevity, we abuse notation thus:
2
2
3121 )]([1
Pr?
?
???? K
rd
xdnn
ENT
char
network ?
???
???
?
???
???
?
????
????
???
?
??
E
rd
xdnn
ENT
char
network
???
????
???
?
??
3121 )]([1
??? E
Source Moving Along A Line
B
?
A
dB
S0 S1dN
dW d(x)
rd
dddd
ddddd
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
rdd
nn
ENT
char
rectnetwork
11
.
??
?
????
???
????
????
???
????
????
???
????
????
???
???W
W
W
WWW
WNrect dd
ddd
dddd
ddddd
dddddddd 2
231
4
433
43
2134321 lnlnln2)(4
121
Simulation Results
Source in a Semi-Circle
dW?dR
dR
dB
rdd
nn
ENT
char
tornetwork
sec11
.
??
?
??
??
??
???
????
? ???
?))((3
ln2
22
33
sectorWBWB
B
WRBWRBR
ddddd
ddddddd
d ??? Rdd32
circle-semi?
Simulation Results
Bounding Lifetime for Sources in Arbitrary Regions: Partitioning Theorem
Rj, pj
B
1
1 )()(
?
????
????
?? ?
P
j j
jnetwork RT
pRT
Partitioning Relation:
Lifetime bound forregion Rj
Step III
B
?
Single SourceTopology information Degenerate R
Including Topology
? Topology insensitive bounds can be grossly unfair in scenarios where the user does not have deployment control
? Topology: Graph of the network
? Flavor 1: Accept a graph and solve the problem exactly
? Flavor 2: Accept a probabilistic description of a graph and produce a p.d.f. of the lifetime bound
The Role Assignment Problem: Jargon
? Node Roles: ?Sense, Relay, Aggregate, Sleep?
? Role Attributes:?Sense: Destination?Relay: Source and Destination?Aggregate: Source1, Source2, Destination?Sleep: None
? Feasible Role Assignment: An assignment of roles to nodes such that valid and non-redundant sensing is performed
B
?
d A
Feasible Role Assignment
B
2
1
34
5
7
6
8
9
10
11
1213
14
15
FRA: 1 ? 5 ? 11 ? 14 ? B
Infeasible Role Assignment (Redundant)
B
Infeasible Role Assignment (Invalid)
B
Infeasible Role Assignment (Invalid)
B
Infeasible Role Assignment (Invalid)
B
Infeasible Role Assignment (Redundant)
B
Feasible Role Assignment
B
2
1
34
5
7
6
8
9
10
11
1213
14
15
FRA: 1 ? 5 ? 11 ? 14? B;2 ? 3 ? 9 ? 14 ? B
Infeasible Role Assignment
B
Enumerating FRAs (Collinear Networks)
? Collinear networks: All nodes lie on a line
? Flavor being considered: Sensor given, no aggregation (Max Lifetime Multi-hop Routing)
? Property: Self crossing roles need not be considered
B12345
B12345
B12345
Enumerating Candidate FRAs
? Property allows reduction of candidate FRAs from (N-1)! to 2N-1
B12345
R0: 1 ? BR1: 1 ? 2 ? BR2: 1 ? 3 ? BR3: 1 ? 4 ? BR4: 1 ? 5 ? BR5: 1 ? 2 ? 3 ? BR6: 1 ? 2 ? 4 ? B R7: 1 ? 2 ? 5 ? BR8: 1 ? 3 ? 4 ? BR9: 1 ? 3 ? 5 ? BR10: 1 ? 4 ? 5 ? BR11: 1 ? 2 ? 3 ? 4 ? BR12: 1 ? 2 ? 3 ? 5 ? BR13: 1 ? 2 ? 4 ? 5 ? BR14: 1 ? 3 ? 4 ? 5 ? BR15: 1 ? 2 ? 3 ? 4 ? 5 ? B
Collaborative Strategy
? Collaborative strategy is a formalism that precisely captures the mechanism of gathering data
? Is characterized by specifying the order of FRAs and the time for which they are sustained
? A collaborative strategy is feasible iff it ends with non-negative energies in the nodes
R2, ?0 R13, ?1 R15, ?2
R0, ?3
R2, ?4 R6, ?5
R8, ?6
R5, ?7
R11, ?8 R2, ?9 R11, ?10
B12345
Canonical Form of a Strategy
? Canonical form: FRAs are sequenced in order. Some FRAs might be sustained for zero time
? It is always possible to express any feasible collaborative strategy in an equivalent canonical form
Ra0, ?0 Ra1, ?1 Ra2, ?2
Ra3, ?3
Ra4, ?4 Ra5, ?5
Ra6, ?6
Ra7, ?7
Ra8, ?8 Ra9, ?9 Ra10, ?10
R0, ?’0R1, ?’1
R2, ?’2R3, ?’3
R4, ?’4
R5, ?’5
R6, ?’6R8, ?’8
R7, ?’7 R9, ?’9 R10, ?’10
R11, ?’11
R12, ?’12
R13, ?’13
R14, ?’14
R15, ?’15
Canonical Form
The Role Assignment Problem
? How to assign roles to nodes to maximize lifetime?
