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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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Cluster-based Routing with Compression

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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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Near-Optimal Clustering

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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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NP-Hardness

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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

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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)