Information transfer in wireless networks for distributed sensing and control

1. Information transfer in wireless networks for distributed sensing and control Sanjeev Kulkarni, P. R. Kumar, John Tsitsiklis, and Sergio Verdu Dept. of Electrical and Computer Engineering, and Coordinated Science Lab University of Illinois, Urbana-Champaign SensorWeb MURI Review Meeting, Sep 22, 2003 Phone 217-333-7476, 217-244-1653 (Fax) Email prkumar@uiuc.eduWeb http://black.csl.uiuc.edu/~prkumar MURI Review: SensorWebData Fusion in Large Arrays of Microsensors Sep 22, 2003

2. Sensor web networks • Networks with large numbersof sensors • Potentially large number ofinformation gathering nodes • Connected by wireless medium • Possibly low power nodes • IT-3: Wireless networks,Network communication andinformation theory • RCA 2&3: Fundamental limits on fusion,Network Info Theory, Tradeoffs in local vs. global processing

3. The oncoming convergence • Sensor networks • Nodes can sense • Nodes can compute • Nodes can communicate • Also can actuate • ~ 1950 — 2000 and continuing:Substantial progress in several individual disciplines • Computation: ENIAC (1946), von Neumann (1944), Turing,.. • Sensing and inference: Fisher, Wiener (1949),… • Actuation/Control: Bode, Kalman (1960),… • Communication: Shannon (1948), Nyquist,… • Signal Processing: FFT, Cooley-Tukey (1965),… • ~ 2000 — onwards • A gradual fusion of all these fields • Example: Increasingly hard to separate signal processing from communication • Larger grand unification of sensing, inference, communication and computation

4. Outline • Problems at the interfaces of communication, computation and inference • Harvesting statistics from a sensor field • Towards a general problem • Energy limited communication over fading channels • Optimal usage of energy limited nodes • Networks of nodes with fading channels • Throughput of wireless/hybrid networks

5. Problems at the interface

6. Harvesting statistics from a sensorfield: Communication and computing • Example: • n sensors with computation and wireless communication capabilities • Sensor i has a measurement xi • Compute the mean of the set of sensor measurements • More generally • Calculate a symmetric function f(x1, x2, … , xn) • Data-centric paradigm: Identity of sensor not relevant, only the value • Includes most statistical functions of interest

7. Model for fusion over a sensor network • Observations are generated periodically at sensors with some frequency • Observations belong to some fixed finite set • Protocol model of wireless communication • Packet successfully received if there is no nearby interfering transmission • Or if SINR is above a threshold • A symmetric function of each set of observations must be communicated to some fixed node designated as the fusion center • What is the maximum rate of sensor fusion?

8. Preliminary result (GK ‘03) • Key idea: Function value depends only on the type of the observation sequence • Type of a sequence is the vector of frequencies of each value • Theorem: The maximum frequency at which types can be gathered at a fusion node in a random network is • Outline • Type can be represented by O(log n) bits • Tessellate by cells with log n nodes in each cell • Gather data at a local fusion center cum relay node in each cell • Takes O(log n) time • Daisy chain cumulative types from cell to cell • Takes O(log n) time • Pipeline scheduling

9. Specific functions of interest • The Mean of the sensor readings (GK ‘03) • Converse is also true • Mean requires log n bits of information • So the maximum frequency of sensor fusion is indeed • Strategy is Tessellation and Daisy chaining • The Maximum of the sensor readings (GK ‘03) • This does not require full knowledge of types • If information can be conveyed by collisions, then Max can be computed at frequency O(1) • Optimal strategy is: Scheduled Broadcast • Nodes with Largest possible broadcast • Then second largest • … • What can we say in general?

