Performance Oriented Association in Cellular Networks with - - PowerPoint PPT Presentation

Performance Oriented Association in Cellular Networks with Technology Diversity

Abishek Sankararaman, Jeong-woo Cho, François Baccelli.

Outline

• Motivation and Background.
• Main mathematical model.
• Summary of Results.
• Conclusions

MVNOs (Mobile Virtual Network Operators) These operators pool together various wireless technologies (for eg. 2G, 3G, 4G LTE) to create a service.

Background

A UE can dynamically chose between the various technologies depending on whichever yields higher instantaneous benefit.

A new model of cellular service by

Background

Leveraging the presence and control of multiple wireless technologies operating on non-overlapping bandwidth.

Google Fi - The different cellular operators and WiFi operate

• n separate bands.

MVNOs - Multiple orthogonal technologies (For ex. 3G and 4G LTE).

What the Paper is About ? Technology Diversity

Leveraging the presence and control of multiple wireless technologies operating on non-overlapping bandwidth.

Google Fi - The different cellular operators and WiFi operate

• n separate bands.

MVNOs - Multiple orthogonal technologies (For ex. 2G, 3G and 4G LTE).

Technology Diversity - A framework to leverage and evaluate the benefits from the diversity of wireless technologies.

What the Paper is About ? Technology Diversity

Which Base-Station/Access Point must a UE associate with ? A principled way to exploit the diversity in the network.

The Problem we study - Base Station Association

The “optimal” BS Association is not an obvious choice

First Guess - Connect to the nearest BS irrespective of the type

• f BS it is.

First Guess - Connect to the nearest BS irrespective of the type

• f BS it is.

Nearest BS

SNR > SNR

(Basis for nearest BS association.)

Signal Interference + Noise

SINR =

The “optimal” BS Association is not an obvious choice

First Guess - Connect to the nearest BS irrespective of the type

• f BS it is.

Nearest BS

SNR > SNR

(Basis for nearest BS association.)

But bandwidths are not overlapping and thus interference only from one type of Base Stations.

Signal Interference + Noise

SINR =

The “optimal” BS Association is not an obvious choice

First Guess - Connect to the nearest BS irrespective of the type

• f BS it is.

Nearest BS

SINR < SINR SNR > SNR

(Basis for nearest BS association.)

Thus, in this example But bandwidths are not overlapping and thus interference only from one type of Base Stations.

Signal Interference + Noise

SINR =

The “optimal” BS Association is not an obvious choice

Mathematical Framework - Network Model

Consider a network comprised of different technologies.

T

The BS/APs of technology is distributed as a independent Poisson Point Process with intensity

i

φi ⊂ R2 λi

Denote by to be the closest point to the origin and by Consider a network comprised of different technologies.

T

jth

Xi

j ∈ φi

ri

j = ||Xi j||

Mathematical Framework - Network Model

The BS/APs of technology is distributed as a independent Poisson Point Process with intensity

i

φi ⊂ R2 λi

Denote by to be the closest point to the origin and by Consider a network comprised of different technologies.

T

There is a typical user at the origin of the Euclidean plane who wishes to associate to a BS. The BS/APs of technology is distributed as a independent Poisson Point Process with intensity

i

φi ⊂ R2 λi

jth

Xi

j ∈ φi

ri

j = ||Xi j||

Palm theory connects the viewpoint of a single user to the average performance experienced by the users in the network.

Mathematical Framework - Network Model

X2

1

X1

1

T = 2 φ1 = φ2 =

with intensity with intensity

λ1 λ2

Mathematical Framework - Network Model

Independent fading from the jth nearest BS of technology i to the typical user - BS of technology transmits at power i Pi Signal from BS of technology i attenuated with distance as given by the function li(·) : R+ → R+ Hi

j

Mathematical Framework - Signal Model

r1

j

r2

j

(Signal Power from the BS) (Signal Power from the BS)

P2H2

j l2(r2 j)

P1H1

j l2(r1 j)

Independent fading from the jth nearest BS of technology i to the typical user - BS of technology transmits at power i Pi Signal from BS of technology i attenuated with distance as given by the function li(·) : R+ → R+ Hi

j

Mathematical Framework - Signal Model

Performance Metrics

(Non-overlapping bandwidths =

⇒Interference from only one technology) pi(·) : R+ → R+

• For each technology , denote by bounded non-increasing

functions denoting the reward function. i SINRi,j SINR of the typical UE when it associates to the jth nearest BS of technology i.

