Bounding the Lifetime of Sensor Networks

1
Bounding the Lifetime of Sensor Networks

• Manish Bhardwaj
• Massachusetts Institute of Technology
• November 2001

Acknowledgments Timothy Garnett, Anantha
Chandrakasan
2
Data Gathering Wireless Networks A Primer
Sensor
Relay
Aggregator
Asleep
R
3
Wireless Sensor Networks

• Sensor Types Low Rate (e.g., acoustic and
  seismic)
• Bandwidth bits/sec to kbits/sec
• Transmission Distance 5-10m (lt 100m)
• Spatial Density
• 0.1 nodes/m2 to 20 nodes/m2
• Node Requirements
• Small Form Factor
• Required Lifetime gt year

4
Step I
Single Source No topology information (only
N) Degenerate R (Fixed Source)
5
Step II
Single Source No topology information (only
N) Resides over R with a certain PDF
R
6
Step III
Single Source Topology information Degenerate R
7
Step IV
Single Source Topology information Degenerate R
Aggregation
8
Step V
Multiple Fixed Sources Topology information
Degenerate R
9
Step VI
Single Source Topology information Resides over
R with a certain PDF
R
10
Step VII
Single Moving Source Topology information
Specified Trajectory
R
11
Step VIII
Multiple Moving Sources Topology information
Specified Trajectories
R
12
Preview of Tools

• Energy Conservation Arguments
• Simple properties of convex functions
• LLN
• Linear Programming
• Transformation of Programs
• Network Flow Formulations
• Miscellaneous tricks

13
Step I
Single Source No topology information (only
N) Degenerate R (Fixed Source)
14
Functional Abstraction of DGWN Node
Sensor Analog Pre-Conditioning
A/D
Sensor Core
Computational Core
Communication Collaboration Core
15
Energy Models
16
Step I

• 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

17
Preliminaries Minimum-Energy Links and
Characteristic Distance
D meters
B
A
Source
Sink
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

• Problem Find a solution that minimizes the
  measure

18
Claim I Optimal Solution is Collinear w/
Non-Overlapping Link Projections
B
A
S
ST
A
B

• 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) lt measure(S)
  (QED)
• Result holds for any radio function monotonic in
  d
• Reduces to a 1-D problem

19
Claim II Optimal Solution Has Equal Hop Distances
d1
d2
S
A
B
(d1d2)/2
B
A
ST

• 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) lt
  measure(S) (recall that 2f((x1x2)/2) lt
  f(x1)f(x2) for a convex function f) (QED)

20
Optimal Solution
D/K

• Measure of the optimal solution
  -a12KPrelay(D/K)
• Prelay convex ? KPrelay(D/K) is convex
• The continuous function xPrelay(D/x) is minimized
  when

?

• Hence, the K that minimizes Plink(D) is given by

?
21
Corollary Minimum Energy Relay
D meters
B
A
Source
Sink

• It is not possible to relay bits from A to B at a
  rate r using total link power less than

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

22
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
Distance

• 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

23
Digression The Optimum Power-Control Problem

• What is the best way to quantize the radio energy
  curve(for a given number of levels)?

Or?
Distance
24
Maximizing Lifetime
r
A
d

• Problem Using N nodes what is maximum sensing
  lifetime one can ever hope to achieve?

25
Take I
r
A
d
26
Take II
r
d
A
d/K
27
Take III
r
A
d2
d1
Need an alternative approach to bound lifetime
28
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

?

• If the network lifetime is Tnetwork, then

1000 node network, 2 J on a node has the
potential to listen to human conversations 1 km
away for 128 hours
29
Simulation Results
30
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

x3
x2
x1
xk-1
xk1
xk

• Assumption E, T chosen such that k gtgt 1

31
Step II
Single Source No topology information (only
N) Resides over R with a certain PDF
R
32
Bounding Strategy
r
d(x)
A
R
33
Bounding Strategy
34
Bounding Strategy

• Bound depends on region only via Ed(x)
• For brevity, we abuse notation thus

35
Source Moving Along A Line
S0
S1
dN
A
dW
d(x)
dB
36
Simulation Results
37
Source in a Rectangular Region
dW
A
y
dB
B
dW
x
dN
38
Simulation Results
39
Source in a Semi-Circle
dR
dW
dB
dR
?
40
Simulation Results
41
Bounding Lifetime for Sources in Arbitrary
Regions Partitioning Theorem
Rj, pj
Partitioning Relation
Lifetime bound for region Rj
42
Step III
Single Source Topology information Degenerate R
43
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

