abishek sankararaman phd qualifying exam supervisor

Spatial Stochastic Models for Wireless and Data Networks

PhD Qualifying ExamAbishek Sankararaman

Supervisor - François Baccelli

IntroductionEmerging trends in networking bring about new design challenges.

Online Social NetworksLarge scale wireless networks

Dynamics in wireless networks

Diversity due to multiple operators Community Detection

1

1. Dynamics on Wireless Networks!

!

2. Clustering in a Planted-Partition network.!

!

!3. Base-station association in multi-operator networks.

Contents of the Proposed Thesis

Theme - !• What are the questions of interest in these applications ? !• What are tractable mathematical models to answer these questions ?

Completed and Proposed Work

Completed Work.

Ongoing and Proposed Work

2

Spatial Birth-Death Wireless Network Model!

Motivation

NOT Ad-Hoc

Understanding spatio-temporal dynamics of ad-hoc networks

Ad-Hoc Networks are those without any centralized infrastructure!

Ad-Hoc

3

Motivation

Ad-Hoc Networks are those without any centralized infrastructure!

Ad-Hoc

D2D networks considered as one of the key enabling technologies in 5G

Understanding spatio-temporal dynamics of ad-hoc networks

Example - Overlaid Device-to-Device (D2D) Networks

Consider these in this talk

3

Spatio-Temporal Dynamics

Understanding the interplay of space-time interactions is crucial for design

time

space

Spatial Component - Interference

Temporal Component - Traffic Patterns

Wireless Spectrum is a space-time shared resource

4

Outline

A novel stochastic framework for modeling spatio-temporal dynamics

Use the framework to derive insight for design of networks• How to dimension the network to keep it stable ?!!• How do design parameters affect various performance metrics ?

5

!However, little is understood on the spatio-temporal interactions.!!1. Static spatial setting ! ![Gupta et al. 00][Baccelli et al. 03][Andrews et al. 07][De-Veciana et al. 08]![Haenggi et al. 09]! (Does not precisely capture interactions through traffic arrivals)!!2. Flow-based queuing models !![Bonald et al. 06][Srikant et al. 07][Shah et al, 09][Shakkottai et al. 07]![De-Veciana et al. 08]! (Does not capture precisely, the information-theoretic interactions)!

!

Prior WorkAd-hoc networks have been studied for a long time !

We provide a framework to capture interactions in space and time.

6

Stochastic Network Model - Overview

• A link exits after the Tx completes file transfer.

• Each Tx has a file to transmit to its Rx.

• - Compact space. S = [�Q,Q]⇥ [�Q,Q]

• Links (Tx-Rx pairs) - Line segments of length . T

• Links arrive randomly in space at some time.

S

7

�t tConfiguration of links at time

Schematic

Spatial Birth-Death(SBD) Process

Increasing Time

(Time = t)

S

8

!!

3. Links exit after the Tx completes file transfer.!

4. The speed of file transfer - Shannon rate under Interference as Noise.

Dynamics - Statistical Assumptions

!2. Each Tx has an i.i.d. exponentially distributed file of mean bits

1. Links arrival - Poisson Point Process on with intensity R⇥ S �

L

!

Tx as PPP and Rx distributed uniformly and independently around the TX

9

S�t Configuration of links at time t

y1y2

y3y4

Network Model - DynamicsConfiguration

Set of Transmitter locations

�

Rx = {x1

, · · · , xn} Set of Receiver locations

� = {(x1, y1), · · · , (xn, yn)}

�Tx = {y0

, · · · , yn}x1

x2

x3x4

Given and two positive constants, ,l(·) : R+ ! R+ B > 0,N0 > 0 8 i 2 [1, n]

Rate of file transfer - bits per secondR(xi,�) = B log2

✓1 +

l(T )

N0 + I(xi,�)

◆

I(xi,�) :=X

z2�Tx\{yi}

l(||z � xi||) is the interference seen at inxi

configuration �

10

Spatial Birth-Death(SBD) Process

A link with receiver at location and transmitter at location born at!time with file of size will leave the system at time

x

yb L

x

d

Implicit Equation

Schematic

Increasing Time

(Time = t)

S

11

d = inf

⇢t � b :

Zt

u=b

R(x,�u

)du � L

x

�

Notation

Increasing Time

(Time = t)

S

SNt The number of links at time t

�Tx

t := {y1

, · · · , yNt}

�t := {(x1, y1), · · · , (xNt , yNt)} Configuration at time t

Set of transmitter locations at time t

�

Rx

t := {x1

, · · · , xNt} Set of receiver locations at time t

x1

x2

x3

x4

y1

y2

y3

y4

Model Parameters�

|S|

l(·) : R+ ! R+

BN0

Space-time arrival rate of links

Area of the network

Signal path-loss function

Positive constant representing bandwidthPositive constant representing thermal noise

L

T

Average File Size

Link length

12

Mean Delay ? Delay Distribution ?