(Non-overlapping bandwidths =

⇒Interference from only one technology) pi(·) : R+ → R+

• For each technology , denote by bounded non-increasing

functions denoting the reward function. i SINRi,j SINR of the typical UE when it associates to the jth nearest BS of technology i.

• If the typical UE connects to the jth nearest BS of technology

i, then it receives a reward of pi(SINRi,j

0 )

Performance Metrics

X2

2

The reward received by the UE in this example is p2(SINR2,2

0 )

T = 2 φ1 = φ2 =

with intensity with intensity

λ1 λ2

Performance Metrics

X2

2

The reward received by the UE in this example is p2(SINR2,2

0 )

T = 2 φ1 = φ2 =

with intensity with intensity

λ1 λ2 Examples of common reward functions

• Coverage
• Average Achievable Rate

pi(x) = 1(x ≥ βi) pi(x) = Bi log2(1 + x)

Performance Metrics

Goal- Design association schemes exploiting available network “information” at the UE, that maximize expected reward of a typical UE.

Information at the UE

Examples of Information that a UE can know -

• Nearest BS of all technologies.
• Nearest BS of all technologies.
• Instant fading and the distance to the nearest BS
• Noisy estimate of the instant fading from the nearest BS
• f each technology.

k k Goal- Design association schemes exploiting available network “information” at the UE, that maximize expected reward of a typical UE. k

Information at the UE

How Information affects Optimal Association

SINR < SINR

When averaged across fading

SINR < SINR

When averaged across fading

How Information affects Optimal Association

Sudden very good signal which the UE can sense.

SINR < SINR

When averaged across fading

How Information affects Optimal Association

Sudden very good signal which the UE can sense.

In this case, UE should associate to

SINR < SINR

When averaged across fading

How Information affects Optimal Association

Information at the UE

Notion of Information at the typical UE formalized through the notion of filtrations of a sigma algebra.

Notion of Information at the typical UE formalized through the notion of filtrations of a sigma algebra.

(Ω, F, P)

The Probability space containing the independent RVs and {φi}T

1=1

{Hi

j}i∈[1,T ],j∈N

• Information at the UE - Formalization

Notion of Information at the typical UE formalized through the notion of filtrations of a sigma algebra.

(Ω, F, P)

The Probability space containing the independent RVs and {φi}T

1=1

{Hi

j}i∈[1,T ],j∈N

• FI ⊆ F

The information sigma-algebra.

• Information at the UE - Formalization

Notion of Information at the typical UE formalized through the notion of filtrations of a sigma algebra.

(Ω, F, P)

The Probability space containing the independent RVs and {φi}T

1=1

{Hi

j}i∈[1,T ],j∈N

• This is measurable function

denoting the association scheme.

• π : Ω → [1, T] × N

FI

FI ⊆ F

The information sigma-algebra.

• Information at the UE - Formalization

Examples of Information that a UE can know -

• Nearest BS of all technologies.
• Nearest BS of all technologies.
• Instant fading and the distance to the nearest BS
• Noisy estimate of the instant fading from the nearest BS
• f each technology.

k k k

Information at the UE - Examples

Examples of Information that a UE can know -

• Nearest BS of all technologies.
• Nearest BS of all technologies.
• Instant fading and the distance to the nearest BS
• Noisy estimate of the instant fading from the nearest BS
• f each technology.

k k k

Information at the UE - Examples

FI = σ

• {ri

1}T i=1

Examples of Information that a UE can know -

• Nearest BS of all technologies.
• Nearest BS of all technologies.
• Instant fading and the distance to the nearest BS
• Noisy estimate of the instant fading from the nearest BS
• f each technology.

k k k

Information at the UE - Examples

FI = σ

• {ri

1}T i=1

• FI = σ
• {ri

k}T i=1

Examples of Information that a UE can know -

• Nearest BS of all technologies.
• Nearest BS of all technologies.
• Instant fading and the distance to the nearest BS
• Noisy estimate of the instant fading from the nearest BS
• f each technology.

k k k

Information at the UE - Examples

FI = σ

• {ri

1}T i=1

• FI = σ
• {ri

k}T i=1

• FI = σ
• {ri

k}T i=1, {Hi j}j∈[1,k],i∈[1,T ]

