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