1. When is Stable ? �t

Design implications in determining how much traffic to off-load to D2D.

Questions of interest

!2. When is stable, can one say something about the steady-state ?�t

Increasing Time

(Time = t)

S

13

�t A Markov Process

Corollary: !! is always unstable for the popular power law path-loss function !

for all since l(r) = r�↵ ↵ > 2

�t Z

x2S||x||�↵

dx = 1

(a.k.a. unstable)admits no stationary regime.

admits an unique stationary regime. Under further assumptions that is bounded, r ! l(r)

� > �c =) �t

� < �c =) �t

(1)

(2)

Let . Then,

Theorem -Phase Transition for Stability

�

c

=Bl(T )

ln(2)LRx2S

l(||x||)dx

14

Let be any non-increasing function. Then B(·) : R+ ! R+Theorem :

Clustering in Steady State

E0�0

The law of the configuration when viewed from a “typical” receiver. The Palm measure.

Configuration in steady-state

E0�0

2

4X

z2�Tx

0

\{y1

}

B(||z||)

3

5 � E

2

4X

z2�Tx

0

B(||z||)

3

5

y1y2

y3y4

x1

x2

x3x4

�0 sample of steady state configuration of links.

y1y2

y3y4

x1

x2

x3x4

15

�0

0 5 10 150

5

10

15

Clustering in Steady State

Corollary

B(r) = 1(r R)

(Clustering)

Proof

Take

Let be any non-increasing function. Then B(·) : R+ ! R+Theorem :

E0�0

2

4X

z2�Tx

0

\{y1

}

B(||z||)

3

5 � E

2

4X

z2�Tx

0

B(||z||)

3

5

Let , i.e. each link is a point. Then mean number of points around a typical point in steady state is larger than around a typical location of the network.

T = 0

16

Configuration in steady-state�0

What is the average density of links in steady-state ?

Mean Field Second order !Approximation

where is the smallest solution to the following fixed point equationIs

� =

�L

C log2

⇣1 +

l(T )N0+Is

⌘

I

s

= �L

Z

x2S

l(||x||)C log2

⇣1 +

l(T )N0+Is+l(||x||)

⌘

�

Mean- Delay = �

�(Little’s Law)

Formulas for Performance17

0.4 0.5 0.6 0.7 0.8 0.9 1

0

10

20

30

40

λ / λc

β

Simulations

Second Order Heuristic

Poisson Heuristic

Poisson heuristic - Neglect all correlations and assume steady-state is a Poisson Point Process.!!Second-Order heuristic - Our formula we gave in the previous slide.

Result - Formulas for Performance18

Proof Idea - Instability

Rate-Conservation - “On average, what comes in is what goes out”

Total speed at which!bits depart.

Total speed at !which bits arrive �|S|L = E

2

4X

x2�0

R(x,�0)

3

5

For example

KEY IDEA

Apply Rate-Conservation to “Total Interference in Network”.

If dynamics stable, then it must satisfy rate-conservation laws.

19

Proof Idea - InstabilityX

x2�0

I(x,�0) =X

x2�0

X

y2�0

l(||x� y||)Total Interference :=

Average increase in total !interference due to arrival

= Average decrease in total!interference due to departure

PASTA and linearity !of expectation

= 21

L

E

2

4X

x2�0

R(x,�0)I(x,�0)

3

5

Property of the minimum of independent !exponential random variables

2�|S|�Z

x2S

l(||x||)dx

20

Proof Idea - Instability

= 21

L

E

2

4X

x2�0

R(x,�0)I(x,�0)

3

5

This equality implies

� > �c =) unstable

(*)

Thus from (*)

2�|S|�Z

x2S

l(||x||)dx

R(x,�) = C log2

✓1 +

l(T )

N0 + I(x,�)

◆

xC log2

✓1 +

a

b+ x

◆ Ca log2(e)

Definition

and fact

Follows from

1)

2)

� Cl(T )

ln(2)LRx2S

l(||x||)dx= �

c

21

Summary and Contributions

• A new stochastic framework for Spectrum-Sharing!• Rate-Conservation argument for a class of Interacting Queuing Systems!• Proof of clustering!

