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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Wide Ranging Applications
Structural monitoring
Disaster management
Bio-habitat monitoring
Military surveillance
Industrial monitoring
Home/building security
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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
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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:
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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
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
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Cluster-based
Routing with Compression
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Analysis
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Data from s nodes is compressed sequentially before routing to the sink.
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We can derive expressions for the energy cost as a function of the cluster
size s:
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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
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Suggests the existence of a near-optimal cluster (about 15) that is insensitive to correlation level!
Near-Optimal Clustering
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Can formalize the notion of near-optimality using a maximum difference
metric:
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We can then derive an expression for the near-optimal cluster size:
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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
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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
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Largest source of energy consumption is keeping the radio on (even if idle).
Particularly wasteful in low-data-rate applications.
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Solution: Globally synchronized duty-cycled sleep-wakeup cycles. E.g. SMAC (Ye, Heidemann ‘02)
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Another Problem: increased sleep latency
time
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Setup/Assumptions
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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.
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Nodes sleep on all other slots unless they have a packet to transmit.
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We assume low traffic so that only sleep latency is dominant and there is
low interference/contention.
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General Problem Formulation
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The per-hop sleep delay is the difference between reception slots of
neighboring nodes
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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 shortestcost-path delay between any two
points in the network
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NP-Hardness
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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
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• Ring: sequential slot assignment has best possible delay diameter
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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
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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
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Sleep schedules should be intelligently designed to enable low-latency
routing while maintaining energy efficiency
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Ongoing work looks at adaptively assigning these schedules depending on
current flows in the network (rather than worst-case over all possible flows)
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