1. CENS, April 20 20071 Modeling Wireless Sensor Networks Bhaskar Krishnamachari Ming Hsieh Department of Electrical Engineering USC Viterbi School of Engineering

2. 2 Overview Mathematical modeling provides fundamental insights into: 1.Link layer behavior 2. Protocol design 3. Scaling and architecture

3. 3 1. Modeling Link Layer Behavior in 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. 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. 5 Link Quality Variation with Distance From Woo et al. ‘03

6. 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. 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. 8 Expectation and Variance of Packet Reception Rate Justifies the presence of “long links”

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

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

11. 11 SINR-view of Interference

12. 12 Feasibility of Concurrent Transmissions P 1 g 11 /(P 2 g 21 +N) ≥  P 2 g 22 /(P 1 g 12 +N) ≥  S1R1 R2S2 g11 g12 g21 g22

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14. 14 Linear Topology Case Counter-intuitive “embedding” of simultaneous conversations S1R1R2S2

15. 15 2. MAC Design for Scalable Data 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.

16. 16 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

17. 17 p-persistent CSMA Model … : Idle Slot : Collision Slot : Successful Slot epoch … … … ……

18. 18 Delay and Energy Expressions 1.Average expected epoch delay 2.Average expected epoch energy consumption ξ R : Energy Consumption per node per time slot in the Receive State ξ T : Energy Consumption per node per time slot in the Transmit State

19. 19 Optimality Delay Energy If ξ R = ξ T, the same transmission probability optimizes both delay and energy simultaneously.

20. 20 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

21. 21 Receiver Feedback Enhancement Receiver performs measurement and broadcast Window update rule: All contending nodes change the window size simultaneously

22. 22 Results

23. 23 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.

24. 24 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

25. 25 Receiver capacity interference model – source rates from node’s sub-tree and its interfering neighbors’ sub- trees must not exceed available bandwidth Problem Formulation

26. 26 Validating Capacity Model

27. 27 Bottleneck rate turns out to be the minimum supply/demand ratio: This can be calculated easily given the tree, interference graph, and receiver bandwidths P1: Solving for Fairness

28. 28 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

29. 29 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 sum-price

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31. 31 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… Subgradient Optimization

32. 32 Good News Numerical evidence: setting all shadow prices to 1 provides near-optimal solutions in one iteration!

33. 33 Resulting Heuristic 1. First determine and allocate min rate to all sources 2. Give rank to each source that is inversely proportional to the number of downstream receivers whose bandwidth it consumes; 3. Allocate saturating rates to flows, in rank order

34. 34 Simulation Results CDF of difference from optimal solution

35. 35 Ongoing Work Test-bed Implementation Cross-layer extensions

36. 36 4. Fundamental Scaling Laws for 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.

37. 37 Race between increasing supply and demand: - Energy and storage - Application-specific event and query traffic The winner of this race determines scalability. In a Nutshell

38. 38 N nodes deployed in a 2D area with constant density for time T m atomic events and q i queries for the i th event, all uniformly distributed Can create r i replicas for event i to reduce search cost (at the expense of increased replication cost) Preliminaries

39. 39 Data-Centric Querying Approaches Unstructured: expanding ring searches, random walks. Structured: Geographic Hash Table, DIFS, DIM

40. 40 Energy Cost Scaling C replication = c 1 r : # of copies of an event N : # of nodes C search ( unstructured ) = c 2 C search ( structured ) = c 3 EVENTEVENT REPLICATIONUNSTRUCTURED QUERYSTRUCTURED QUERY

41. 41 Energy Optimization Formulation S : total storage size m : the total number of events q i : the query rate for i th event r i : the number of copies of i th event C s (r i ) : the expected minimum search cost of i th event C r (r i ) : the expected replication cost of i th event C r (r) = c 1 C s (r) = c 2

42. 42 Optimization Solution Minimizer The Optimized Total Cost (inactive constraint) (active constraint) q i : # of queries for event i N : # of nodes S : total storage size m : # of events

43. 43 Optimal Total Cost Simplified, assuming : q : # of queries per event N : # of nodes S : total storage size m : # of events if

44. 44 Illustration of Energy Scaling m : # of events q : # of queries per event

45. 45 I - Storage and Energy Scalability Results Energy Condition The energy requirement per node is bounded if and only if mq 1/2 = O(N 1/4 )  Energy constraint is stricter than storage constraint m : # of events q : # of queries per event N : # of nodes Storage Condition A network scales efficiently with bounded storage per node if mq 1/2 = o(N 3/4 )

46. 46 II - Fixed Energy Budget Results S – successful operation region N : # of nodes e: per-node energy budget

47. 47 III - Network Lifetime Scaling Results Network Lifetime as a function of Network Size

48. 48 Summary Only certain classes of applications can be sustained in arbitrarily large sensor networks. Specifically, if mq 1/2 = O(N 1/4 ) for unstructured networks, and mq 2/3 = O(N 1/2 ) for structured networks: a.The network can operate with bounded energy and storage per node. b.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.

49. 49 Final Thoughts “In theory, theory and practice are the same; in practice, they’re different.”

50. 50 Thanks