A generative model for locations of wireless-links ?! Clustering is usually assumed in models [Dhillon et.al 16,17] and in ! 3GPP simulation standards !

• Mean-field heuristic for delay

Possible Extensions

• Algorithm design - A link can transmit only if the transmitter senses that! the interference is low, otherwise save power and do not interfere with ! others

22

Open Questions

• Delay Tails

P[D0 > x] = ?

Very useful for guaranteeing quality of service.

Delay experienced by a typical link in steady state.

• Network Scalability ?

Will the stability result hold if the size of the network was infinite

23

Ongoing work with Sergey Foss, Heriot-Watt University, Edinburgh and Institute of Mathematics, Novosibirisk, Russian academy of sciences.

An Interacting Queuing model on Zd

Proposed Future Work

Start with a discrete system before generalizing to the continuum system

Spatial Birth-Death Dynamics

If no, then the protocol is not scalableEvery arriving link transmits at full power till completion of file transfer

24

Increasing Time

(Time = t)

S

Does the spatial birth death dynamics have an unique stationary and ergodic invariant distribution on the infinite domain ?Rd

SINR at a queue.

25

Queues at every integer.

0 321-2 -1Z

�

Departure Probability

Independent Bernoulli arrival.

Discrete Time

Time

Independent Departures

An Interacting Queue Model

xi(t), i 2 Z, t 2 NLength of queue i at time t

In this figure, the departure probabilities are -!!! 3/4 for queue 0!! 0 for queue 1

xi(t)

xi�1(t) + xi(t) + xi+1(t)log(1 + SINR)Replaced by SINR

Discrete Time Dynamics 26

0 321-2 -1Z

�

xi(t)

xi�1(t) + xi(t) + xi+1(t)Departure Probability

Independent Bernoulli arrival.Time

Independent Departures

xi(t+ 1) = max(xi(t) +Ai(t)�Di(t), 0)

Di(t) ⇠ Bernoulli

✓xi(t)

xi�1(t) + xi(t) + xi+1(t)

◆Ai(t) ⇠ Bernoulli(�) independent across i and t.

independent across i and t.

i 2 Z, t 2 N


Discrete Time


SINR at a queue.

27

0 321-2 -1Z

Time


xi(t+ 1) = max(xi(t) +Ai(t)�Di(t), 0)

Di(t) ⇠ Bernoulli

✓xi(t)

xi�1(t) + xi(t) + xi+1(t)

◆Ai(t) ⇠ Bernoulli(�) independent across i and t.

independent across i and t.

i 2 Z, t 2 N

Conjecture - � <

1

3but not necessarily ergodic in high dimensions.Stability region is

Discrete Time Dynamics

Independent Poisson arrival.

0 321-2 -1Z

�Time


Continuous Time

Length of queue i at time t

Continuous Time Dynamics

xi(t), i 2 Z, t 2 R

xi(t)

xi�1(t) + xi(t) + xi+1(t)Instantaneous Rate of Departure SINR at queue i.

• Each Queue has an independent Poisson arrivals.!• The instantaneous rate of departure from queue i at time t is given by

�

xi(t)

xi�1(t) + xi(t) + xi+1(t)

28

!!

3. Average queue length for any invariant solution is ✓1

�� 3

◆�1

N

(Uniqueness of the invariant distribution due to monotonicity)

!2. If , and all queues are initially empty, then the queue lengths ! converge weakly to a non-degenerate distribution on

� <1

3

In Continuous Time, we can show that

(Existence of an invariant distribution)

Partial Progress

1. The dynamics is well defined and exists.(Coupling from the Past argument)

29

• The one dimensional discrete space model is the “bottleneck”. Can easily generalize to continuum space

• Future Work - Analyze the model in either discrete or continuous time. ! ! Is it always ergodic ? Interesting if the answer is no.!!We believe it could shed light on understanding delay tails in the finite model.

Future Proposed Work 30

Community Detection on an Euclidean Random Graph

A population partitioned into groups

• Identifying ‘groups’ of objects in a population given ! indirect information on group memberships.

Community Detection - Abstract Definition 31

A population

Community Detection - Examples

1. People on an Online Social Network grouped according to whether or ! not they like or dislike a particular product or content.!!2. Proteins classified into groups based on their functional behavior.!!3. Grouping Base-Stations based on similarities in traffic pattern.

partitioned into groups

• Identifying ‘groups’ of objects in a population given ! indirect information on group memberships.