Examples of Information that a UE can know -

• Nearest BS of all technologies.
• Nearest BS of all technologies.
• Instant fading and the distance to the nearest BS
• Noisy estimate of the instant fading from the nearest BS
• f each technology.

k k k

Information at the UE - Examples

FI = σ

• {ri

1}T i=1

• FI = σ
• {ri

k}T i=1

• FI = σ
• {ri

k}T i=1, {Hi j}j∈[1,k],i∈[1,T ]

• where

FI = σ ⇣ σ({ri

k}T i=1) ∪ F

0⌘

F

0 ⊆ σ

• {Hi

j}j∈[1,k],i∈[1,T ]

Performance Metric for Association

This is measurable function denoting the association scheme.

• π : Ω → [1, T] × N

FI

FI ⊆ F

The information sigma-algebra.

• (Ω, F, P)

The Probability space containing the independent RVs and {φi}T

1=1

{Hi

j}i∈[1,T ],j∈N

Recall - pi(·) : R+ → R+ - The non-increasing reward function of technology i

This is measurable function denoting the association scheme.

• π : Ω → [1, T] × N

FI

FI ⊆ F

The information sigma-algebra.

• (Ω, F, P)

The Probability space containing the independent RVs and {φi}T

1=1

{Hi

j}i∈[1,T ],j∈N

pi(·) : R+ → R+ - The non-increasing reward function of technology i

Rπ = E[pπ(0)(SINRπ

0 )]

Performance of policy is then

π

Recall -

Performance Metric for Association

Optimal Association

For any association scheme , the expected reward is

π

Rπ = E[pπ(0)(SINRπ

0 )]

Thus, the optimal association is then obviously the one that maximizes expected reward i.e. π∗

I = arg sup π Rπ I

(sup over all measurable functions )

π : Ω → [1, T] × N

FI

For any association scheme , the expected reward is

π

Rπ = E[pπ(0)(SINRπ

0 )]

Optimal Association

Proposition:

π∗

I = arg

sup

i∈[1,T ],j∈N

E[pi(SINRi,j

0 )|FI]

Thus, the optimal association is then obviously the one that maximizes expected reward i.e. π∗

I = arg sup π Rπ I

(sup over all measurable functions )

π : Ω → [1, T] × N

FI

For any association scheme , the expected reward is

π

Rπ = E[pπ(0)(SINRπ

0 )]

a.s.

Optimal Association

Optimal Association - Comparison of Information

a.s. The optimal association under information is given by FI π∗

I

π∗

I = arg

sup

i∈[1,T ],j∈N

E[pi(SINRi,j

0 )|FI]

a.s. The optimal association under information is given by FI π∗

I

π∗

I = arg

sup

i∈[1,T ],j∈N

E[pi(SINRi,j

0 )|FI]

Denote by the performance of the optimal association as Rπ∗

I

= E[pπ∗(0)(SINRπ∗

0 )]

Rπ∗

I

where

Optimal Association - Comparison of Information

a.s. The optimal association under information is given by FI π∗

I

π∗

I = arg

sup

i∈[1,T ],j∈N

E[pi(SINRi,j

0 )|FI]

Denote by the performance of the optimal association as Rπ∗

I

= E[pπ∗(0)(SINRπ∗

0 )]

Rπ∗

I

where Theorem - “More information gives better performance”

FI1 ⊆ FI2 = ⇒ Rπ∗

I1 ≤ Rπ∗ I2

Optimal Association - Comparison of Information

a.s. The optimal association under information is given by FI π∗

I

π∗

I = arg

sup

i∈[1,T ],j∈N

E[pi(SINRi,j

0 )|FI]

Denote by the performance of the optimal association as Rπ∗

I

= E[pπ∗(0)(SINRπ∗

0 )]

Rπ∗

I

where Theorem - “More information gives better performance”

FI1 ⊆ FI2 = ⇒ Rπ∗

I1 ≤ Rπ∗ I2

Comparison of schemes without messy computation !