? Same as: Which collaborative strategy maximizes lifetime?
? Same as: How long should each of the FRAs be sustained for maximizing lifetime (i.e. determine the ?’ks)?
? Solved via Linear Programming:
NiiEkiPFRAN
kk
k
???
?
??
1 ,)(),(
0
1
?
?
??
FRAN
kk
1
max ?
iiEkikiP
kk
nodein energy Initial - )(FRA in nodeby dissipatedPower - ),(
FRA in spent Time - th
th?
subject to:
Objective:
[Non-negativity of role time]
[Non-negativity of residual energy]
Example
B123
dchar dchar/2 dchar/2
R0: 1 ? BR1: 1 ? 2 ? BR2: 1 ? 3 ? BR3: 1 ? 2 ? 3 ? B
Total Lifetime
Persistent
R0: 0.09R1: 0.23R2: 0R3: 1.0
1.32
Optimal
R0: 0R1: 0.375R2: 0.375R3: 0.625
1.38
Min-hop
R0: 0.25R1: 0R2: 0R3: 0
0.25
Min-Energy
R0: 0R1: 0R2: 1.0R3: 0
1.0
7 Node Non-Collinear Network
? General N-node network with specified sensor has ?e(N-1)!? FRAs
? 326 FRAs for a 7 node network!
1 ? 3 ? 2 ? B (32%)1 ? 3 ? 2 ? 6 ? B (20%)1 ? 5 ? 2 ? 4 ? 6 ? B (19%)1 ? 5 ? 2 ? B (18%)
Attack Strategy
? Polynomial time separation oracle + Interior point method
? Transformation to network flows
? Key observation (motivated by Tassiulas et al.)
Broad class of RA problems can be transformed to network flow problems
Network flow problems solved in polynomial time
Flow solution ? RA solution in polynomial time
Equivalence to Flow Problems
B123
B123
R0: 0 (0)R1: 0.375 (3/11)R2: 0.375 (3/11)R3: 0.625 (5/11)
1.375 (11/11)
f1?2: 8/11f1?3: 3/11f1?B: 0f2?3: 3/11f2?B: 5/11f3?B: 6/11
3/113/113/11
3/113/115/11 5/11
3/11 + 5/11
3/11
3/11
5/11
3/11 + 3/11
Role Assignment View
Network Flow View
?
Equivalent Flow Program
Extensions to k-of-m Sensors
? Set of potential sensors (S), |S| = m
? Contract: k of m sensors must sense
? Flow framework easily extended ?Total net volume emerging from nodes in S is now k?Constraints to prevent monopolies?Constraints to prevent consumption
B
S
k of m sensors Program (additional constraints)
2-Sensor Example
? Sensing time divided equally between 1a and 1b
? Note the complete change in optimal routing strategy
B123
R0: 0 (0)R1: 0.375 (3/11)R2: 0.375 (3/11)R3: 0.625 (5/11)
1.375 (11/11)
3/11
3/115/11
B
1a
23
R0: 0.246 (2/15)R1: 0.615 (5/15)R2: 1.0 (8/15)R3: 0 (0)
1.816 (15/15)
2/15
8/155/15
1b
Single Sensor Lifetime 1.375 s
2 Sensor Lifetime 1.816 s
Step IV
B
?