31

Graph as Information

The data is structured as follows -

Membership Information - Encoded as a labeled edge of the graph.

An useful sub-class of the general problem

Population - Represented as nodes of a graph.

‘Stochastic Block Model’ - The simplest toy model to study this class of problems.

32

The simplest case, SBM(n,a,b) , is a random graph

Population of size n

Color uniformly and independently

Conditional on the colors, draw an !edge between two members with probability!! - a if they have same colors.! - b if they have different colors.!

Stochastic Block Model (SBM) 33

n 2 N a, b 2 [0, 1]

1. Sparse - (Finite Average Degree)

The SBM is either

!2. Non-Sparse - Average Degree goes to infinity as .

SBM for applications

n ! 1

The sparse SBM is ‘Tree-Like’ around any typical vertex !

Not very convincing in practice.

[Mossel, Neeman, Sly ’12]

34

Social networks are Sparse and transitive!Sparsity - Dunbar’s number : !!An average human being cannot have more than 200 relationships at any point of time. This bound is a fundamental cognitive limitation, not a limitation of resources.

If i and j are friends, j and k are friends then i and k are likely to be friends

Models for Social Network

Transitivity

i

j k

35

1. The members of a social network are points in a ‘Latent Social Space’.! This is typically an unobservable abstract space, but in certain applications, it can be geographic or some feature space (age, income).!!2. Conditional on the location in this latent space, edges are drawn

independently at random depending on the Euclidean distance.!

Latent Space Model

Our Network Model - The simplest ‘planted version’ of the above.

36

A class of models introduced by ![Hoff, Raftery, Handcock, 02], [Handcock, Raftery, Tantrum, 07].

• The random graph is parametrized by the following.!•

!•

!

� 2 R+

fin

(·), fout

(·) : R+ ! [0, 1] s.t 8r � 0 , fin

(r) � fout

(r)

Random Graph Model

Simplest possible starting point. Statistically the most general.

d � 2

• Locations of the points as the support of a Poisson Point Process.

37

Random Graph Model

, PPP of intensity � �with an additional point at the origin.

38

Random Graph Model

Color uniformly and independentlyZ0 2 {blue, red}color of origin.


39

Random Graph Model


Two points at distance are connected with probability!• - if they have same colors.!• - if they have opposite colors.

rfin(r)fout

(r)

Z0 2 {blue, red}color of origin.


Conditional on the location and !colors, draw edges independently. !

40

Random Graph Model


, PPP of intensity � �


8r � 0, fin

(r) � fout

(r)

Thus edge structure can have some information about the colors.


rfin(r)fout

(r)

with an additional point at the origin.

Z0 2 {blue, red}color of origin.

40

Random Graph Model - AlternativelyPlace points independently and !uniformly

�n�pn

2,

pn

2

�

Place an extra point at the origin.

pn

pn

41

Random Graph Model - Alternatively


pn

pn

Place points independently and !uniformly

�n�pn

2,

pn

2

�


42

Random Graph Model - Alternatively



rfin(r)fout

(r)

pn

pn

Place points independently and !uniformly �

pn

2,

pn

2

�

Ignore Edge Effects



n ! 1Let

�n

43

For

Denote by -

r

r 2 R+, � 2 (0, 1]

(G(�)r ,�(�)

r )

Community Detection Problem

The graph and the location of !points with the color of points!revealed with probability !for those points at distance!larger than r

�

44

We say community detection is solvable if

, there exists a random variable

which is measurable with respect to satisfying

9 ✏ > 0 such that for every


r 2 R+, � 2 (0, 1] ⌧ (�)r 2 {blue, red}

�⇣G(�)

r ,�(�)r

⌘

lim inf�!0

lim infr!1

P0[⌧ (�)r = Z0] =1

2+ ✏

r

(G(�)r ,�(�)

r )

Can you learn the color of the typical point!given the graph, position of all points !and some labels far away ?!!Does information flow from infinity ?

Definition

45

Note that for every and � > 0

(Maximum Likelihood by truncating the !graph for example)

r

P0[⌧ (�)r = Z0] >1

2


r

46

Note that for every and � > 0

(Maximum Likelihood by truncating the !graph for example)

r

P0[⌧ (�)r = Z0] >1

2

Question - Is ?

