information transfer in wireless networks for distributed sensing and control
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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.
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