1. Finite-Horizon Energy Allocation and Routing Scheme in Rechargeable Sensor Networks Shengbo Chen, Prasun Sinha, Ness Shroff, Changhee Joo Electrical and Computer Engineering & Computer Science and Engineering

2. 2 Introduction [Rechargeable sensor network] Environment monitoring Earthquake, structural, soil, glacial Unattended Operability for long periods Battery with renewable energy (like solar or wind) Challenge: energy allocation Sensor Network without replenishment: full battery is desirable feature Sensor Network with replenishment: no opportunity to harvest energy

3. 3 Introduction(cont’) [Rechargeable sensor network] M r(t)B(t)e(t)B(t+1) M: Battery size B(t): Battery level at time slot t e(t): allocated energy at time slot t r(t): harvested energy at time slot t

4. 4 Motivation Rate-power function Nondecreasing and strictly concave Data transmission with spending units of energy How to design

5. 5 Motivation(cont’) Example 1: r(1)=4, r(2)=2, r(3)=0 e*(1)=2, e*(2)=2, e*(3)=2 Average replenishment rate is the best because of Jensen’s inequality r(1)r(2)

6. 6 Motivation(cont’) Example 2: r(1)=2, r(2)=0, r(3)=4 r(1)r(3) Average replenishment rate is infeasible r(2)

7. 7 Problem Statement Sensor Network with renewal energy Assumption No interference from other nodes Problem: throughput maximization where, is the amount of data from source to the destination at time slot t

8. 8 Problem Statement (cont’) Convex optimization problem Joint energy allocation and routing Complex due to the “time coupling property” Concave rate-power function

9. 9 Related Literatures Finite horizon A. Fu, E. Modiano and J. Tsitsiklis, 2003. Dynamic programming Infinite horizon L. Lin, N. B. Shroff, and R. Srikant, 2007 Asymptotically optimal competitive ratio M. Gatzianas, L. Georgiadis, and L. Tassiulas, 2010. Maximize a function of the long-term rate per link L. Huang, Neely Asymptotically optimal

10. 10 Three-step Approach One node with full knowledge of replenishment profile One node with estimation of replenishment profile Multiple-node network

11. 11 Three-step Approach One node with full knowledge of replenishment profile One node with estimation of replenishment profile Multiple-node network

12. 12 One node with full knowledge of replenishment profile Finite time horizon: T time slots Assumption: replenishment profile is known Constraints: Cumulative used no greater than cumulative harvested Residual no greater than the battery size

13. 13 Result 1 Theorem 1: The energy allocation scheme, satisfying s(t) = S(t) − S(t − 1), is the optimal energy allocation scheme Shortest path S(t): curve that connects two points (0, 0) and (T,K) in the domain D with least Euclidean length time Cumulative Energy R(t) R(t)-M T K D

14. 14 Three-step Approach One node with full knowledge of replenishment profile One node with estimation of replenishment profile Multiple-node network

15. 15 One node with estimation of replenishment profile Assumption relaxed Replenishment profile is unknown Estimation replenishment rate Actual replenishment rate

16. 16 Online algorithm Theorem 2: The throughput U of the online algorithm, achieves fraction of the optimal throughput time Cumulative Energy R(t) T K (1-β 1 )R(t) (1+β 2 )R(t) 1. Calculate e(t) from the lower-bound of the estimated replenishment profile by the shortest- path solution 2. The allocated energy is determined as e(t) = e(t) + r(t) − r(t)

17. 17 Three-step Approach One node with full knowledge of replenishment profile One node with estimation of replenishment profile Multiple-node network

18. 18 Heuristic scheme: NetOnline Throughput maximization Decouple energy allocation and routing ： Energy allocation of each node follows the online algorithm Routing:

19. 19 Result 3 Theorem 3: The heuristic scheme is optimal if all nodes have the same replenishment profile and perfect estimation.

20. 20 Simulations

21. 21 Simulations (cont’)

22. 22 Simulations (cont’) NRABP: Infinite-horizon based scheme in Gatzianas’s paper

23. 23 Future work Considering interference in the model Replenishment rate is known with some distribution, what is the best strategy? Infinite horizon but only finite period of estimation

24. 24