A metric to quantify the presence of ‘signal’ in the graph.


r

47

lim inf�!0

lim infr!1

P0[⌧ (�)r = Z0] >1

2

8fin

(·), fout

(·) and d ,Proposition (Easy Consequence of the Definitions)

48

r r

Independently remove points

Monotonicity

Transfer solution

What is non-trivial is to show that the problem can be solved for some

increasing makes the problem easier.�

�

Our Result

and

9

such that 8r � 0 , fin

(r) � fout

(r)8fin

(·), fout

(·), d � 2

{r � 0 : fin

(r) 6= fout

(r)} has positive Lebesgue measure, such that

•

•

Community Detection can’t be solved.

Our Algorithm solves the problem.

�lower

�upper

< 1

RemarkZ

r�0fin(r)r

d�1dr < 1If (finite average degree), then �lower

> 0

Prove this result by coupling and percolation arguments.

� < �lower

=)

� > �upper =)

49

�lower

>

✓Z

r�0(f

in

(r)� fout

(r))rd�1dr

◆�1

A Non-Trivial Phase Transition in the sparse regime.

• if they have the! same color.

Consider the simple example of fin(r) = a1rR fout

(r) = b1rR

and

Algorithm Idea

Key Idea - 1-hop neighborhood of ‘nearby’ points have a lot of signal.

The number of common neighbors is Poisson with mean

R

R

↵R , ↵ < 2

• if they have different colors.

50

�c(↵)Rd

✓a2 + b2

2

◆

�c(↵)Rdab

Both are of order �

Same color -

Opposite color -

Set threshold -

Pairwise-Classify(x,y)!• IF # (common neighbors) < T, DECLARE color(x) = color(y).!• ELSE DECLARE color(x) color(y).

P(Mis-classifying a given pair of nearby points) e�c0�

Chernoff

Algorithm Idea

R

R

↵R , ↵ < 2T (↵) = c(↵)Rd�

✓a+ b

2

◆2

6=

51

�c(↵)Rd

✓a2 + b2

2

◆

�c(↵)Rdab

Algorithm IdeaIterate the pairwise-classify to paths and ‘propagate’ a revealed label to the origin.

Naive guess - Take majority of the !propagated symbol from all paths to the origin.

Exponentially many paths !!!Almost all paths will have errors !

Obvious Roadblocks

r

52

Tesselate into grids of sideRd R/4

Cell Good if!

1. At-least the mean number of points!

2. No inconsistencies in pairwise! checks with all neighboring cells

1� ✏

Algorithm Idea

Same

Same

DifferentExample of Inconsistent output of !Pairwise-Classify

Classify cells to be Good or Bad.

53

Algorithm Idea

Majority of all good paths.

Propagate label down to origin only through A-Good cells

Polynomial time to search and!compute majority.

54

Proposed Future Work

• A new model of a random graph.!• Problem Definition is new and the non-trivial phase transition indicates

that the problem is posed at the correct scale.

• A Lower and Upper bound to the community detection problem.!• A novel spatial algorithm based on percolation coupling.!• Lower bound based on another percolation coupling arguments.

Future Work• Relax the assumption that spatial locations are known. !• Sharp Phase-Transitions in some regimes of the problem. !

• Help characterize and design ‘optimal’ algorithms.

Contributions

55

Base-station association in multi-operator networks.!

Joint work with Jeong-woo Cho, KTH Stockholm, Sweeden.

A new model of cellular service by

Background

Goal of the Project-!!How to leverage the presence of multiple cellular network operating on !separate frequencies.

56

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 Association57

Summary of Results• Model the network as a collection of independent PPPs with a typical

user located at the origin.

• A notion of information at the user as filtrations of a probability space.

• A notion of optimal association based on information.

• A heuristic and practical scheme - “Max-Ratio associate”.

• This heuristic is asymptotically optimal under a certain asymptotic.

• Use classical methods from Wireless Stochastic Geometry to evaluate performance gains due to optimal association. Technology Diversity.

58

Conclusion and Summary of the TalkWe presented two new stochastic models for problems in networking.

Spatial Birth Death Networks


New Analysis using Rate-Conservation arguments.Future Work - Consider the infinite domain.

New problem definition.Future Work - Relax some assumptions and consider different scaling regimes.

59

Publications1. “Spatial Birth Death Wireless Networks” with F. Baccelli, accepted for

publication in IEEE Transactions on Information Theory.!!2. “Performance Oriented Association in Large Cellular Networks with

Technology Diversity”, with J.woo-Cho and F.Baccelli in International Teletraffic Congress (ITC) 2016.

Other Paper

1. “CSMA k-SIC, A Class of Distributed MAC protocols and their performance evaluation” with F. Baccelli in INFOCOM 2015.

60

Thank You

Back-Up Slides

Spatial Birth-Death Wireless Network Model!