10. ≥ ≥ ≥ Towards a system theory for inference over sensor networks • Hierarchy of problems • Fusion over a wired network • Nodes have correlated observations • Distinct non-interfering links • The fusion node needs to gather all the information • Slepian-Wolf Result: The rate region is: X Y Z

11. Ho Additional complexities ofinference over wireless networks • Wireless nodes • There are no independent links: Sources share channel • Multiple access problem • Source-channel separation does not work • Points not in the intersection of the Slepian-Wolf rate region and multiple access channel rate region may be achievable • Also, sensors can communicate with each other and thus cooperate • Also nodes need not know the “hidden hypothesis” which is to be inferred • And the number of nodes may be large • Little is known at present • Some ongoing work with possibly some new results at our next meeting

12. Energy limited communication

13. Fading channel Optimal energy allocationwith fading channel • How many bits can be transferred over a fading channel when the source has a fixed amount of energy? • Elements of problem • Channel changes randomly • The transmitter has a fixed amount of energy • Energy constraint - not power • Though power can also be constrained • There is a time deadline • Questions of interest • Given the current channel state, should we use the channel or wait for a better channel? • If we do use the channel, how much energy should we use? • How many bits can be transferred with the given energy before the deadline?

14. Optimal energy allocationwith fading channel • Model • Energy Constrained Transmitter • Finite amount of total available energy • Fading channel • At each time, the channel can be in a different “state” (channel quality) • When channel is in good state, more data can be sent per unit of energy • May want to delay transmission when channel is bad • Maximizing data throughput • Given a certain amount of energy, schedule transmissions to maximize the amount of data transmitted within a given deadline • Can be viewed as maximizing the “capacity” of the channel, subject to energy limitations • Minimizing energy consumption • Given a certain amount of data that must be sent by a deadline, schedule transmissions to minimize the amount of energy consumed • Technical approach: Dynamic Programming (DP) • Alvin Fu, Eytan Modiano, and John N. Tsitsiklis, “Optimal Energy Allocation for Delay-Constrained Data Transmission over a Time-Varying Channel”, Infocom 2003

15. DP formulation(Throughput maximization) • Formulation DP Recursion:

16. DP solution • Optimal policy characterized by thresholds: transmit when channel quality exceeds a threshold • Threshold depends on time and available energy • Efficiently computed • Intuition • Save energy for timeslots that are of good quality • As we get closer to deadline, threshold for spending energy decreases • When available energy is smaller, threshold is higher

17. Example: Throughput maximization (Rayleigh fading) N = 50 slotsto send data Etotal = 95 units P = 10

18. Example (continued) Optimal policy (dynamic threshold) vs. fixed threshold policy

19. Networks with limited energy nodes formed over fading channels

20. The energy cost of transporting information over a wireless network • What is the energy cost of information transport over a non-fading channel? • How many joules does a bit-meter require? • Are there fundamental requirements on the required energy consumption? • Yes (Xie & K ‘02) • Bits x Meters ≤ C · Joules • C = The minimum energy requiredto transport one bit one meter • C is a function of the attenuation properties of the medium

21. AWGN Attenuation over the distance from i to j Fading Transmission of node i Networks over fading channels • Node j’s reception • How much information can be transmitted over such networks?

22. The case of relatively high attenuation (Xue and K ‘03) • If path loss exponent d > 3 • Then (XK ‘03) • c1n ≤ Total information transmitted over the network ≤ c2n • The optimal way to transport energy to save energy is multihop transport from node to node • Ongoing work • When path loss exponent is low, then coherent cooperation can sometimes achieve huge transfers for low energy - but that is difficult under fading

23. A deterministic approach to wireless network capacity (KV ‘02)

24. A Deterministic Approach • Don’t introduce randomness at the outset. • Understand key issues from a deterministic viewpoint. • Recover random results as special cases. • Advantages • Intuition. • More tractable approach. • Stronger results. • Framework for further extensions.

25. Good and Bad Arrangements • If users are too concentrated, shouldn’t expect good throughput. • “Effective” area is quite small. • Too much interference for each bit-meter. • Good arrangements should have the users spread out in some sense. • How can we quantify this and exploit good arrangements?

