Lecture 10:
Wireless Network Capacity
Anish Arora
CIS788.11J
Introduction to Wireless Sensor Networks
Goals
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Transmission Rates
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Information Capacity versus Network Capacity
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Impact of Wireless Link Model:
use short links
•
Impact of Traffic Pattern:
use beamforming
use local traffics
•
Impact of Mobility:
spread across nodes
•
Impact of Duty Cycling:
spread across time
2
Slides use some material from
Rahul Mangaram
Nitin Vaidya
Roger Watenhofer
3
Bit Error Rate
•
BER = Errors / Total number of bits
 Error means reception of “1” when “0” transmitted, or vice versa
•
•
Noise is the main factor of BER performance – signal path
loss, circuit noise, …
Packet Error/Reception Rate
 incorrectly received data packets / total # of received packets
 for packet of length n bits, this probability is
assuming bit errors are independent of each other
•
For small bit error probabilities, approximately
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4
Bit errors and SINR
Bit errors depend essentially on strength of received signal compared to
the corruption sources

Captured by signal to noise and interference ratio (SINR)
SINR allows to compute bit error rate (BER) for a given modulation
Also depends on data rate (# bits/symbol) of modulation
 E.g., for simple DPSK, data rate corresponding to bandwidth:


For QPSK and AWGN noise,
where Eb/N0 is energy per bit to noise power spectral density ratio, erfc(z)=
Thermal Noise
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Thermal Noise
 white noise since it contains the same level of power at all
frequencies
 kTB, where
 k is the Boltzmann’s constant = 1.381e-21 W / K / Hz,
 T is the absolute temperature in Kelvin, and
 B is the bandwidth
•
At room temperature, T = 290K, thermal noise power spectral
density
 kT = 4.005e-21 W/Hz or
–174 dBm/Hz
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Receiver Sensitivity
•
•
The minimum input signal power needed at receiver input
to provide adequate SNR at receiver output to do data
demodulation
SNR depends on
 Received signal power
 Background thermal noise at antenna (Na)
 Noise added by the receiver (Nr)
•
Pmin = SNRmin ×(Na +Nr)
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Noise Figure
Noise Figure (F) quantifies the increase in noise caused by
the noise source in the receiver relative to input noise
F = SNRinput/SNRoutput = (Na + Nr)/Na
Pmin = SNRmin×(Na + Nr) = SNRmin×F ×Na
Example: if SNRmin = 10 dB,
F = 4 dB,
BW = 1 MHz
Pmin= 10 + 4 -174 + 10×log(106) = -100 dBm
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802.15.4 - Modulation Scheme
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2.4 GHz PHY
 250 kb/s (4 bits/symbol, 62.5 kBaud)
 Data modulation is 16-ary orthogonal O-QPSK
 16 symbols are ~orthogonal set of 32-chip PN codes
•
868 MHz/915 MHz PHY
 Symbol rate
 868 MHz band: 20 kbps (1bit/symbol, 20 Kbaud)
 915 MHz band: 40 kbps (1bit/symbol, 40 Kbaud)
 Spreading code is 15-chip
 Data modulation is BPSK
 868 MHz: 300 Kchips/s
 915 MHz: 600 Kchips/s
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802.15.4 - PHY Communication Parameters
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Transmit power
 Capable of at least 0.5 mW
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Transmit center frequency tolerance