Back-Up Slides

Rate-Conservation - “On average, what comes in is what goes out”

Total speed at which!bits depart.

Total speed at !which bits arrive

Assume is the steady-state point process on with intensity !for the dynamics to guess the phase-transition point.

�0 �S

Using the definition of Spatial Palm probability, the above simplifies to

�|S|L = E

2

4X

x2�0

R(x,�0)

3

5

Intuition for Phase Transition

�L = �CE0�0

log2

✓1 +

l(T )

N0 + I(0,�0)

◆�

“On average, speed of arrival of bits equals speed of departure of bits.”

�0� ! 1Assume, that as , i.e. at the brink of instability - is Poisson !

(1)


�L = �CE0�0

log2

✓1 +

l(T )

N0 + I(0,�0)

◆�

“On average, speed of arrival of bits equals speed of departure of bits.”

Thus (1) simplifies to give

Under the assumption, concentration of interference holds, i.e.

�0

X

y2�0

l(||y||) ⇡ E[X

y2�0

l(||y||)] = �

Z

y2Sl(||x||)dx

� ! 1Assume, that as , i.e. at the brink of instability - is Poisson !

(1)


�L = �CE0�0

log2

✓1 +

l(T )

N0 + I(0,�0)

◆�

�L = �C log2

1 +

l(T )

N0 + �

Rx2S l(||x||)dx

!= f(�)

�0� ! 1Assume, that as , i.e. at the brink of instability - is Poisson !

The rate-conservation can be simplified to the following.

0 10 20 300

0.5

1

1.5

`

f(`)

The function ` A f(`)Assymptote hc

�L = f(�)

We need for the equationto hold.� < �c


�L = �C log2

1 +

l(T )

N0 + �

Rx2S l(||x||)dx

!= f(�)

0.4 0.6 0.8 10

5

10

15

20

25

h / hc

`

Poisson HeuristicSimulation

As expected, performs poorly.

�L = �E"log2

1 +

1

N0 +P

y2�0l(||y||)

!#

=

�f

ln(2)

Z 1

z=0

e�N0z(1� e�z

)

ze��

f

Rx2S(1�e

�zl(||x||))dxdz

The largest solution to the above fixed point equation gives a value for mean number of links in steady-state.

Poisson Heuristic

Rate-Conservation Equation

(By the Poisson assumption)

1. Any single tagged link interacts with a static non-random environment!!!!!2. Pairs of points are not independent

I

�s =�L

C log2

⇣1 +

1N0+I

⌘

Conditional on two receivers at and , they each “see” an interference of x

y

Pair of receivers -

The Mean-Field Approximation

I +

Z 2⇡

✓=0l(||x� y + Te

j✓||) d✓2⇡

They “see” each other and an independent environment.

19

Proof Idea - Clustering

FKG Inequality

= 21

L

E

2

4X

x2�0

R(x,�0)I(x,�0)

3

52�|S|�Z

x2S

l(||x||)dx

= 21

L

E[�0(S)]E

2

4 1

E[�0(S)]

X

x2�0

R(x,�0)I(x,�0)

3

5

21

LE[�0(S)]E0

�0[R(0,�0)]E0

�0[I(0,�0)]

= 21

LE[�0(S)]E0

�0[R(0,�0)I(0,�0)]

Pick one uniformly at random

Rearranging the inequality above gives the result.

23

Subset Monotonicity

y1y2

y3y4

x1

x2

x3x4

y1y2

y3

x1

x2

x3

�

�

� ✓ � =) 8x 2 �

Rx

,

I(x,�) I(x, �)

R(x,�) � R(x, �)

Configuration � = {(x1, y1), · · · , (xn, yn)}

R(xi,�) = B log2

✓1 +

l(T )

N0 + I(xi,�)

◆

Set of Transmitter locations�

Rx = {x1

, · · · , xn} Set of Receiver locations

�Tx = {y0

, · · · , yn}

and I(xi,�) =X

z2�Tx\{yi}

l(||xi � z||)

11

Proof Idea - Stability

Discretize continuum space.

