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 B R

3. 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

4. Step I r B Single Source No topology information (only N)
Degenerate R (Fixed Source)

5. Step II r R B Single Source No topology information (only N)
Resides over R with a certain PDF R

6. Step III r B
Single Source
Topology information
Degenerate R

7. Step IV r B Single Source Topology information Degenerate R
Aggregation

8. Step V r B
Multiple Fixed Sources
Topology information
Degenerate R

9. Step VI r R B Single Source Topology information
Resides over R with a certain PDF R

10. Step VII r R B Single Moving Source Topology information
Specified Trajectory R

11. Step VIII r r R B 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 r B Single Source No topology information (only N)
Degenerate R (Fixed Source)

14. Functional Abstraction of DGWN Node
“Raw”
Sensor
Data
Processed
Sensor
Data
Radio+
Protocol Processor
Analog
Sensor Signal
DSP+RISC
+FPGA etc.
Analog Pre-Conditioning
Sensor+
A/D
Sensor
Core
Computational
Core
Communication &
Collaboration Core

15. Energy Models Etx = a11+ a2dn d n = Path loss index
Transmit Energy Per Bit
Transceiver Electronics
Startup Energy
Power-Amp
Erx = a12
Receive Energy Per Bit
Erelay = a11+a2dn+a12 = a1+a2dn
Prelay = (a1+a2dn)r d
Relay Energy Per Bit
Esense = a3
Sensing Energy Per Bit
Eagg = a4
Aggregation Energy Per Bit

16. 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

17. Preliminaries: Minimum-Energy Links and Characteristic Distance
D meters B A
Sink
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 –
Problem: Find a solution that minimizes the measure

18. Claim I: Optimal Solution is Collinear w/ Non-Overlapping Link ProjectionsB 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

19. Claim II: Optimal Solution Has Equal Hop DistancesB A
(d1+d2)/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) < measure(S) (recall that 2f((x1+x2)/2) < f(x1)+f(x2) for a convex function f) (QED)

20. Optimal Solution A B
D/K
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: 
Hence, the K that minimizes Plink(D) is given by: 

21. Corollary: Minimum Energy Relay
D meters B A
Sink
Source
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 B
N nodes available
Problem: Using N nodes what is maximum sensing lifetime one can ever hope to achieve?
Motivating Example …

25. Take I r A d B

26. Take II r d A B
d/K

27. Take III r A d2 B d1
Need an alternative approach to bound lifetime …

28. Bounding Lifetime  Claim: At any instant in an active network: r A d
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 x1 x2 x3
xk-1 xk
xk+1 … …
Assumption: E, T chosen such that k >> 1

31. Step II r R B Single Source No topology information (only N)
Resides over R with a certain PDF R

32. Bounding Strategy r
d(x) A B R

33. Bounding Strategy

34. Bounding Strategy Bound depends on region only via E[d(x)]
For brevity, we abuse notation thus:

35. Source Moving Along A LinedN S1 A dW
d(x) B dB

36. Simulation Results

37. Source in a Rectangular RegiondW A y dB B dW x dN

38. Simulation Results

39. Source in a Semi-CircledR dW r dB dR 

40. Simulation Results

41. Bounding Lifetime for Sources in Arbitrary Regions: Partitioning Theorem
Rj, pj B
Partitioning Relation:
Lifetime bound for
region Rj

42. Step III r B
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: Jargond A
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 Assignment11 6 1 5 2 15 13 12 8 7 B 14 4 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 Assignment11 6 1 5 2 15 13 12 8 7 B 14 4 3 9 10
FRA: 1  5  11  14  B;
2  3  9  14  B

52. Infeasible Role Assignment

53. Enumerating FRAs (Collinear Networks)B 5 4 3 2 1
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 B 1 2 3 4 5 B 1 2 3 4 5

54. Enumerating Candidate FRAsB 5 4 3 2 1
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
Note that FRAs might be repeated (e.g. R2 here)
R5, t7 B 5 4 3 2 1

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, t’0
R2, t’2
R6, t’6
R7, t’7
R9, t’9
R10, t’10
R12, t’12
R14, t’14
R1, t’1
R3, t’3
R5, t’5
R8, t’8
R11, t’11
R13, t’13
R4, t’4
R15, t’15
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 t’ks)?
Solved via Linear Programming:
Objective:
subject to:
[Non-negativity of role time]
[Non-negativity of residual energy]

58. Example dchar dchar/2 dchar/2 B 3 2 1 Min-hop Min-Energy Persistent
0.25
Min-Energy
R0: 0
R1: 0
R2: 1.0
R3: 0
1.0
Persistent
R0: 0.09
R1: 0.23
R2: 0
R3: 1.0
1.32
Optimal
R0: 0
R1:
R2:
R3:
1.38
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!
1  5  2  4  6  B (19%)
1  5  2  B (18%)
1  3  2  6  B (20%)
1  3  2  B (32%)

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)
R1: (3/11)
R2: (3/11)
R3: (5/11) B 3 2 1
1.375 (11/11)
5/11
5/11
3/11
3/11
Network Flow View
3/11 + 3/11
3/11
3/11 + 5/11
f12: 8/11
f13: 3/11
f1B: 0
f23: 3/11
f2B: 5/11
f3B: 6/11 B 3 2 1
5/11
3/11

62. Equivalent Flow Program

63. Extensions to k-of-m SensorsB
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 Sensing time divided equally between 1a and 1b
Single Sensor Lifetime s
3/11
R0: (0)
R1: (3/11)
R2: (3/11)
R3: (5/11) B 3 2 1
1.375 (11/11)
5/11
3/11
2/15
2 Sensor Lifetime s
R0: (2/15)
R1: (5/15)
R2: (8/15)
R3: (0) 1a B 3 2 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 r B Single Source Topology information Degenerate R
Aggregation

67. Extensions to AggregationB 3 2 1
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 Aggregation energy per bit taken as 180 nJ3 2 1
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 (1.596 for 0 nJ/bit, 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 2-Level Flat General 9 8 B 10 3 8 9 1 2 3 4 5 6 711 3 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 P+1 commodities
P-2
P-1 P

71. Constraints Non-aggregating, non-sensing nodes Aggregating 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 r B
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
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
(sA, sB, sC)
T(sA, sB, sC) 1 
Fraction Vector Space
Lifetime Space
1 determines 
2 and  determine number of trials

79. Step VI r R B Single Source Topology information
Resides over R with a certain PDF R

80. 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 …

81. Key: Partitioning R Partition into sub-regions (a through k)1 c B 3 g e f d 2 l j i h k 4 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
Growth of number of regions
For fixed density and r, grows linearly with the number of nodes

83. Step VII r R B Single Moving Source Topology information
Specified Trajectory R

84. Dealing with TrajectoriesB
r(t) R
Is an absolute trajectory feasible?
How can one maximize the lifetime if the trajectory is relative?

85. Simple extension … Calculate fraction of time spent in every regionB 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 SourcesB R
Same strategy as for single source
Time spent in region summed over all sources

87. Recall …
Sensor
Relay
Aggregator
Asleep r B 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