•
±40 ppm
Receiver sensitivity (packet error rate < 1%)
 −85 dBm @ 2.4 GHz band
 −92 dBm @ 868/915 MHz band
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Receiver Selectivity
 2.4 GHz: 5 MHz channel spacing, 0 dB adjacent channel requirement
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Channel Selectivity and Blocking
 915 MHz and 2.4 GHz band: 0 dB rejection of interference from
adjacent channel
 30 dB rejection of interference from alternate channel
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Rx Signal Strength Indication Measurements
 Packet strength indication
 Clear channel assessment
 Dynamic channel selection
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802.15.4: Receiver Noise Figure Calculation
•
•
•
•
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Channel Noise bandwidth is 1.5 MHz
Transmit Power is 1mW or 0 dBm
Thermal noise floor is –174 dBm/Hz X 1.5 MHz = –112 dBm
Total SNR budget is 0 dBm –(–112 dBm) = 112 dBm
To cover ~100 ft. at 2.4 GHz results in a path loss of 40 dB
 i.e. Receiver sensitivity is –85 dBm
•
Required SNR for QPSK is 12.5 dB
 802.15.4 packet length is 1Kb
 Worst packet loss < 1%, (1 –BER)1024= 1 –1%, BER = 10–5
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Receiver noise figure requirement
 NF = Transmit Power – Path Loss – Required SNR – Noise floor
= 0 + 112 –40 –12.5 = 59.5 dB
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•
The design spec is very relaxed
Low transmit power enables CMOS single chip solution at low cost
and power!
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Information (or Channel or Transmission) Capacity
Capacity maximizes time average bit rate, optimizing over all coding strategies
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Information Theoretic Concept of Capacity
L
Capacity Region Λ = Set of all end-to-end
rate vectors (or matrices) achievable over
a network
l
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Results known for point-to-point links
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Results known for small 1-hop systems (broadcast/MAC)
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In terms of SNR
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Shannon-Hartley Theorem
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channel capacity , the tightest upper bound on information
rate (excluding error correcting codes) of arbitrarily low bit error rate
data that can be sent with a given average signal power S through an
additive white Gaussian noise channel of power N, is:
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C is the channel capacity in bits per second
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B is the bandwidth of the channel in hertz
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S is the total received signal power over bandwidth, in watts
•
N is the total noise or interference power over bandwidth, in watts
•
S/N is the signal-to-noise ratio (SNR) expressed as a linear power ratio
(not as logarithmic decibels).
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Shannon’s Theorem: Example
•
•
For SNR of 0, 10, 20, 30 dB, one can achieve C/B of 1, 3.46,
6.66, 9.97 bps/Hz, respectively
Example:
 Consider the operation of a modem on an ordinary telephone
line. The SNR is usually about 1000. The bandwidth is 3.4
KHz. Therefore:
C = 3400 X log2(1 + 1000)
= (3400)(9.97)
≈34 kbps
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Protocol Model (k can send reliably when j sends if)
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Physical (SINR) Model
Received signal power from sender
Power level
of sender u
Noise
Received signal power from
all other nodes (=interference)
Path-loss exponent
Minimum signalto-interference
ratio
Distance between
two nodes
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Example: Protocol vs. Physical Model
C
B
A
4m
1m
D
2m
Assume a single frequency
NO
Protocol Model
YES
With power control
Is spatial reuse possible?
Let =3, =3, and N=10nW
Transmission powers: PB= -15 dBm and PA= 1 dBm
SINR of A at D:
SINR of B at C:
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Terminology
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From Roger Watenhofer
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Network Capacity Measures
Throughput capacity
 Number of packets successfully delivered per time
 Dependent on the traffic pattern
 E.g.: What is the maximum achievable rate, over all
protocols, for a random node distribution and a random
destination for each source?
Transport capacity
 A network transports one bit-meter when one bit has
been transported a distance of one meter
 What is the maximum achievable rate, over all node
locations, and all traffic patterns, and all protocols?
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Why make the distinction?
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Transport Capacity
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Transport Capacity
•
•
•
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n nodes are arbitrarily located in a unit disk
We adopt the protocol model with R=2, that is a
transmission is successful if and only if the sender is at
least a factor 2 closer than any interfering transmitter. In
other words, each node transmits with the same power,
and transmissions are in synchronized slots
Quiz: What configuration and traffic pattern will yield the
highest transport capacity?
Idea: Distribute n/2 senders uniformly in the unit disk.
Place the n/2 receivers just close enough to senders so as
to satisfy the threshold
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Transport Capacity: Example
sender
receiver
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Transport Capacity: Understanding the example
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Sender-receiver distance is  (1/√n).
Assuming channel bandwidth W [bits], transport capacity is
(W√n) [bit-meter], or per node:  (W/√n) [bit-meter]
•
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Can we do better by placing the sourcedestination pairs more carefully? No,
having a sender-receiver pair at distance d
inhibits another receiver within distance up
to 2d from the sender. In other words, it kills
an area of  (d2)
d
We want to maximize n transmissions with distances d1, d2, …, dn given
that the total area is less than a unit disk. This is maximized if all di =
 (1/√n). So the example is asymptotically optimal
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More capacity results
The throughput capacity of an n node random network is (
W
)
n log n
I.e., there exist constants c and c’ such that
lim
Pr[c
n 
W
is feasible ]  1
n log n
W
lim
Pr[c'
is feasible ]  0
n 
n log n
Transport capacity:
1
 Per node transport capacity decreases with n
 Maximized when nodes transmit to neighbors
Throughput capacity:
1
 For random networks, decreases with
n log n
 Near-optimal when nodes transmit to neighbors
Result improved by Franceschetti et al to :  (W/√n)
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Convergecast Capacity
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•
•
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Single sink/collector node (potential bottleneck)
Information theoretic network transmission capacity (node
capacity) scales not as Θ(1) but as Θ(log (n))
Idea:

Each node talks to closely located nodes, which is efficient given node
density

Relay nodes cooperate to transmit the information to collector using a
beamformer, to get logarithmic increase in received power, and
therefore, the capacity
H. El Gamal, "On the Scaling Laws of Dense Wireless Sensor Networks: The Data Gathering Channel,"
IEEE Trans. Inform. Theory, vol. 51, no. 3, pp. 1229-1234, Mar. 2005
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Broadcast Capacity
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Network transport capacity scales not as Θ(1), but as Θ(log (n))
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Similar idea as convergecast: two phases
(i) source broadcasts the message;
(ii) close-by neighbors of source retransmit the message
with log (n) scaling factor
•
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A. Keshavarz-Haddad, V. Ribeiro, and R. Riedi, “Broadcast Capacity in Multihop Wireless Networks”,
Proceedings of the 12th Annual International Conference on Mobile Computing and Networking
(MobiCom '06). ACM, New York, NY, USA, 239-250, 2006
B. Sirkeci-Mergen, Michael Gastpar, ``On the Broadcast Capacity of High Density Wireless networks'',
2007 Information Theory and Applications Workshop, San Diego, CA, January 2007
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Capacity in the presence of mobility
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Results are based on an idealized setup
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Assume a central scheduler
 At time t, scheduler chooses the senders and their power
levels
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Goal: under random motion patterns
 Show that long term throughput remains constant as
number of users increases

Caveat:
 long term throughput averaged over node mobility time-scale
 delays of same order can occur
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Mobile Nodes w/o Relaying
•
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Can mobile nodes achieve a throughput of O(1) per S-D pair
by not relaying at all?
Answer: number of simultaneous long range communications
is limited by interference in physical model
Positions of nodes t,j at time t
 | X (t )  X
iS ( t )
i

j
 /2
(t ) |  2
 L

S(t) – Set of source nodes scheduled for successful transmission
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Mobile nodes without relaying
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Without relaying the achievable throughput per S-D pair goes
to 0 at least as fast as
n
1

1 a / 2
Distance attenuation factor
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Mobile nodes with relaying
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What is the problem with direct transmission to S-D pairs?
 Transmissions are long range => interference limits the
number of concurrent transmissions
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How can we increase throughput?
 Constrain transmission to nearest neighbors
 Use lower transmission power to avoid interference
 Cannot wait for nearest neighbor to come close by, time 1/n
– vanishes at time goes by
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Spread out packets along a large number of relay nodes
 Nodes temporarily buffer packets while they move
 Ensure that every node will have packets to send to its
nearest neighbor at any time
 Cannot do this with direct transmission alone
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Main idea
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•
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spread traffic stream between s and d over large number
of intermediate relay nodes (all others can be relays)
each packet goes through a relay node that temporarily
buffers the packet until final delivery to d is possible
as node location processes are independent, stationary,
and ergodic, it is sufficient to relay only once
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Scheduling Policy & Theorem
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Assume that time is divided into slots
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Fix a sender density parameter
  (0,1)
nS  n senders and nR  n  nS receivers
•
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Select the n S sender receiver pairs where interference is
small enough to make transmission possible
Theorem
The number of feasible sender-receiver pairs is O(n)
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2-phase scheduling policy
Apply a 2-phase interleaved scheduling policy:
1) Source sends to relay (odd slots)
2) Relay sends to destination (even slots)
Direct transmission to destination is also allowed if destination is close enough
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1 hop vs. 2-hop routes
Theorem: Number of feasible sender receiver pairs is O(n)