44
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

45
Feasible Role Assignment
11
1
6
2
5
15
12
13
7
8
4
14
3
9
10
FRA 1 ? 5 ? 11 ? 14 ? B
46
Infeasible Role Assignment (Redundant)
47
Infeasible Role Assignment (Invalid)
48
Infeasible Role Assignment (Invalid)
49
Infeasible Role Assignment (Invalid)
50
Infeasible Role Assignment (Redundant)
51
Feasible Role Assignment
11
1
6
2
5
15
12
13
7
8
4
14
3
9
10
FRA 1 ? 5 ? 11 ? 14 ? B 2 ? 3 ? 9 ?
14 ? B
52
Infeasible Role Assignment
53
Enumerating FRAs (Collinear Networks)
1
2
3
4
5

• 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

54
Enumerating Candidate FRAs
1
2
3
4
5

• 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
55
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, t0
R13, t1
R15, t2
R2, t4
R6, t5
R11, t8
R2, t9
R11, t10
R0, t3
R8, t6
R5, t7
4
5
1
2
3
56
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
Ra8, t8
Ra9, t9
Ra10, t10
Ra6, t6
Ra3, t3
Ra7, t7
R0, t0
R2, t2
R6, t6
R7, t7
R9, t9
R10, t10
R12, t12
R14, t14
R1, t1
R3, t3
R5, t5
R8, t8
R11, t11
R13, t13
R4, t4
R15, t15
Canonical Form
57
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 tks)?
• Solved via Linear Programming

Objective
subject to
Non-negativity of role time
Non-negativity of residual energy
58
Example
dchar
dchar/2
dchar/2
3
1
2
R0 1 ? B R1 1 ? 2 ? B R2 1 ? 3 ? B R3 1 ?
2 ? 3 ? B
Total Lifetime
59
7 Node Non-Collinear Network

• General N-node network with specified sensor has
  ?e(N-1)!? FRAs
• 326 FRAs for a 7 node network!

60
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
61
Equivalence to Flow Problems
Role Assignment View
3/11
3/11
3/11
R0 0 (0) R1 0.375 (3/11) R2 0.375
(3/11) R3 0.625 (5/11)
1
2
3
1.375 (11/11)
5/11
5/11
3/11
3/11
Network Flow View
3/11 5/11
3/11
3/11 3/11
f1?2 8/11 f1?3 3/11 f1?B 0 f2?3
3/11 f2?B 5/11 f3?B 6/11
1
2
3
3/11
5/11
62
Equivalent Flow Program
63
Extensions to k-of-m Sensors
S

• 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

64
k of m sensors Program (additional constraints)
65
2-Sensor Example
3/11
Single Sensor Lifetime 1.375 s
R0 0 (0) R1 0.375 (3/11) R2 0.375
(3/11) R3 0.625 (5/11)
1
2
3
1.375 (11/11)
5/11
3/11
2/15
2 Sensor Lifetime 1.816 s
R0 0.246 (2/15) R1 0.615 (5/15) R2 1.0
(8/15) R3 0 (0)
1a
2
3
1b
1.816 (15/15)
5/15
8/15

• Sensing time divided equally between 1a and 1b
• Note the complete change in optimal routing
  strategy

66
Step IV
Single Source Topology information Degenerate R
Aggregation
67
Extensions to Aggregation
1
2
3

• 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)

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
Non-Aggregating FRAs
Aggregating FRAs
68
Aggregation Example
1
2
3
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

69
Aggregation Flavors
9
8
B
10
3
9
8
1
2
5
6
7
3
4
3
4
11
8
4
1
2
5
6
7
1
2
5
6
7
2-Level
Flat
General
70
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 P1 commodities

0
P-2
P-1
P
71
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

72
What can I say
73
Step V
Multiple Fixed Sources Topology information
Degenerate R
74
Multiple Sources
B

• Constraints non-trivial due to possible overlaps

75
Key Virtual Nodes
B

• Constraints as before (but using virtual nodes
  when there are overlaps)
• Virtual nodes connected via an overall energy
  constraint

76
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

77
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

78
Picture for well-behaved programs
T(sA, sB, sC)
(sA, sB, sC)
?1
?
Fraction Vector Space
Lifetime Space

• ?1 determines ?
• ?2 and ? determine number of trials

79
Step VI
Single Source Topology information Resides over
R with a certain PDF
R
80
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

81
Key Partitioning R
b
1
c
3
B
e
g
f
d
2
j
i
h
l
4
k
5
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!

82
Reduction to discrete probabilistic source
B
R

• Growth of number of regions
• For fixed density and r, grows linearly with the
  number of nodes

83
Step VII
Single Moving Source Topology information
Specified Trajectory
R
84
Dealing with Trajectories
B
r(t)
R

• Is an absolute trajectory feasible?
• How can one maximize the lifetime if the
  trajectory is relative?

85
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

86
Multiple Moving Sources
B
R

• Same strategy as for single source
• Time spent in region summed over all sources

87
Recall
Sensor
Relay
Aggregator
Asleep
R
88
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