26. Squarelets and Conditions • Assume source destination distances grow as O(n). • Split area into smaller “squarelets” of size • is such that no squarelet is empty. • is the max number of nodes in any squarelet. Nodes in squarelets sufficiently far away (depending on Δ) can transmit simultaneously without interference.

27. Scheduling Algorithm • Visit each equivalence class of squarelets, schedule nodes one after another until all nodes with packets to transmit/relay are done. • When a node is scheduled, it sequentially transmits all packets being relayed plus one new packet (if it’s a source). • If final destination is in same squarelet or one of four neighboring squarelets, transmit directly to final destination. • Otherwise, transmit to a node (for relay) in one of four neighboring squarelets • Which neighbor? How to avoid congestion?

28. A Result fromParallel/Distributed Computing • k x k permutation routing (each processor is source and destination of exactly k packets). • Array of j x j processing units, that can transmit to/receive from its 4 neighbors simultaneously. • How to route packets with minimal steps and queue lengths? • Theorem 1 (Kunde ’91, et at.) k x k routing in an j x j mesh can be performed deterministically in 1/2jl + o(kj) steps with maximum queue size k. (Further, every routing algorithm takes at least 1/2 kj steps.)

29. Wireless Throughput Results • Theorem 2 A throughput bit-meters/sec is achievable. • Special Cases: • Nodes “evenly spread”: Throughput W(√n). • Iid Random Nodes: Throughput • Users on Roads: Throughput • Users in Neighborhoods: Throughput

30. Heterogeneous/Hybrid Networks: A Possible Model • Fixed geographic area, slotted time. • n wireless nodes as before. • Wired Infrastructure: • access points at fixed locations • access points provide interface between wired and wireless realms • Wireless side: each access point is just another wireless node • Wired side: Packets that enter one access point are ready for exit at any other access point in the next time slot. • Queues may be necessary for entry to and exit from wired infrastructure. • Throughput of such a network?

31. Throughput Results • Squarelet structure and assumptions as before. • Create (overlay) cells of size such that every cell has an access point (base station). • Upper bound: Throughput is no more than • Lower Bound: Can achieve throughput • Upper and lower bounds match in various settings. • In various settings, a wired infrastructure helps only if and base stations are no more than apart.

32. Other events • U. S. Army interactions • Panel Member ONR/ARL Workshop on Sensor Networks: Theory and Military Application. Aug 27, 2003, Cornell University, Ithaca, NY. • Plenary Talks • NCCR Annual Workshop On Mobile Information and Communication Systems Annual Workshop 2003, February 13, 2003, Zurich, Switzerland. • WiOpt'03: Modeling and Optimization in Mobile and Ad Hoc and Wireless Networks , March 3 - 5, 2003, Sophia-Antipolis, France. • PWC 2003: The Eighth International Conference on Personal Wireless Communications, September 23-24, 2003, Venice, Italy. • IEEE TENCON'2003, October 15-17, 2003, Bangalore, India • Invited Talks • International Workshop on Stochastic Models and IV International Workshop on Retrial Queues, Cochin, India, December 17-21, 2002. • IUTAM Symposium on Nonlinear Stochastic Dynamics, Allerton Park, Monticello, Illinois, USA, August 26-30, 2002. • IEEE Information Theory Workshop, Bangalore, India, October 20-25, 2002. • DIMACS Workshop on Network Information Theory , March 17 - 19, 2003, Rutgers University, Piscataway, NJ. • The 2nd International Workshop on Information Processing in Sensor Networks (IPSN '03), , April 22-23, 2003, Palo Alto Research Center (PARC), Palo Alto, California, USA. • Probability and Statistical Mechanics in Information Science, May 20 - July 20, 2003, Centro Di Ricerca Matematica, Ennio De Giorgi, Scuola Normale Superiore, Pisa, Italy. • Sensor Networks: Theory and Military Applications, Aug 27, 2003, Cornell University, Ithaca, NY.