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. B r 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 (< 100m) • Spatial Density • 0.1 nodes/m2 to 20 nodes/m2 • Node Requirements • Small Form Factor • Required Lifetime: > year

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

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

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

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

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

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

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

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

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

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

16. B r 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 • 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 – Source Sink K-1 nodes available • Problem: Find a solution that minimizes the measure

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

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

20. B A Optimal Solution 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 • It is not possible to relay bits from A to B at a rate r using total link power less than: Source Sink 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 d2behavior Energy/bit Perfect power control d4behavior 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 • Complex path loss behavior • Not a problem! • Energy/bit can be made linear • Equal hops still best strategy • But … Dchar varies with distance

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. N nodes available B Maximizing Lifetime r • Problem: Using N nodes what is maximum sensing lifetime one can ever hope to achieve? A d

25. B Take I r A d

26. B Take II r d A d/K

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

28. r A d B 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 xk-1 xk+1 x1 xk … … • Assumption: E, T chosen such that k >> 1

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

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

33. Bounding Strategy

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

35. B r Source Moving Along A Line S0 S1 dN A dW d(x) dB

36. Simulation Results

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

38. Simulation Results

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

40. Simulation Results

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

42. B r 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. r A d B 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. B Feasible Role Assignment 11 1 6 2 5 15 12 13 7 8 14 4 3 9 10 FRA: 1  5  11  14  B

46. B Infeasible Role Assignment (Redundant)

47. B Infeasible Role Assignment (Invalid)

48. B Infeasible Role Assignment (Invalid)

49. B Infeasible Role Assignment (Invalid)

50. B Infeasible Role Assignment (Redundant)