Long-term throughput between any two nodes = probability
that 2 nodes are a feasible node pair O(1/n) per theorem
Throughput over direct route O(1/n)
 Single hops routes alone O(1/n)

In 2-hop routes there are n-2 routes
 Total average throughput per S-D pair is O(1)
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Main result
Theorem: The two-phased algorithm achieves a throughput
per S-D pair of O(1) i.e. there exists a constant c>0 such
that
lim Pr{l (n)  cR is feasible}  1
n 
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Capacity and Delay Tradeoffs
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There is a minimum critical delay to achieve capacity results
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Capacity achieving strategy yields O(N) delay
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Redundant transmission protocol can achieve O( N )
delay at expense of reducing throughput to O(1 /
•
•
N)
M. J. Neely and E. Modiano, “Capacity and Delay Tradeoffs for Ad-Hoc Mobile Networks”, Proceedings
of the First International Conference on Broadband Networks (BROADNETS), 2004
X. Wang, L. Fu, X. Tian, Y. Bei, Q. Peng, X. Gan, H. Yu, J. Liu, "Converge-Cast: On the Capacity and
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Delay Tradeoffs," IEEE Transactions on Mobile Computing, 99(1), 2011
Network Capacity in Directional Link Network
•
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Higher fidelity of the physical layer yields better by
allowing antenna sharing for coherent relaying and
interference subtraction or for MIMO beamforming
With a sender gain of A and receiver gain of B, an AB
gain is possible
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Network Capacity with MIMO Links
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Create nulls for up to N-2 other nodes to increase capacity
R. Mudumbai, D.R. Brown, U. Madhow, and H.V. Poor, “Distributed Transmit Beamforming:
Challenges and Recent Progress”, Communications Magazine, 47, 2, 102-110. February 2009
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Asymptotic Scalability for Local Traffics
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Per node capacity with power-law distributed traffic with
exponent greater than 2 scales as O(1)
if exponent is
•
< 1, then GK result for uniform traffic
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= 1, then it is O(ln(n)/√n)
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< 2, then it is O(n 2 )
•
•
 2
= 2, then O(1/ln(n))
> 2 then scales as O(1)
J. Li, C. Blake, D. S. J. De Couto, C. Hu, H. I. Lee, and R. Morris. Capacity of ad hoc wireless
networks. In In ACM Mobicom, pages 61–69, 2001
•
“Scalability of Mobile Ad Hoc Networks: Theory vs. Practice”, by R. Ramanathan, R. Allan, P. Basu, J.
Feinberg, G. Jakllari, V. Kawadia, S. Loos, J. Redi, C. Santivanez and J. Freebersyse, in The 2010
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Military Communications Conference
Duty Cycled Transport Capacity
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Ignore short links, assume all links are global
 i.e., network is 1-hop
•
•
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each node is up with duty cycle
per node throughput capacity
provided
bps
i.e., each node gets a private copy of the channel until the
network capacity is reached
[Jing Li, Wenjie Zeng, A]
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Relationship between Capacity and Network Overhead
Fixed Size
Capacity
Traditional P2P
Long Link — Broadcast
O n
2
Application Specific Networking
O n
 
O n 
O n2
 n
O n 
 3 2
O n
O 1
O
NLO/Capacity
 2
 n
Long Link — Arbitrary Traffic
Virtual Hierarchy
•
O
NLO

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O log( n)  n
O log( n)  n
 
O n2
O n

O  log( n)  n 
O  log(n ) 
“Hierarchical Cooperation achieves Optimal Capacity Scaling in Ad hoc Networks”, by A. Özgür, O.
Lévêque, D. N. C. Tse, , IEEE Trans. Inf. Theory, 2007
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References
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P. Gupta and P. R. Kumar, “The Capacity of Wireless Networks,” IEEE Transactions on
Information Theory, vol. 46, no. 2, pp. 388-404, Mar. 2000
“Scaling Laws for Ad Hoc Wireless Networks: An Information Theoretic Approach” by F.
Xue and P. R. Kumar, in Foundations and Trends in Networking, vol. 1, no. 2, 2006, pp.
145-270
"Mobile Ad hoc Networking and the IETF — IETF 69", by I. D. Chakeres and J. P.
Macker, in ACM SIGMOBILE Mobile Computing and Communications Review (MC2R),
2007
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