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Modeling Wireless Sensor Networks

Modeling Wireless Sensor Networks. Bhaskar Krishnamachari Ming Hsieh Department of Electrical Engineering USC Viterbi School of Engineering. Overview. Mathematical modeling provides fundamental insights into: Link layer behavior 2. Protocol design 3. Scaling and architecture .

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Modeling Wireless Sensor Networks

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  1. Modeling Wireless Sensor Networks Bhaskar Krishnamachari Ming Hsieh Department of Electrical Engineering USC Viterbi School of Engineering CENS, April 20 2007

  2. Overview • Mathematical modeling provides fundamental insights into: • Link layer behavior 2. Protocol design 3. Scaling and architecture

  3. 1. Modeling Link Layer Behaviorin Low Power Wireless Networks Marco Zuniga, Bhaskar Krishnamachari, "An Analysis of Unreliability and Asymmetry in Low-Power Wireless Links", ACM Transactions on Sensor Networks, accepted to appear, 2007. Dongjin Son, Bhaskar Krishnamachari, John Heidemann, “Experimental Analysis of Concurrent Packet Transmissions in Low-Power Wireless Networks”, Sensys ’06 + Ongoing work

  4. Two simplified models form the basis of >95% of the literature on wireless networks: X Circular radio range with perfect reception within & zero reception outside Collision with simultaneous transmissions within range

  5. Link Quality Variation with Distance From Woo et al. ‘03

  6. An Explanatory Model • Basic idea: compose the following two functions (a) SNR versus distance with (b) PRR versus SNR Marco Zuniga, Bhaskar Krishnamachari, "An Analysis of Unreliability and Asymmetry in Low-Power Wireless Links", ACM Transactions on Sensor Networks, accepted to appear, 2007.

  7. Bimodal PRR Distribution • A majority of the links are either good (above 90%) or bad (below 10%), matching empirical findings (e.g., Cerpa et al. ’05)

  8. Expectation and Varianceof Packet Reception Rate Justifies the presence of “long links”

  9. Models Incorporated into simulators: • TOSSIM (Berkeley) • Castalia (NICTA, Australia) • Standalone code at http://ceng.usc.edu/~anrg/downloads.html

  10. Concurrent Transmissions Reality: SINR makes the difference X Conservative protocol assumption: always a collision Son, Krishnamachari, Heidemann, “Experimental Analysis of Concurrent Packet Transmissions in Low-Power Wireless Networks”, Sensys ‘06

  11. SINR-view of Interference

  12. g11 S1 R1 g12 g21 S2 R2 g22 Feasibility of Concurrent Transmissions P1g11/(P2g21+N) ≥  P2g22/(P1g12+N) ≥ 

  13. Counter-intuitive “embedding” of simultaneous conversations Linear Topology Case S2 S1 R1 R2

  14. 2. MAC Design for ScalableData Collection Kiran Yedavalli, Bhaskar Krishnamachari, "Enhancement of the IEEE 802.15.4 MAC Protocol for Scalable Data Collection in Dense Sensor Networks", USC Computer Engineering Technical Report CENG-2006-14, November 2006.

  15. State of the Art: IEEE 802.15.4 • Specifies both PHY and MAC layers for low-power, low-rate embedded wireless networks. • The MAC protocol is a slotted CSMA with binary exponential back-off • 256 nodes allowed by standard

  16. p-persistent CSMA Model epoch … … … … … … : Idle Slot : Collision Slot : Successful Slot

  17. Delay and Energy Expressions • Average expected epoch delay • Average expected epoch energy consumption ξR: Energy Consumption per node per time slot in the Receive State ξT: Energy Consumption per node per time slot in the Transmit State

  18. Optimality • Delay • Energy If ξR = ξT, the same transmission probability optimizes both delay and energy simultaneously.

  19. A Useful Optimality Criterion When the number of contending nodes is high, this provides sensitive feedback that can be used to adapt the access rate

  20. Receiver Feedback Enhancement • Receiver performs measurement and broadcast • Window update rule: • All contending nodes change the window size simultaneously

  21. Results

  22. 3. Fair and Efficient Rate Control for Data Gathering Avinash Sridharan and Bhaskar Krishnamachari, "Maximizing Network Utilization with Max-Min Fairness in Wireless Sensor Networks," to be presented at 5th Intl. Symposium on Modeling and Optimization in Mobile, Ad Hoc, and Wireless Networks (WiOpt), April 2007.

