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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Six is a magic number? QuickTime™ and a

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