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Mathematical Modeling and Algorithms for Wireless Sensor Networks Bhaskar Krishnamachari Autonomous Networks Research Group Department of Electrical Engineering-Systems USC Viterbi School of Engineering http://ceng.usc.edu/~anrg 1 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, 2001. • Research pioneered at USC/ISI 2 Wide Ranging Applications Structural monitoring Disaster management Bio-habitat monitoring Military surveillance Industrial monitoring Home/building security 3 Note: images used may be copyrighted. Used here for limited educational purposes only. Not intended for commercial or public use. 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 4 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 5 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 2004. [IPSN ‘04 Best Student Paper Award] 6 Spatial Correlation Model Entropy of single source H1 Number of nodes n A parameterized expression for the joint entropy of n linearly placed equally spaced nodes Inter-node spacing d Correlation level c 7 Cluster-based Routing with Compression 8 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: 9 Cluster-based routing + compression 10 Suggests the existence of a near-optimal cluster (about 15) that is insensitive to correlation level! 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 sno = 14 (which explains the results shown). 11 Near-Optimal Clustering 12 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. 13 2. Delay Efficient Sleep Scheduling Lu, Sadagopan, Krishnamachari, Goel “Delay Efficient Sleep Scheduling in Wireless Sensor Networks,” IEEE Infocom 2005. [2005 USC EE-Systems Dept. Best Student Paper Award] 14 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. SMAC (Ye, Heidemann ‘02) • Another Problem: increased sleep latency time 1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6 15 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. 16 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) 17 DESS Problem Formulation Given a graph G, assign one of k reception slots to each node to minimize the maximum shortestcost-path delay between any two points in the network 18 NP-Hardness 19 Special Cases: Tree, Ring 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 2 1 0 1 0 2 2 1 0 • Ring: sequential slot assignment has best possible delay diameter 20 of (1 - 1/k)*n 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 21 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 22 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) 23