✏

✏

• Construct an “Upper-Bound” discrete state ! space Markov Chain

• Conclude Stability of this by using the norm ! as the Lyapunov function.

l1

• Let ✏ ! 0

24


Consider points interacting with function !instead of

l✏(·, ·)l(·)

✏

✏


Assume link length = 0.



l✏(·, ·)l(·)

✏

✏



l✏(x, y) � l(||x� y||) 8x, y 2 S

l✏(·, ·) Constant inside a cell

•

•



l✏(·, ·)l(·)

✏

✏



l✏(x, y) � l(||x� y||) 8x, y 2 S


•

•

# of points in the cells is MarkovX(✏)t 2 Nn✏

+



l✏(·, ·)l(·)

✏

✏



l✏(x, y) � l(||x� y||) 8x, y 2 S


•

•

# of points in the cells is MarkovX(✏)t 2 Nn✏

+

X(✏)t < �t=)Monotonicity


✏

✏

• # of points in the cells is MarkovX(✏)t 2 Nn✏


✏

✏


Trick -!

Max-Lyapunov function in the fluid scale


✏

✏


Trick -!


X(✏)t < �t

Thus stable stable.X(✏)t =)

Have

�t


✏

✏


Trick -!


X(✏)t < �t

Thus stable stable.X(✏)t =)

✏ ! 0Let

Have

�t


Back-Up Slides

Sample an instance of SBM(n,a,b)

Erase the colors

Can you reconstruct the partition back ?

Community Detection on SBM 37

Other Possible Definition

Cluster just the nodes in a !ring of radius n.

n


Ignore edge effects.n

Cluster just the nodes in a !ring of radius .r

a.s. clustering finite sets.


Can you recover strictly larger than half the fraction of nodes as ? r ! 1

n

Ignore edge effects.

Cluster just the nodes in a !ring of radius .r

a.s. clustering finite sets.

Conjecture - This is equivalent to the our formulation.

Base-station association in multi-operator networks.!

Joint work with Jeong-woo Cho, KTH Stockholm, Sweeden.

Back-Up Slides

A new model of cellular service by

Background

Goal of the Project-!!How to leverage the presence of multiple cellular network operating on !separate frequencies.

56

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 Association57

First Guess - Connect to the nearest BS irrespective of the type of BS it is.

Nearest BS

SNR > SNR

Signal

Interference + NoiseSINR =

The “optimal” BS Association is not an obvious choice

(Basis for nearest BS association.)

First Guess - Connect to the nearest BS irrespective of the type of BS it is.

Nearest BS

SINR < SINR

SNR > SNR

Thus, in this example

But frequency are separate.

Signal

Interference + NoiseSINR =

The “optimal” BS Association is not an obvious choice

(Basis for nearest BS association.)

Mathematical Framework - Network Model

Number of networks.T

BS of each network form an independent PPP.

Evaluate the performance of a single “typical” user.

X21

X11

T = 2�1 =

�2 =

with intensity

with intensity

�1

�2

Mathematical Framework - Network Model

Number of networks.T

BS of each network form an independent PPP.

Evaluate the performance of a single “typical” user.

Mathematical Framework - Signal Model

Fading from jth nearest BS of operator i to the typical user -

Transmit Power of operator - i Pi

Signal of operator i attenuated with distance by li(·) : R+ ! R+

Hij

r1jr2j

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

P1H1j l1(r

1j )P2H

2k l2(r

2k)

Performance MetricsNon-overlapping bandwidths

SINRi,j0 SINR from the jth nearest BS of technology i.


pi(·) : R+ ! R+For technology denote the reward function.i



pi(·) : R+ ! R+For technology denote the reward function.i


User gets a reward of when associated with jth nearest !BS of technology i.

pi(SINRi,j0 )

X22

The reward received by the UE in this example is p2(SINR2,20 )

T = 2�1 =

�2 =

with intensity

with intensity

�1

�2

Performance Metrics

X22

T = 2�1 =

�2 =

with intensity

with intensity

�1

�2

Examples of common reward functions!• Coverage !• Average Achievable Rate

pi(x) = 1(x � �i)

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

Performance Metrics

The reward received by the UE in this example is p2(SINR2,20 )

Information at the UE

Goal- How to exploit learnt“information” for increased reward ?

Examples of Information that a UE can know - !• Nearest BS of all technologies.!

!• Instant fading and the distance to the nearest BS!

!• Noisy estimate of the instant fading from the nearest BS of each

technology.

k

k

k

Information at the UE

Goal- How to exploit learnt“information” for increased reward ?

How Information affects Optimal Association

SINR < SINRWhen averaged across fading

SINR < SINR


When averaged across fading

Sudden very good signal which the UE can sense.

SINR < SINR



Sudden very good signal which the UE can sense.

