1

On the Performance of Slotted Aloha with Capture Effect

in Wireless Networks

On the Performance of Slotted Aloha with Capture Effect

in Wireless Networks

Arash Behzad and Julan HsuProfessor Mario Gerla

CS218 ProjectUCLA

December 1, 2003

{abehzad,julan}@ee.ucla.edu

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System ModelSystem Model

Assumptions: Slotted ALOHA Omni directional antennas Half-duplex radios Immobile nodes and fixed topological configuration (single access net)

Assumptions: Slotted ALOHA Omni directional antennas Half-duplex radios Immobile nodes and fixed topological configuration (single access net)

Objective: Analysis of the throughput performance of the Slotted Aloha medium access control for an arbitrary topology under variations of transmission probability (q) and transmission power level (P)

Objective: Analysis of the throughput performance of the Slotted Aloha medium access control for an arbitrary topology under variations of transmission probability (q) and transmission power level (P)

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Highlights of PresentationHighlights of Presentation

Highlights of this presentation:

i. Major interference models in wireless networks and their features

ii. Some asymptotic theoretical results

iii. Power control and capture effect

iv. Preliminary simulation results

v. Conclusions

Highlights of this presentation:

i. Major interference models in wireless networks and their features

ii. Some asymptotic theoretical results

iii. Power control and capture effect

iv. Preliminary simulation results

v. Conclusions

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I. Two Core Interference Models in Wireless Networks and their Features

I. Two Core Interference Models in Wireless Networks and their Features

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1. Protocol Interference Model1. Protocol Interference Model

Assuming all nodes employ a common transmission range rc, transmission from

node i1 to node j1 is successful if

and for every other node i2 transmitting in the same time slot

rc and ri are commonly known as communication range and interference range, respectively.

Assuming all nodes employ a common transmission range rc, transmission from

node i1 to node j1 is successful if

and for every other node i2 transmitting in the same time slot

rc and ri are commonly known as communication range and interference range, respectively.

,),( 11 crjid

.),( 12 irjid

j1

j2

j3i3

i1

i2

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Let be the subset of nodes simultaneously transmitting at some time instant (time slot) employing an identical transmit power level. Then the transmission from a node , , is successfully received by a node , , if and only if

whereby is the distance betweennodes and , N is the ambient noise power level, and is the path loss exponent.

Let be the subset of nodes simultaneously transmitting at some time instant (time slot) employing an identical transmit power level. Then the transmission from a node , , is successfully received by a node , , if and only if

whereby is the distance betweennodes and , N is the ambient noise power level, and is the path loss exponent.

2. Physical Interference Model 2. Physical Interference Model

Wii kk ,

1i Wi 1

1j Wj 1

,),(/

),(/

11

11c

Wkik

k jidPN

jidP

),( 1jid k

1jki

j1

j2

j3i3

i1

i2

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Disadvantages of Protocol Interference Model: Aggregate Effect of Interference

Disadvantages of Protocol Interference Model: Aggregate Effect of Interference

-Illustration of a transmission scenario, which is feasible based on Protocol Interference Model and is infeasible based on Physical Interference Model. Note that all receivers are out of the “interference range” ri of non-associated transmitters. This problem can be resolved by Protocol Interference Model via considering a larger interference range in the expense of losing some spatial reuse.

j1

j2

j3i3

i1

i2

Feasible based on Protocol Interference Model, butinfeasible based on Physical Interference Model

Feasible based on Protocol Interference Model, butinfeasible based on Physical Interference Model

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-Illustration of a transmission scenario, which is feasible based on Physical Interference Model (assuming i2 is sufficiently close to j2) and is infeasible based on Protocol Interference Model, since j2 is in the interference range of i1.

j1i1

i2

j2

Infeasible based on Protocol Interference Model, butfeasible based on Physical Interference Model

Infeasible based on Protocol Interference Model, butfeasible based on Physical Interference Model

Disadvantages of Protocol Interference Model: Capture Effect

Disadvantages of Protocol Interference Model: Capture Effect

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KEY ASSUMPTIONS:I. Protocol Interference ModelII. Identical transmit power level (no power control) III. Identical probability of transmission for all nodes

KEY ASSUMPTIONS:I. Protocol Interference ModelII. Identical transmit power level (no power control) III. Identical probability of transmission for all nodes

Conventional Slotted Aloha in Wireless Networks

Conventional Slotted Aloha in Wireless Networks

4

AP/BN/BS

2

6 3

5

1

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Interference Model Assumptions: Protocol Model

Interference Model Assumptions: Protocol Model

Based on the Protocol Interference Model, transmission from node k to AP is successfully received if and only if it is the only transmission in the underlying slot. Why?

