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Bounding the Lifetime of Sensor Networks Via Optimal Role Assignments. Manish Bhardwaj, Anantha Chandrakasan Massachusetts Institute of Technology June 2002. B. r. Data Gathering Wireless Networks: A Primer. Sensor. Relay. Aggregator. Asleep. R. Network Characteristics. - PowerPoint PPT Presentation
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Bounding the Lifetime of Sensor Networks Via Optimal Role Assignments
Manish Bhardwaj, Anantha Chandrakasan
Massachusetts Institute of Technology
June 2002
Data Gathering Wireless Networks: A Primer
B
R
SensorRelayAggregatorAsleep
Network Characteristics
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
Maximizing network lifetime is a key challenge
Functional Abstraction of DGWN Node
A/D
Se
nso
r+A
nal
og
Pre
-Co
nditi
on
ing
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
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, ’0
R1, ’1
R2, ’2
R3, ’3
R4, ’4
R5, ’5
R6, ’6
R8, ’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
iiE
kikiP
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
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)
f12: 8/11f13: 3/11f1B: 0f23: 3/11f2B: 5/11f3B: 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
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
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
B
A
B
C
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
“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