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

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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

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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 …

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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

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“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