On the Optimal SINR in Random Access Networks with Spatial Re-Use
Navid Ehsan and R. L. CruzUCSD
On Public Speaking
• The 85% Rule
• Should I be talking now?
An Analogy…
The bottom line, almost…
Horizontal throughput (bit meter/sec) versus link reliability
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Model (“infinite density”)
• Slotted system, users distributed throughout infinite plane
• In each slot, the set of transmitting users forms a 2-D Poisson point process with spatial intensity
(includes re-transmissions)• Each transmission is to a fixed
receiver at distance r
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Model, cont’d
• Flat fading channel model. Power attenuation between two points separated by distance x is
l(x) = (1 + A x)-
path loss exponent, > 2 A = constant (we later assume A=1)
Model, cont’d
• Each active transmitter transmits with power P
• Thermal noise power at each receiver is 2
• Assume interference from different transmitters are uncorrelated
Model, cont’d
• Total interference from all transmissions at a given receiver at position x:
I = i P l( | yi - x | )
– random sum of received powers– => interference power in each slot is random
– approximate I as Gaussian, can get mean and
variance of I from Campbell’s theorem
Model, cont’d
• Signal to Interference and Noise Ratio
SINR = = Pl(r ) / (2 + I )
• SINR in each slot is random
Model, cont’d
• Target SINR: target
– If target then transmission is successful, otherwise it is not successful
• Information rate: = W log2 (1 + target ) (Shannon)
– Assumes noise + interference is Gaussian
– W = Bandwidth, assume = 1 Hz.
Optimization Problem
• Horizontal Throughput per unit area:– J = max{ r Psucc : r , target }
– Psucc = Prob ( > target )
• Theorem
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Optimal Parameters * = , * = 0,
target =0 (- dB) , P*succ=1, r* = 1/[A(a-1)].
= G, (offered info load per unit area)
• Optimal load:
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Finitely Dense NetworksModel
• Location of nodes in each slot is a 2D Poisson point process with intensity 0 .
• Each node transmits with probability / 0 in each slot, so that set of transmitting nodes in each slot is a 2D Poisson point process with intensity .
• Psucc = ( 1 - / 0 )Prob{ > target }
• 0 ≤ ≤ 0 ==> is finite
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J*( ) as a function of for various values of 0
The bottom line…
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Horizontal throughput (bit meter/sec) versus target SINR
0 = 30
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Horizontal throughput (bit meter/sec) versus target SINR
0 = 15,60
The bottom line, almost…
Horizontal throughput (bit meter/sec) versus link reliability
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