Optimal Association - Comparison of Information

Theorem - “More information gives better performance”

FI1 ⊆ FI2 = ⇒ Rπ∗

I1 ≤ Rπ∗ I2

Proof -

Rπ∗

I2 = E

" sup

i∈[1,T ],j≥1

E[pi(SINRi,j

0 )|FI2]

#

Optimal Association - Comparison of Information

Theorem - “More information gives better performance”

FI1 ⊆ FI2 = ⇒ Rπ∗

I1 ≤ Rπ∗ I2

Proof -

Rπ∗

I2 = E

" sup

i∈[1,T ],j≥1

E[pi(SINRi,j

0 )|FI2]

#

= E " E " sup

i∈[1,T ],j≥1

E[pi(SINRi,j

0 )|FI2]

• FI1

##

Tower Property

Optimal Association - Comparison of Information

Theorem - “More information gives better performance”

FI1 ⊆ FI2 = ⇒ Rπ∗

I1 ≤ Rπ∗ I2

Proof -

Rπ∗

I2 = E

" sup

i∈[1,T ],j≥1

E[pi(SINRi,j

0 )|FI2]

#

= E " E " sup

i∈[1,T ],j≥1

E[pi(SINRi,j

0 )|FI2]

• FI1

## ≥ E " sup

i∈[1,T ],j≥1

E  E[pi(SINRi,j

0 )|FI2]

• FI1

#

Tower Property Jensen’s Inequality

Optimal Association - Comparison of Information

Theorem - “More information gives better performance”

FI1 ⊆ FI2 = ⇒ Rπ∗

I1 ≤ Rπ∗ I2

Proof -

Rπ∗

I2 = E

" sup

i∈[1,T ],j≥1

E[pi(SINRi,j

0 )|FI2]

#

= E " E " sup

i∈[1,T ],j≥1

E[pi(SINRi,j

0 )|FI2]

• FI1

## ≥ E " sup

i∈[1,T ],j≥1

E  E[pi(SINRi,j

0 )|FI2]

• FI1

# = E " sup

i∈[1,T ],j≥1

E h pi(SINRi,j

0 )|FI1

i#

Tower Property Jensen’s Inequality Tower Property

Optimal Association - Comparison of Information

Theorem - “More information gives better performance”

FI1 ⊆ FI2 = ⇒ Rπ∗

I1 ≤ Rπ∗ I2

Proof -

Rπ∗

I2 = E

" sup

i∈[1,T ],j≥1

E[pi(SINRi,j

0 )|FI2]

#

= E " E " sup

i∈[1,T ],j≥1

E[pi(SINRi,j

0 )|FI2]

• FI1

## ≥ E " sup

i∈[1,T ],j≥1

E  E[pi(SINRi,j

0 )|FI2]

• FI1

# = E " sup

i∈[1,T ],j≥1

E h pi(SINRi,j

0 )|FI1

i#

= Rπ∗

I1

Tower Property Jensen’s Inequality Tower Property

Optimal Association - Comparison of Information

Optimal Association - Associate to the Nearest BS

Lemma: If the information is independent of FI then, arg sup

j≥1

E[pi(SINRi,j

0 )|FI] = 1

for all . i σ

• {Hi

j}j≥1,i∈[1,T ]

In particular, in the absence of fading information, the

• ptimal strategy is to associate to the nearest BS of the best

technology.

Optimal Association - Associate to the Nearest BS

Lemma: If the information is independent of FI then, arg sup

j≥1

E[pi(SINRi,j

0 )|FI] = 1

for all . i σ

• {Hi

j}j≥1,i∈[1,T ]

Examples of Association Policies and Computations

Some examples of association policies.

• Optimal Association under no information.
• Optimal Association under nearest BS.
• Max-Ratio Association.
• Optimal Association under complete network information.

Max-Ratio Association

Optimal Association scheme great in theory. But it is a parametric algorithm - depends on the distribution of the point-processes, fading powers !

Optimal Association scheme great in theory. But it is a parametric algorithm - depends on the distribution of the point-processes, fading powers ! We propose a “data-dependent” association scheme called Max-Ratio association.

Max-Ratio Association

Optimal Association scheme great in theory. But it is a parametric algorithm - depends on the distribution of the point-processes, fading powers ! We propose a “data-dependent” association scheme called Max-Ratio association. πm =  arg max

i∈[1,T ]

ri

1

ri

2

, 1

• Associate to the nearest BS of the technology yielding the

maximum ratio. (Oblivious to the statistical assumptions on the network.)