Based on the Protocol Interference Model, transmission from node k to AP is successfully received if and only if it is the only transmission in the underlying slot. Why?

1)1(1

}1Pr{)(

nqqn

xnTH

4

AP

2

6 3

5

1

nq /1*

Is it fair?Is it fair?

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Interference Model Assumptions:Physical Model

Interference Model Assumptions:Physical Model

What if we consider the Physical Interference Model? The throughput under Protocol Interference Model is higher or under the Physical Interference Model? Why?

What if we consider the Physical Interference Model? The throughput under Protocol Interference Model is higher or under the Physical Interference Model? Why?

4

AP

2

6 3

5

1

}Pr{)1(1

}Pr{}1Pr{)(

1 captureqqn

capturexnTH

n

?*q

Does q=1/n still work?Does q=1/n still work?

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II. Some Asymptotic Theoretical Results

II. Some Asymptotic Theoretical Results

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Consider an arbitrary (symmetric/asymmetric) topology. Based on the Physical Interference Model, the probability of success for transmission from ik to AP (Ck) is equal to

where Yr is a 0-1 Bernoulli variable defined as follows:

Consider an arbitrary (symmetric/asymmetric) topology. Based on the Physical Interference Model, the probability of success for transmission from ik to AP (Ck) is equal to

where Yr is a 0-1 Bernoulli variable defined as follows:

.,0

,1

otherwise

slotunderlyingtheintransmitsinodeifY rr

)1(})),(/(

),(/Pr{

1

APidPYN

APidPC

r

n

krr

r

kk

Probability of Successful Transmission Probability of Successful Transmission

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Estimation of Aggregate Interference by L-F Central Limit Theorem

Estimation of Aggregate Interference by L-F Central Limit Theorem

Equation (1) can be written as

The term (i.e. the aggregate interference) is a linear

combination of independent Bernoulli variables. Conclusively, based on a generalization of the Central Limit Theorem (the Lindeberg-Feller Central Limit Theorem) we have

where n is a sufficiently large number.

Equation (1) can be written as

The term (i.e. the aggregate interference) is a linear

combination of independent Bernoulli variables. Conclusively, based on a generalization of the Central Limit Theorem (the Lindeberg-Feller Central Limit Theorem) we have

where n is a sufficiently large number.

),(/1

APidY r

n

krr

r

)],,(/)1(),,(/[~),(/ 2

111

APidqqAPidqNAPidY r

n

krr

r

n

krr

r

n

krr

r

)2()},(/1/)],(/[Pr{1

APidPNAPidYC kr

n

krr

rk

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Theorem 1Theorem 1

Theorem 1. Consider an arbitrary (symmetric/asymmetric) topology with large number of nodes operating under a Slotted Aloha medium access control. Based on the Physical Interference Model, the probability of success for transmission ikAP can be calculated as

where Q(.) is the Q-function, n is an arbitrarily large number and

Theorem 1. Consider an arbitrary (symmetric/asymmetric) topology with large number of nodes operating under a Slotted Aloha medium access control. Based on the Physical Interference Model, the probability of success for transmission ikAP can be calculated as

where Q(.) is the Q-function, n is an arbitrarily large number and

)),(( qfQC kk

.1,0,

),(/)1(

),(/),(/1/

)(2

1

1

qq

APidqq

APidqAPidPN

qf

r

n

krr

r

n

krr

k

k

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Derivation of Aggregate ThroughputDerivation of Aggregate Throughput

The aggregate throughput (per slot) can be calculated as the following:

where Xr is a 0-1 Bernoulli variable defined as

Clearly, (why?; are X i’s independent?)

The aggregate throughput (per slot) can be calculated as the following:

where Xr is a 0-1 Bernoulli variable defined as

Clearly, (why?; are X i’s independent?)

,][)(1

n

rrXEqTH

.,0

,1

otherwise

slotunderlyingtheinlysuccessfultransmitsinodeifx rr

.)1()(1

n

rrXPqTH

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Theorem 2Theorem 2

Theorem 2. Consider an arbitrary (symmetric/asymmetric) topology with large number of nodes operating under a Slotted Aloha medium access control. Based on the Physical Interference Model, the aggregate throughput can be calculated as

where Q(.) is the Q-function, n is arbitrarily large number and

Theorem 2. Consider an arbitrary (symmetric/asymmetric) topology with large number of nodes operating under a Slotted Aloha medium access control. Based on the Physical Interference Model, the aggregate throughput can be calculated as

where Q(.) is the Q-function, n is arbitrarily large number and

)3(,))(()(1

n

rk qfqQqTH

)4(.1,0,

),(/)1(

),(/),(/1/

2

1

1

qq

APidqq

APidqAPidPN

f

r

n

krr

r

n

krr

k

k

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Based on Theorem 2, what happens to TH as q1? Would it be still fair?

