Information transfer in wireless networks for distributed sensing and control

Sanjeev Kulkarni, P. R. Kumar, John Tsitsiklis, and Sergio VerduDept. of Electrical and Computer Engineering, andCoordinated Science LabUniversity of Illinois, Urbana-Champaign

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 MicrosensorsSep 22, 2003

SensorWeb MURI Review Meeting, Sep 22, 2003

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

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

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

Problems at the interface

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

. npermutatioany for

)),,...,,((),...,,( 2121

ππ nn xxxfxxxf =

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?

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

O

1logn

⎛

⎝ ⎜

⎞

⎠ ⎟

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?

O

1logn

⎛

⎝ ⎜

⎞

⎠ ⎟

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:

),(

),|(),|(

YXHRR

XYHRYXHR

yx

yx

?+

??≥ ≥

≥

X

Y

Z

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

Ho

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

Energy limited communication

Optimal energy allocationwith fading channel

How many bits can be transferred over a fading channel when the source has a fixed amount of energy?

Fading channel

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?

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

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

DP formulation(Throughput maximization)

Chooseenergyconsumptione1,...,en to:

maxT =E qteti=1

n

∑⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥

subjectto: eti=1

n

∑ ≤Etotal

DP Recursion:

J k(ak,qk) =0≤ek ≤max{ak ,P}max qkek +Eq J k+1(ak −ek,qk+1)[ ]{ }

ek =energyconsumptionattimek

ak =available energy at time k

qk =channel quality at time k

P= transmitter power limit

Formulation

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

Example: Throughput maximization (Rayleigh fading)

N = 50 slotsto send data

Etotal = 95 units

P = 10

Example (continued)

Optimal policy (dynamic threshold) vs. fixed threshold policy

Networks with limited energy nodes formed over fading channels

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

Networks over fading channels

Node j’s reception

How much information can be transmitted over such networks?

Yj (t) =β

ρijδ

i≠j∑ Kijl (t)⋅Xi(t−τij −l)

l=0

∞∑

⎛

⎝ ⎜

⎞

⎠ ⎟ +Zj (t)

Attenuation over the distance

from i to j

Fading

AWGN

Transmission of node

i

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

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

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.

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?

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.

ns

nc

sn ×sn

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?

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.)

Wireless Throughput Results

Theorem 2 A throughput bit-meters/sec is achievable.

Special Cases: Nodes “evenly spread”: Throughput (√n). Iid Random Nodes: Throughput Users on Roads: Throughput Users in Neighborhoods: Throughput

) log/ nn(

) log / nn(

) log / nn(

)( / nn cns

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?

na

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.

wn×wn

)( naO n +

) ( 22 nnnnn wcnscns +

nan >4/11 n

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.