Max-Ratio Association

πm =  arg max

i∈[1,T ]

ri

1

ri

2

, 1

• Associate to the nearest BS of the technology yielding the

maximum ratio. Information - FI = σ ✓ ∪T

i=1

ri

1

ri

2

◆

Max-Ratio Association

πm =  arg max

i∈[1,T ]

ri

1

ri

2

, 1

• Associate to the nearest BS of the technology yielding the

maximum ratio. This scheme trades off high signal power with that of interference power. Information - FI = σ ✓ ∪T

i=1

ri

1

ri

2

◆

Max-Ratio Association

Theorem - Let the noise powers be 0 and the reward function for all technologies are the same, i.e. . Consider, the family of power law path-loss functions , where . Let be any integer larger than or equal to 2. If the information at the UE is the distance to the nearest BS of each technology, i.e. , then Max-Ratio is asymptotically almost surely optimal i.e.

Max-Ratio Association - Asymptotic Optimality

pi(·) = p(·)∀i {l(α)(·)}α>2 l(α)(x) = x−α k k FI = σ(∪T

i=1(ri 1, · · · , ri k))

π∗

α α→∞

− − − − →  arg max

i∈[1,T ]

ri

1

ri

2

, 1

• a.s.

2.5 3 3.5 4 4.5 5 5.5 6 6.5 7

Path-loss Exponent α

2 4 6 8 10 12

Average Achievable Rate

Opt-Association (1 BS) Opt-Association (2 BS) Random BS Association Nearest BS Association Max-Ratio Association

Max-Ratio Association - Finite Scale Behavior

π∗

α α→∞

− − − − →  arg max

i∈[1,T ]

ri

1

ri

2

, 1

• a.s.

For , Max-Ratio is nearly optimal.

α ≥ 5

Formulas for Performance - Generalized Association

Denote by , the dimensional vector corresponding to the information about technology ri L i

Denote by , the dimensional vector corresponding to the information about technology ri L i Denote by as the PDF of the random variable and by as the associated CDF. fi(·) πi(ri, λi) Fi(·) Generalized Association - and i∗ = arg max

i∈[1,T ] πi(ri, λi)

by , the index of the BS to associate to if .

i∗ = i

ji(ri, λi)

Formulas for Performance - Generalized Association

Denote by , the dimensional vector corresponding to the information about technology ri L i Denote by as the PDF of the random variable and by as the associated CDF. fi(·) πi(ri, λi) Fi(·) Generalized Association - and i∗ = arg max

i∈[1,T ] πi(ri, λi)

by , the index of the BS to associate to if .

i∗ = i

ji(ri, λi)

Randomness

Formulas for Performance - Generalized Association

Denote by , the dimensional vector corresponding to the information about technology ri L i Denote by as the PDF of the random variable and by as the associated CDF. fi(·) πi(ri, λi) Fi(·) Example - The Max-Ratio and the Optimal Association fit into this performance evaluation framework. Generalized Association - and i∗ = arg max

i∈[1,T ] πi(ri, λi)

by , the index of the BS to associate to if .

i∗ = i

ji(ri, λi)

Randomness

Formulas for Performance - Generalized Association

Generalized Association - and i∗ = arg max

i∈[1,T ] πi(ri, λi)

by , the index of the BS to associate to if .

i∗ = i

ji(ri, λi)

ri

L i The dimensional vector corresponding to information

• n technology

fi(·) πi(ri, λi) Fi(·) The PDF and CDF of ,

Rπ

I = T

X

i=1

Z

r2Rl E[pi(SINRi,ji(r,λi)

)|r]fi(r)

T

Y

j=1,j6=i

Fj(πj(r, λj))dr

Theorem -

Formulas for Performance - Generalized Association

Generalized Association - and i∗ = arg max

i∈[1,T ] πi(ri, λi)

by , the index of the BS to associate to if .

i∗ = i

ji(ri, λi)

ri

L i The dimensional vector corresponding to information

• n technology

fi(·) πi(ri, λi) Fi(·) The PDF and CDF of ,

Rπ

I = T

X

i=1

Z

r2Rl E[pi(SINRi,ji(r,λi)

)|r]fi(r)

T

Y

j=1,j6=i

Fj(πj(r, λj))dr

Theorem - Corollary - Closed form formulas for Max-Ratio and certain optimal association schemes.

Formulas for Performance - Generalized Association

Conclusions

Provide a systematic framework to design and evaluate association schemes that exploits the available information and technology diversity

Provide a systematic framework to design and evaluate association schemes that exploits the available information and technology diversity Max-Ratio Algorithm - A simple and almost optimal algorithm for association.

Conclusions

Thank you for your time.