Single SourceTopology information Degenerate R Aggregation
Extensions to Aggregation
? Flavor: 1 and 2 must sense, aggregation permitted
? Roles increase from 2N-1 to 3.(2N-2)2 (for N-node collinear network with two assigned sensors)
B123
R0: 1 ? B; 2 ? BR1: 1 ? 2 ? B; 2 ? BR2: 1 ? 3 ? B; 2 ? BR3: 1 ? 2 ? 3 ? B; 2 ? BR4: 1 ? B; 2 ? 3 ? BR5: 1 ? 2 ? B; 2 ? 3 ? BR6: 1 ? 3 ? B; 2 ? 3 ? BR7: 1 ? 2 ? 3 ? B; 2 ? 3 ? BR8: 1 ? 2? B; 2? BR9: 1 ? 2? 3 ? B; 2? 3 ? BR10: 1 ? 3? B; 2 ? 3? BR11: 1 ? 2 ? 3? B; 2 ? 3? B
??Aggregating FRAs
Non-Aggregating FRAs
Aggregation Example
? Aggregation energy per bit taken as 180 nJ
? Total lifetime is 1.195 (1.596 for 0 nJ/bit, 0.8101 for ? nJ/bit)
? It is NOT optimal for network to aggregate ALL the time
? The aggregator roles shifts from node to node
R10: 1 ? 3? B; 2 ? 3? B (20%)
R6: 1 ? 3 ? B; 2 ? 3 ? B (20%)
R8: 1 ? 2? B; 2? B (56%)
B123
Aggregation Flavors
11
10
9
8
1 2
3
4
5 6 7
8
1 2 5 6 73 4
B
8
1
9
2
3 4
5 6 7
General Flat 2-Level
Flat and 2-Level are Poly-Time
? Key Idea: Multicommodity Flows
? Two classes of bits:?Bits destined for aggregation?Bits not destined for aggregation? Already aggregated? Never aggregated
? Total of P+1 commodities
0
PP-1
P-2
Constraints
? Non-aggregating, non-sensing nodes ?Conserve all commodities
? Aggregating nodes? (1/k) aggregated-flow is sent out as unagg commodity?No out flows on aggregated commodity
? Sensing nodes?Net agg commodity must match that from other sources
What can I say …
Step V
B
?
Multiple Fixed SourcesTopology information Degenerate R
Multiple Sources
? Constraints non-trivial due to possible overlaps …
B
Key: Virtual Nodes
? Constraints as before (but using virtual nodes when there are overlaps)
? Virtual nodes connected via an overall energy constraint
B
Probabilistic Extension
? Single source, but lives at A, B and C probabilistically ?Discrete source location pmf
? What is the lifetime bound now?
? Previous program except weigh the flow by the probability
BA
B
C
Bounding Strategy: WLLN + Perturbations of Linear Programs
? Claim 1a [WLLN]: With enough trials, the fraction of time spent at A can be made as close to pA as we like
? Claim 1b [WLLN]: With enough trials, the sample fraction vector can be made as close to (pA, pB, pC) as we like?Difference is defined elementwise
? Claim 2: For well behaved linear programs, small perturbations from the constraint parameters cause small perturbations in the optimal
Picture for well-behaved programs
? ?1 determines ?
? ?2 and ? determine number of trials
??1
(sA, sB, sC) T(sA, sB, sC)
? ? 21Pr ?? ??? pTT
Fraction Vector Space Lifetime Space
Step VI
B
?
Single SourceTopology information Resides over R with a certain PDF
R
Extensions to Arbitrary PDFs
? Given topology and the source location pdf how can we derive a lifetime bound?
? No more difficult than the discrete problem …
B
R
Key: Partitioning R
? Partition into sub-regions (a through k)
? Every point in a sub-region has the same S
? Calculate the probabilities of all the sub-regions
? Same as the discrete problem!
i
c
df
eB
b
1
2
3
45 a
g
hj
k
l
R
Reduction to discrete probabilistic source
? Growth of number of regions?For fixed density and ? , grows linearly with the number of
nodes
B
R
Step VII
B
?
Single Moving SourceTopology information Specified Trajectory
R
Dealing with Trajectories
? Is an absolute trajectory feasible?
? How can one maximize the lifetime if the trajectory is relative?
B
R
r(t)
Simple extension …
? Calculate fraction of time spent in every region
? Treat as single source problem with fractional residence
? Find out maximum time (T) possible
? Solves both relative and absolute versions
B
R
Multiple Moving Sources
? Same strategy as for single source?Time spent in region summed over all sources
B
R
?
Recall …
B
R
SensorRelayAggregatorAsleep
“Future Work”
? PDFs of lifetime using PDFs of input graphs
? Lifetime loss in the absence of an oracle?Multiple access issues
? Translating optimal role assignment into feasible data gathering protocols