Bhaskar Krishnamachari Electrical Engineering-Systems University of Southern California

1. Analysis of Energy-Efficient, Fair Routingin Wireless Sensor Networksthrough Non-linear Optimization Bhaskar Krishnamachari Electrical Engineering-Systems University of Southern California Los Angeles, CA 90089 Email: bkrishna@usc.edu Fernando Ord´o˜nez Industrial Systems Engineering University of Southern California Los Angeles, CA 90089 Email: fordon@usc.edu Workshop on Wireless Ad hoc, Sensor, and Wearable Networks, in IEEE Vehicular Technology Conference, October 2003

2. Outline • Introduction • Modeling and Problem Formulation • Experiments and Results • Conclusions

3. Introduction • In the area of WSN there is a significant gap between theory and practice. • Convex non-linear optimization techniques • We treat end-to-end fairness by placing constraints on the maximum percentage of the total information to the sink that each source node can send. • Thus, in a completely fair WSN every node contributes the same percentage of the total information to the sink, and in a completely unfair WSN all the information to the sink could be from a single source node.

4. Modeling and Problem Formulation • n :the number of sensor nodes • :limited energy • :maximum source rate • :the information flow rate on the link between i and j • :transmission power on the link between nodes i and j

5. Modeling and Problem Formulation • C :the per-bit rate reception power for each node • :the noise in the communication channel • :the physical distance between nodes i and j the total amount of data • :that node i sends to a certain fairness proportion of the total information sent to the sink

6. Maximizing Information Extraction

7. Maximizing Information Extraction • Constraint (1a) ensures that for all nodes except the sink, the out-flow from each node is at least as large as the in-flow. • Constraint (1b) is the maximum source rate constraint and says that the outflow from any node is always less than its in-flow plus the maximum rate at which the node obtains information. • Constraint (1c) is the fairness constraint. • Constraint (1d) is the energy constraint.

8. Maximizing Information Extraction • Constraint (1e) is essentially Shannon’s capacity theorem, which relates the maximum flow on a particular channel to the SNR (signal to noise ratio) on that channel. • constraint (1f) ascribes the non-negativity property to the pairwise flow and transmission power variables • LOQO: is a software package for solving general (smooth) nonlinear optimization problems

9. Maximizing Information Extraction

10. Minimizing Energy Usage

11. Minimizing Energy Usage • Constraint (2d) on the total information obtained by the sink

12. Experiments and Results • a sink and four sensor nodes in line (0,0) 1(1,0), 2(2,0), 3(3,0) and 4(4,0) • Uniform fairness constraint across all nodes

13. Experiments and Results

14. Experiments and Results • Problem 1: • Problem 2:

15. Experiments and Results

16. Experiments and Results • The first observation is that the total flow increases monotonically with α. • This is intuitive: as the fairness indicatorαincreases, the fairness constraint is being relaxed, allowing for higher overall flow. • E4 and E6 (α = 0.5) • E1 and E5 overlap ( Energy Constraints) • E2 and E3 coincide with E1(fairness constraints) • E6 • E4 ( the worst case )

17. Experiments and Results

18. Experiments and Results • E3 and E5 (α = 0.3 ) • A critical threshold- this represents the point at which the fairness constraint is no longer tight and the information constraint starts to dominate.

19. Conclusions • We found that higher fairness constraints can result in significant decrease in information extraction and higher energy usage.

20. Future Work • Aggregation • Sensor deployment/placement