Upper Bounds on the Lifetime of Sensor Networks

1. Upper Bounds on the Lifetime of Sensor Networks Manish Bhardwaj, Timothy Garnett, Anantha Chandrakasan Massachusetts Institute of Technology October 2001

2. Outline • Wireless Sensor Networks • Energy Models • The Lifetime Problem • Bounding Lifetime • Extensions • Summary

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 Key Challenge: Maximizing Lifetime

4. B r Data Gathering Wireless Networks: A Primer Sensor Relay Aggregator Asleep R

5. 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

6. 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 Energy Models

7. Defining Lifetime • Three network states: • Active • Failure • Dormant • Possible lifetime definitions: • Cumulative active time • Cumulative active time to first failure

8. B The Lifetime Bound Problem N nodes, Initial energy E J • Bound the lifetime of a network given: • A description of R and the relative position of the base-station • 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) • Spatial distribution of the source (lsource(x,y)) • Expected source rate (r bps) • Note: Bound is topology insensitive r R

9. 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

10. 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

11. 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

12. 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: 

13. 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

14. 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

15. 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 • Answer depends on: • Distribution of distances • Sophisticated non-linear optimization needed for best multi-hop

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

17. B Take I r A d

18. B Take II r d A d/K

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

20. 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:

21. Simulation Results

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

23. Simulation Results

24. r Source in a Rectangular Region dW A y dB B dW x dN 1000 node network, 2 J on a node has the potential to report finite velocity tank intrusions in a sq. km, a km away for more than 7 years!

25. Simulation Results

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

27. Simulation Results

28. B Bounding Lifetime for Sources in Arbitrary Regions: Partitioning Theorem R1, p1 Probability of residing in a sub-region R5, p5 R2, p2 Sub-region R4, p4 R6, p6 R3, p3 R Partitioning Relation: Lifetime bound for region Rj

29. Work completed subsequently … • Factoring in topology • Factoring in source movement • Factoring in aggregation: • Flat aggregation • 2-step hierarchical • Non-constructive approaches don’t seem to work here • Bounds derived by actual construction of the optimal strategy • Strategy (and hence bound) can be derived in polynomial time

30. Summary • Maximizing network lifetime is a key challenge in wireless sensor networks • Using simple arguments based on minimum-energy relays and energy conservation, it is possible to derive tight or near-tight bounds on the lifetime of sensor networks • It is possible to derive extremely sophisticated bounds that factor in the exact graph topology, source movement and aggregation