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
Sensor
Relay
Aggregator
Asleep
r
B
R
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
r
B
Single Source
No topology information (only N)
Degenerate R (Fixed Source)
Step II
r
B
Single Source
No topology information (only N)
Resides over R with a certain PDF
R
Step III
r
B
Single Source
Topology information
Degenerate R
Step IV
r
B
Single Source
Topology information
Degenerate R
Aggregation
Step V
r
B
Multiple Fixed Sources
Topology information
Degenerate R
Step VI
r
B
Single Source
Topology information
Resides over R with a certain PDF
R
Step VII
r
B
Single Moving Source
Topology information
Specified Trajectory
R
Step VIII
r
B
r
Multiple Moving Sources
Topology 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
r
B
Single Source
No topology information (only N)
Degenerate R (Fixed Source)
Analog
Sensor Signal
Sensor+
Analog Pre-Conditioning
Functional Abstraction of DGWN Node
“Raw”
Sensor
Data
Processed
Sensor
Data
A/D
DSP+RISC
+FPGA etc.
Sensor
Core
Computational
Core
Radio+
Protocol Processor
Communication &
Collaboration Core
Energy Models
d
Etx = a11+ a2dn
Transmit Energy Per Bit
1. Transceiver Electronics
2. Startup Energy
n = Path loss index
Power-Amp
Erx = a12
Receive Energy Per Bit
d
Relay Energy Per Bit
Erelay = a11+a2dn+a12 = a1+a2dn
Prelay = (a1+a2dn)r
Sensing Energy Per Bit
Aggregation Energy Per Bit
Esense = a3
Eagg =
a
Step I
r
B

Bound the lifetime of a network given:
 The number of nodes (N) and initial energy in each node (E)
 Node energy parameters (a1, a2, a3), path loss index n
 Source observability radius (r)
 Source rate (r bps)

Note: Bound is topology insensitive
Preliminaries: Minimum-Energy
Links and Characteristic Distance
D meters
B
Sink
A
Source
 
K-1 nodes available

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 –
K
Plink ( D)  a12   Prelay (d i )
i 1

Problem: Find a solution that minimizes the measure
Claim I: Optimal Solution is Collinear
w/ Non-Overlapping Link Projections
S
B
A
ST
B
A

Proof: By contradiction. Suppose a non-compliant
solution S is optimal

Produce another solution ST via the projection
transformation shown

Trivial to prove that measure(ST) < measure(S) (QED)

Result holds for any radio function monotonic in d

Reduces to a 1-D problem
Claim II: Optimal Solution Has Equal
Hop Distances
d1
S
d2
B
A
(d1+d2)/2
ST
B
A

Proof: By contradiction. Suppose a non-compliant
solution S is optimal

Produce solution ST by taking any two unequal adjacent
hops in S and making them equal to half the total hop
length

For any convex Prelay(d), measure(ST) < measure(S) (recall
that 2f((x1+x2)/2) < f(x1)+f(x2) for a convex function f) (QED)
Optimal Solution
D/K
B
A

Measure of the optimal solution: -a12+KPrelay(D/K)

Prelay convex  KPrelay(D/K) is convex

The continuous function xPrelay(D/x) is minimized when:
D
x
n

a1
D

Dchar

a 2 (n  1)
 D  na1 D
min xPrelay   
x
 x  n  1 Dchar
Hence, the K that minimizes Plink(D) is given by:
K opt
 D   D 

 or 

D
D
 char   char 

 na1 D

Plink ( D)  
 a12 r
 n  1 Dchar

Corollary: Minimum Energy Relay
D meters
B
Sink

A
Source
It is not possible to relay bits from A to B at a rate r
using total link power less than:
 na1 D

Plink ( D)  
 a12 r
 n  1 Dchar

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
Digression: Practical Radios

Results hinge only on communication energy versus
distance being monotonically increasing and convex
Overall
radio
behavior
Inflexible
power-amp
d2 behavior
Energy/bit
Perfect
power
control
d4 behavior
Distance
Complex path loss behavior
• Not a problem!
• Energy/bit can be made linear
• Equal hops still best strategy
• But … Dchar varies with distance
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 PowerControl Problem

What is the best way to quantize the radio energy
curve(for a given number of levels)?
Or?
Distance
Maximizing Lifetime
r
d
B

A
N nodes available
 
Problem: Using N nodes what is maximum sensing
lifetime one can ever hope to achieve?
Take I
r
d
B
A
Take II
r
d
A
d/K
B
Take III
r
A
d2
B
d1
Need an alternative approach to bound lifetime …
Bounding Lifetime
r
A
d
B