Based on Theorem 2, what happens to TH as q1? Would it be still fair?

Intuitive Interpretation of Theorem 2Intuitive Interpretation of Theorem 2

Based on equalities (3) and (4), it can be easily shown thatBased on equalities (3) and (4), it can be easily shown that

.

,0

),(//),(/1:,1)(lim 1

1

otherwise

APidqPNAPidkifqTH

r

n

krr

k

q

4

AP

2

6 3

5

1

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III. Power Control and Capture EffectIII. Power Control and Capture Effect

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• For a given topology, every node ik can individually select the transmit power such that its received signal power at AP becomes a constant, say A mW. Clearly, this method supports fairness. What

is the drawback of this approach?

• For a given topology, every node ik can individually select the transmit power such that its received signal power at AP becomes a constant, say A mW. Clearly, this method supports fairness. What

is the drawback of this approach?

Conventional Power ControlConventional Power Control

Under the former approach the probability of capture becomes zero.

Under the former approach the probability of capture becomes zero.

4

AP

2

6 3

5

1

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Assume there are m different transmit power levels. Every node randomly and independently selects a power level for its transmission in the underlying slot.

This method supports fairness and increases the channel utilization. Why?

Assume there are m different transmit power levels. Every node randomly and independently selects a power level for its transmission in the underlying slot.

This method supports fairness and increases the channel utilization. Why?

Power-Controlled Approach for Symmetric Topologies

Power-Controlled Approach for Symmetric Topologies

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Key ObservationKey Observation

As noted before, the aggregate throughput can be calculated as

Key observation: Based on Chebychev’s inequality and Wald’s lemma we have proven that the upper bound for probability of capture is inversely proportional with the variance of power distribution. This is also intuitively correct. Why?

As noted before, the aggregate throughput can be calculated as

Key observation: Based on Chebychev’s inequality and Wald’s lemma we have proven that the upper bound for probability of capture is inversely proportional with the variance of power distribution. This is also intuitively correct. Why?

}Pr{)1(1

)( 1 captureqqn

nTH n

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IV. Preliminary Simulation ResultsIV. Preliminary Simulation Results

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Simulation AssumptionsSimulation Assumptions

Noise power (N) = -90 dBm

Transmit power (P) = 50 mW (unless otherwise specified)

Communication range (rc) = 250 m

Minimum required SINR = 10 dB

Path loss exponent = 4

Number of nodes = 50

Noise power (N) = -90 dBm

Transmit power (P) = 50 mW (unless otherwise specified)

Communication range (rc) = 250 m

Minimum required SINR = 10 dB

Path loss exponent = 4

Number of nodes = 50

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Throughput per Node associated with Approach 1 (Symmetric)

Throughput per Node associated with Approach 1 (Symmetric)

Note that throughput of node k is equal to Note that throughput of node k is equal to )).(( qfqQ k

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Optimal Probability of Transmission associated with

Approach 1

Optimal Probability of Transmission associated with

Approach 1

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Throughput per Node associated with Approach 1

Throughput per Node associated with Approach 1

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Aggregate Throughput associated with Approach 2

(Asymmetric Topology)

Aggregate Throughput associated with Approach 2

(Asymmetric Topology)

P1 = 1 mW; P2 = 100 mWP1 = 1 mW; P2 = 100 mW

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Throughput per Node associated with Approach 2

(Asymmetric Topology)

Throughput per Node associated with Approach 2

(Asymmetric Topology)

P1 = 1 mW; P2 = 100 mWP1 = 1 mW; P2 = 100 mW

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

0 5 10 15 20 25

node ID

TH w ithout pow er control

2 pow er levels

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ConclusionsConclusions

We introduced two methods for increasing the aggregate throughput of a single access net based on the Slotted Aloha MAC: - In approach one no power control were used. The only control knob considered was q, probability of transmission. - In approach two, two knobs were takes in consideration simultaneously: P (transmit power level) and q (probability of transmission) Approach one seems to be more promising, as the results are not asymptotic and a minimum fairness is guaranteed. However, not all systems possess the power control capabilities.