1 Mathematical Modeling and Algorithms for Wireless Sensor Networks Bhaskar Krishnamachari Autonomous Networks Research Group Department of Electrical Engineering-Systems USC Viterbi School of Engineering

2 Wireless Sensor Networks Large scale networks of small embedded devices, each with sensing, computation and communication capabilities. Use of wireless networks of embedded computers “could well dwarf previous milestones in the information revolution” - National Research Council Report: Embedded, Everywhere, Research pioneered at USC/ISI

3 Structural monitoringBio-habitat monitoring Military surveillanceDisaster management Industrial monitoring Note: images used may be copyrighted. Used here for limited educational purposes only. Not intended for commercial or public use. Home/building security Wide Ranging Applications

4 Challenges Scarce energy, low bandwidth Unattended ad-hoc deployment Very large scale High noise and fault rates Dynamic / uncertain environments High variation in application-specific requirements

5 Autonomous Networks Research Group 10 Ph.D. students in EE and CS Primary focus: modeling, analysis, optimization and algorithms for routing and querying in wireless sensor networks. Highlights of ongoing research activities: –experimental studies of wireless link quality –a fundamental theorem concerning random geometric graphs –analysis of routing with compression –linear/non-linear flow optimization formulations of WSN routing –best radio signal strength-based localization technique to date –new querying and search techniques for WSN –algorithms for low latency scheduling and routing

6 1. Impact of Spatial Correlation on Routing with Compression Pattem, Krishnamachari, Govindan, “Impact of Spatial Correlation on Routing with Compression in Wireless Sensor Networks,” IPSN [IPSN ‘04 Best Student Paper Award]

7 Spatial Correlation Model Inter-node spacing d Correlation level c Number of nodes n Entropy of single source H 1 A parameterized expression for the joint entropy of n linearly placed equally spaced nodes

8 Cluster-based Routing with Compression

9 Analysis Data from s nodes is compressed sequentially before routing to the sink. We can derive expressions for the energy cost as a function of the cluster size s: Can then derive an expression for the optimal cluster size as a function of the network size and correlation level:

10 Cluster-based routing + compression Suggests the existence of a near-optimal cluster (about 15) that is insensitive to correlation level!

11 Near-Optimal Clustering Can formalize the notion of near-optimality using a maximum difference metric: We can then derive an expression for the near-optimal cluster size: This is independent of the correlation level, but does depend on the network size, number of sources, and location of the sink. For the above scenario, it turns out s no = 14 (which explains the results shown).

12 Near-Optimal Clustering

13 Summary These results (further extended to 2D scenarios in recent work) indicate that a simple, non-adaptive, cluster-based routing and compression strategy is robust and efficient.

14 2. Delay Efficient Sleep Scheduling Lu, Sadagopan, Krishnamachari, Goel “Delay Efficient Sleep Scheduling in Wireless Sensor Networks,” IEEE Infocom [2005 USC EE-Systems Dept. Best Student Paper Award]

15 Sleep Latency Largest source of energy consumption is keeping the radio on (even if idle). Particularly wasteful in low-data-rate applications. Solution: Globally synchronized duty-cycled sleep-wakeup cycles. E.g. S- MAC (Ye, Heidemann ‘02) Another Problem: increased sleep latency time

16 Setup/Assumptions Each node is assigned one slot out of k to be an active reception slot which is advertised to all neighbors that may have to transmit to it. Nodes sleep on all other slots unless they have a packet to transmit. We assume low traffic so that only sleep latency is dominant and there is low interference/contention.

17 General Problem Formulation The per-hop sleep delay is the difference between reception slots of neighboring nodes Data between any pair of nodes are routed on lowest-delay path between them Goal: assign reception slots to nodes to minimize the worst case end to end delay (delay diameter)

18 DESS Problem Formulation Given a graph G, assign one of k reception slots to each node to minimize the maximum shortest- cost-path delay between any two points in the network

19 NP-Hardness

20 Although problem is NP-hard in general (hence no known polynomial time algorithms), can derive optimal solutions for some special cases with structure Tree: alternate between 0 and k/2. Gives worst delay diameter of dk/2 Ring: sequential slot assignment has best possible delay diameter of (1 - 1/k)*n A constant factor approximation can be obtained in case of the square grid by using the solution for the ring as a building block Special Cases: Tree, Ring

21 Special Case: Grid A solution for the grid is to use an arrangement of concentric rings Can prove that this provides a constant factor approximation

22 Multi-Schedule Solutions If each node is allowed to adopt multiple schedules, then can find much more efficient solutions: Grid: delay diameter of at most d + 8k (create four cascading schedules at each node, one for each direction) Tree: delay diameter of at most d+4k (create two schedules at each node, one for each direction) On general graphs can obtain a O( (d + k)log n) approximation for the delay diameter

23 Summary Sleep schedules should be intelligently designed to enable low-latency routing while maintaining energy efficiency Ongoing work looks at adaptively assigning these schedules depending on current flows in the network (rather than worst-case over all possible flows)