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
Pnetwork  Plink (d )  Psensing


 na1 d

Pnetwork  
 a12 r  a 3r
 n  1 d char

If the network lifetime is Tnetwork, then:
 na1 d


Ei  
 a12 r  a 3 r Tnetwork

i 1

 n  1 d char

N
Tnetwork 
N .E
 na1 d


 a12  a 3 r
 n  1 d char

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
x1

x2
x3
…
xk-1
xk
xk+1
…
Assumption: E, T chosen such that k >> 1
Step II
r
B
Single Source
No topology information (only N)
Resides over R with a certain PDF
R
Bounding Strategy
r
d(x)
A
R
B
Pnetwork ( x)  Plink (d ( x))  Psensing
 na1 d ( x)

Pnetwork ( x)  
 a12  a 3 r
 n  1 d char

 na1 E[d ( x)]

E[ Pnetwork ( x)]  
 a12  a 3 r
 n  1 d char

Bounding Strategy
N
 E  P  P    P T
i 1
i
1
2
k
 na1 di

Pi  
 a12 r  a 3r
 n  1 d char

 P  P    PK
N E  1 2
K

Tnetwork

Tnetwork

N E

Pav
2
PrPav  E[ Pnetwork ( x)]    
K 2


  2
N E
 
Pr Tnetwork  
2
E
[
P
(
x
)]


K

network



Bounding Strategy




2
N E

 
Pr Tnetwork 

2
K



na E[d ( x)]

 1
 a12  a 3 r   


 n  1 d char




Bound depends on region only via E[d(x)]

For brevity, we abuse notation thus:
N E
Tnetwork 
 na1 E[d ( x)]


 a12  a 3 r
 n  1 d char

Source Moving Along A Line
S0
dW
r
dN
A
d(x)
dB
B
Tnetwork 
N .E


 d1  d 2 
2
 d1d 2  d 3d 4  dW ln 




na1
 d 3  d 4   r r

(n  1)d char 
2d N






S1
Simulation Results
Source in a Rectangular Region
r
dW
A
B
y
dB
dW
x
dN
Tnetwork
d rect
1

12d N dW
N .E

d rect
n
a1
r
n  1 d char

 d1  d 2 
 d 4  dW
3
3
  d 3 ln 
4dW (d1d 2  d 3d 4 )  2dW ln 
 d3  d 4 
 d 4  dW


 d 2  dW
3
  d1 ln 

 d 2  dW

  r

Simulation Results
Source in a Semi-Circle
dR
dR
Tnetwork 
dW
dB
N .E
d sector
n
a1
r
n  1 d char
 d R  dW
2d  d B d R dW  d ln 
dB


3( (d B2  dW2 )  d B dW )
3
R
d sector
r
3
B


r

2
d semi -circle  d R  r
3
Simulation Results
Bounding Lifetime for Sources in
Arbitrary Regions: Partitioning Theorem
Rj, pj
B
Partitioning Relation:
 P pj 

Tnetwork ( R )   
 j 1 T ( R ) 
j 

1
Lifetime
bound for
region Rj
Step III
r
B
Single Source
Topology 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
r
d
A
B

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
Feasible Role Assignment
11
1
6
2
5
15
13 12
8
14
7
4
9
B
10
FRA: 1  5  11  14  B
3
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
11
1
6
2
5
15
13 12
8
14
7
4
9
B
10
FRA: 1  5  11  14  B;
2  3  9  14  B
3
Infeasible Role Assignment
B
Enumerating FRAs (Collinear Networks)
5
4
3
2
1
B

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
5
4
3
2
1
5
4
3
2
1
B
B
Enumerating Candidate FRAs
5
4
3
2
1
B

Property allows reduction of candidate FRAs from (N-1)! to 2N-1
R0: 1  B
R1: 1  2  B
R2: 1  3  B
R3: 1  4  B
R4: 1  5  B
R5: 1  2  3  B
R6: 1  2  4  B
R7: 1  2  5  B
R8: 1  3  4  B
R9: 1  3  5  B
R10: 1  4  5  B
R11: 1  2  3  4  B
R12: 1  2  3  5  B
R13: 1  2  4  5  B
R14: 1  3  4  5  B
R15: 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 nonnegative energies in the nodes
R2, t0
R13, t1
R15, t2
R2, t4
R6, t5
R0, t3
R11, t8
R2, t9
R11, t10
R8, t6
R5, t7
5
B
4
3
2
1
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, t0
Ra1, t1
Ra2, t2
Ra4, t4
Ra5, t5
R1, t’1
R2, t’2
R6, t’6
R3, t’3 R5, t’5
R4, t’4
Canonical Form
Ra9, t9
Ra10, t10
Ra6, t6
Ra3, t3
R0, t’0
Ra8, t8
Ra7, t7
R7, t’7
R9, t’9
R8, t’8
R10, t’10
R12, t’12 R14, t’14
R11, t’11
R13, t’13
R15, t’15
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 t’ks)?