  23. Problem Formulation • Allocate rates to each source to (a) ensure fairness, and (b) efficient use of available bandwidth. • Closely related prior work by Rangwala et al. SIGCOMM ’06 – focuses primarily on fairness and proposes a TCP-like AIMD mechanism

  24. Problem Formulation • Receiver capacity interference model – source rates from node’s sub-tree and its interfering neighbors’ sub-trees must not exceed available bandwidth

  25. Validating Capacity Model

  26. P1: Solving for Fairness • Bottleneck rate turns out to be the minimum supply/demand ratio: • This can be calculated easily given the tree, interference graph, and receiver bandwidths

  27. P2: Solving for Efficiency • Duality-based approach based on the classic work on optimization flow control by Low & Lapsley ’99 • Introduce new dual variables (shadow prices) that weigh resource constraints • Yields distributed algorithms with market auction interpretation

  28. sum-price Structure of P2’s Lagrange Dual • Each router sets a price for its bandwidth • The rate for each source depends on sum-price of routers affected by its flow

  29. Subgradient Optimization • Increment the shadow prices in the direction of the negative sub-gradient (determined by source rates) • Choose source-rates to maximize component function (determined by shadow prices) • In general, this could be a very slow iterative process…

  30. Good News • Numerical evidence: setting all shadow prices to 1 provides near-optimal solutions in one iteration!

  31. Resulting Heuristic • First determine and allocate min rate to all sources • Give rank to each source that is inversely proportional to the number of downstream receivers whose bandwidth it consumes; • Allocate saturating rates to flows, in rank order

  32. Simulation Results CDF of difference from optimal solution

  33. Ongoing Work • Test-bed Implementation • Cross-layer extensions

  34. 4. Fundamental Scaling Lawsfor Store and Query Sensor Networks Joon Ahn and Bhaskar Krishnamachari, "Fundamental Scaling Laws for Energy-Efficient Storage and Querying in Wireless Sensor Networks", ACM MobiHoc, May 2006.

  35. In a Nutshell • Race between increasing supply and demand: - Energy and storage - Application-specific event and query traffic • The winner of this race determines scalability.

  36. Preliminaries • N nodes deployed in a 2D area with constant density for time T • m atomic events and qi queries for the ith event, all uniformly distributed • Can create ri replicas for event i to reduce search cost (at the expense of increased replication cost)

  37. Data-Centric Querying Approaches • Unstructured: expanding ring searches, random walks. • Structured: Geographic Hash Table, DIFS, DIM

  38. Energy Cost Scaling • Creplication = c1 • Csearch(unstructured) = c2 • Csearch(structured) = c3 STRUCTURED QUERY UNSTRUCTURED QUERY EVENT REPLICATION EVENT r : # of copies of an event N : # of nodes

  39. Cr(r) = c1 Cs(r) = c2 Energy Optimization Formulation S: total storage size Cs(ri): the expected minimum search cost of ith event Cr(ri): the expected replication cost of ith event m: the total number of events qi: the query rate for ith event ri: the number of copies of ith event

  40. (inactive constraint) (active constraint) Optimization Solution Minimizer qi : # of queries for event i N : # of nodes S : total storage size m : # of events The Optimized Total Cost

  41. if if Optimal Total Cost Simplified, assuming : q : # of queries per event N : # of nodes S : total storage size m : # of events

  42. Illustration of Energy Scaling m : # of events q : # of queries per event

  43. I - Storage and Energy Scalability Results Storage Condition A network scales efficiently with bounded storage per node Energy Condition The energy requirement per node is bounded if and only if mq1/2 = O(N1/4) m : # of events q : # of queries per event N : # of nodes ifmq1/2 = o(N3/4) • Energy constraint is stricter than storage constraint

  44. II - Fixed Energy Budget Results N : # of nodes e: per-node energy budget S – successful operation region

  45. Network Lifetime as a function of Network Size III - Network Lifetime Scaling Results

  46. Summary • Only certain classes of applications can be sustained in arbitrarily large sensor networks. • Specifically, if mq1/2= O(N1/4)for unstructured networks, and mq2/3 = O(N1/2) for structured networks: • The network can operate with bounded energy and storage per node. • The network lifetime does not decrease with network size for a given energy budget. • The results can be reinterpreted to understand how to tier sensor networks into zones with localized queries • These results generalize in a straightforward manner to 1D and 3D deployments. 3D deployments are inherently more scalable.

  47. Final Thoughts “In theory, theory and practice are the same; in practice, they’re different.”

  48. Thanks

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