Reasoning about Performance in
Competition and Cooperation
David Tse
Wireless Foundations
Dept. of EECS
U.C. Berkeley
Microsoft Cognitive Radio Summit
June 5, 2008
Competition and Cooperation
• Cognitive radios:
– compete for resources to transmit their own information
– cooperate with each other to improve performance
• Basic questions:
– What exactly is the resource being competed for?
– What exactly is the value-added of a cooperating radio?
Reasoning about Performance
How does an information theorist go about it?
• formulate a (physical-layer) channel model
• compute capacity
• identify key dependency on channel parameters
Standard PHY-Layer Models
Competition
Cooperation
(interference channel)
(relay channel)
Capture key properties of wireless medium:
•
•
•
Signal strength
Broadcast
Superposition
Unlike p2p capacity, capacity of these networks open for 30 years
New Approach
• Simplify model.
• Reason about performance on simplified model,
• Approximate optimal performance on original model.
Determination of capacity of interference and relay
channels to within 1 bit/s/Hz.
(Etkin,T. & Wang 06, Avestimehr, Diggavi & T. 07)
• In the process, we obtained an interesting abstraction
of the PHY layer.
Capturing Signal Strength
PHY-layer model
Abstraction
Transmit a real number
Least significant bits are
truncated at noise level.
If
p
SNR = 2n we have
n / SNR on the dB scale
Matches approx:
Cawgn (SNR) =
1
2
log(1 + SNR)
Broadcast and Superposition
Broadcast
n1 = 5
n2 = 2
ni / SNRi (dB)
Superposition
n1 = 2
n2 = 5
MSB’s of weak users collide
with LSB’s of strong user.
Competition
PHY-layer model
Abstraction
n
In symmetric case, channel
described by two parameters:
SNR = signal-to-noise ratio
INR = interference-to-noise ratio
n / SNR (dB),
m / INR (dB)
Key coupling parameter:
® :=
m
n
=
log INR
log SNR
m
Capacity as a Function of Coupling
r=
1
frequency-division
1/2
® =
=
®=
r =
2
3
1
3
®=
r =
2
3
2
3
log INR
log SNR
m
n
Cooperation
Abstraction
PHY-Layer Model
nSR
nRD
nSD
C = min (max(nSD ; nSR ); max(nSD ; nR D ))
¡
+
= nSD + min (nSR ¡ nSD ) ; (nR D ¡ nSD )
+
¢
Max-Flow Min-Cut Theorem for General Networks
Theorem:
where
Sc
S
Crelay = min value(S; Sc )
S
value(S; Sc ) = rankG S;S c
Generalization of Ford-Fulkerson Theorem for wireline
networks.
Reasoning about Performance via Abstraction
PHY layer
higher layers
simple abstraction
of channel