Solved via Linear Programming:
t k - Time spent in k th FRA
P(i, k ) - Power dissipated by node i in k th FRA
E (i ) - Initial energy in node i
Objective:
N FRA
max
subject to:
t
k 1
0 tk
N FRA
t
k 1
k
k
[Non-negativity of role time]
P(i, k )  E (i ), 1  i  N
[Non-negativity of residual energy]
Example
dchar
dchar/2
3
dchar/2
2
1
B
Min-hop
R0:
R1:
R2:
R3:
1B
12B
13B
123B
Total Lifetime
R0:
R1:
R2:
R3:
0.25
0
0
0
0.25
Min-Energy
R0:
R1:
R2:
R3:
0
0
1.0
0
1.0
Persistent
R0:
R1:
R2:
R3:
0.09
0.23
0
1.0
1.32
Optimal
R0:
R1:
R2:
R3:
0
0.375
0.375
0.625
1.38
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
1326B
15246B
152B
(32%)
(20%)
(19%)
(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
Role Assignment View
3/11
R0:
R1:
R2:
R3:
0
(0)
0.375 (3/11)
0.375 (3/11)
0.625 (5/11)
1.375 (11/11)
3/11
3
3/11
2
1
B
3/11
5/11
3/11
5/11

Network Flow View
3/11 + 3/11
f12:
f13:
f1B:
f23:
f2B:
f3B:
8/11
3/11
0
3/11
5/11
6/11
3/11
3
3/11 + 5/11
2
B
5/11
3/11
1
Equivalent Flow Program
Extensions to k-of-m Sensors
S
B

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
k of m sensors Program (additional constraints)
2-Sensor Example
3/11
Single Sensor Lifetime 1.375 s
R0:
R1:
R2:
R3:
0
(0)
0.375 (3/11)
0.375 (3/11)
0.625 (5/11)
1.375 (11/11)
3
2
B
3/11
2 Sensor Lifetime 1.816 s
R0:
R1:
R2:
R3:
0.246 (2/15)
0.615 (5/15)
1.0 (8/15)
0
(0)
1.816 (15/15)
1
5/11
2/15
1a
3
2
1b
B
8/15
5/15

Sensing time divided equally between 1a and 1b

Note the complete change in optimal routing strategy
Step IV
r
B
Single Source
Topology information
Degenerate R
Aggregation
Extensions to Aggregation
3
2
B

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)
Non-Aggregating FRAs
Aggregating FRAs


R0: 1  B; 2  B
R1: 1  2  B; 2  B
R2: 1  3  B; 2  B
R3: 1  2  3  B; 2  B
R4: 1  B; 2  3  B
R5: 1  2  B; 2  3  B
R6: 1  3  B; 2  3  B
R7: 1  2  3  B; 2  3  B
R8: 1  2  B; 2  B
R9: 1  2  3  B; 2  3  B
R10: 1  3  B; 2  3  B
R11: 1  2  3  B; 2  3  B
1
Aggregation Example
3
2
1
B
R8: 1  2  B; 2  B (56%)
R10: 1  3  B; 2  3  B (20%)
R6: 1  3  B; 2  3  B (20%)

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
Aggregation Flavors
9
8
B
10
3
9
8
1
11
8
2
3
4
5
6
7
3
4
4
1
1
2
5
General
6
2
5
7
Flat
2-Level
6
7
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
P-2
P-1
P
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
r
B
Multiple Fixed Sources
Topology information
Degenerate R
Multiple Sources
B

Constraints non-trivial due to possible overlaps …
Key: Virtual Nodes
B

Constraints as before (but using virtual nodes when
there are overlaps)

Virtual nodes connected via an overall energy
constraint
Probabilistic Extension
C
B
A
B

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
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
T(sA, sB, sC)
(sA, sB, sC)

1


Pr T  Tp  1   2
Fraction Vector Space

1 determines 

2 and  determine number of trials
Lifetime Space
Step VI
r
B
Single Source
Topology information
Resides over R with a certain PDF
R
Extensions to Arbitrary PDFs
B
R

Given topology and the source location pdf how can
we derive a lifetime bound?

No more difficult than the discrete problem …
Key: Partitioning R
b
1
B
g
c
3
e
f
2
j
l
5
k
i
d
h
4
a
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!
Reduction to discrete probabilistic source
B
R

Growth of number of regions
fixed density and r, grows linearly with the number of
nodes
 For
Step VII
r
B
Single Moving Source
Topology information
Specified Trajectory
R
Dealing with Trajectories
B
r(t)
R

Is an absolute trajectory feasible?

How can one maximize the lifetime if the trajectory is
relative?
Simple extension …
B
R

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
Multiple Moving Sources
B
R

Same strategy as for single source
 Time
spent in region summed over all sources
Recall …
Sensor
Relay
Aggregator
Asleep
r
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