1 mathematical modeling and algorithms for wireless sensor networks bhaskar krishnamachari...
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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
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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
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Structural monitoring Bio-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
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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
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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
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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]
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Spatial Correlation Model
Inter-nodespacing d
Correlationlevel c
Number ofnodes n
Entropy of single source H1
A parameterized expression for the joint entropy of n linearly placed equally spaced nodes
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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:
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Cluster-based routing + compression
Suggests the existence of a near-optimal cluster (about 15) that is insensitive to correlation level!
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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).
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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.
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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]
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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
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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.
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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)
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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
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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
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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
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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