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
Sensor
Relay
Aggregator
Asleep
r
B
R
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
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
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
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
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
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
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
“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