In this case, UE should associate to

SINR < SINR



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

(⌦,F ,P) The Probability space containing the random elements ! and {�i}T1=1 {Hi

j}i2[1,T ],j2N-

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

FI ✓ F The information sigma-algebra.-

Information at the UE - Formalization


j}i2[1,T ],j2N-

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

FI ✓ F The information sigma-algebra.-

Information at the UE - Formalization


j}i2[1,T ],j2N-

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

This is measurable function denoting the association scheme.

-⇡ : ⌦ ! [1, T ]⇥ N FI

For any policy , the expected reward ⇡ R⇡ = E[p⇡(0)(SINR⇡0 )]

Optimal Association


Optimal Association

Optimal Policy - ⇡⇤I = arg sup

⇡R⇡

I

(sup over all measurable functions )⇡ : ⌦ ! [1, T ]⇥ NFI


Optimal Association

Optimal Policy - ⇡⇤I = arg sup

⇡R⇡

I

(sup over all measurable functions )⇡ : ⌦ ! [1, T ]⇥ NFI

Proposition:

⇡⇤I = arg sup

i2[1,T ],j2NE[pi(SINRi,j

0 )|FI ] a.s.

Optimal Association

R⇡⇤

I = E[p⇡⇤(0)(SINR⇡⇤

0 )]Let Optimal Reward

Optimal Association

R⇡⇤

I = E[p⇡⇤(0)(SINR⇡⇤

0 )]Let Optimal Reward

Comparison of schemes without messy computation !

Theorem - “More information gives better performance”

FI1 ✓ FI2 =) R⇡⇤

I1 R⇡⇤

I2

Computation of Performance

Explicit Formula for the Reward ?

Show few natural examples.

Example 1 - No Information

The User knows nothing about the network.

Optimal Association - Connect to the nearest BS of technology.! wherei⇤ i⇤ = arg max

iı[1,T ]E[pi(SINRi,1

0 )]

NO Technology Diversity Gain

Example 2 - Complete Network Information

(An unrealistic example !! )

A UE knows the instantaneous fading and the distances to all BS of all technologies.

⇡⇤ = arg supi2[1,T ],j�1

pi(SINRi,j0 ) (Exhaustive Search Algorithm)

Maximum Technology Diversity Gain

Example 3 - Nearest BS distances known

The UE knows the nearest BS distance of all technologies.

pi(x) = 1(x � �i)

The optimal association is to connect to the nearest BS of !technology wherei⇤

Let the reward function be

i⇤ = arg max

i2[1,T ]e�µN0(r

i1)

↵P�1i

exp

�2⇡�i

Z

u�ri1

1

1 + ��1i (u/ri1)

↵udu

!

Some Technology Diversity Gain

Optimal Association - depends on the statistical description of the network.

We propose a “data-dependent” association.

⇡m =

arg max

i2[1,T ]

ri1ri2

, 1

�

Associate to the nearest BS of the technology yielding the maximum ratio.

(Oblivious to the statistical assumptions on the network.)

Example 4 - Max-Ratio Association

⇡m =

arg max

i2[1,T ]

ri1ri2

, 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 = �

✓⇢ri1ri2

�◆

Max-Ratio Association - A Pragmatic Scheme

Theorem - Under the following conditions!• All reward functions are identical.!!• Path-loss function is power law i.e. !• The nearest BS distance is known to the UE for! any , i.e. ! Then, the following almost sure convergence takes place.!!

!

!

Max-Ratio Association - Asymptotic Optimality

pi(·) = p(·)8i

l

(↵)(x) = x

�↵

k

⇡⇤↵

↵!1��!arg max

i2[1,T ]

ri1ri2

, 1

�

FI = ��{ri1, · · · , rik}i2[1,T ]

�k � 2

Max-Ratio is optimal Policy in “harsh” wireless environments. Moreover, no gain obtained by spending resources at a UE in !learning the network.

2.5 3 3.5 4 4.5 5 5.5 6 6.5 7

Path-loss Exponent α

0

2

4

6

8

10

12

Ave

rage A

chie

vable

Rate

Opt-Association (1 BS)

Opt-Association (2 BS)

Random BS Association

Nearest BS Association

Max-Ratio Association

Max-Ratio Association - Finite Scale Behavior

⇡⇤↵

↵!1��!arg max

i2[1,T ]

ri1ri2

, 1

�a.s.

For , Max-Ratio !is nearly optimal.

↵ � 5

Summary and Contributions

• Framework for Base-Station association.

• Characterization of optimal solutions.

• Simple heuristic scheme that is asymptotically optimal.

A point in a crowded region of space is slowed down and in turn slows!down others near it.

Intuitively, expect some form of clustering in steady state which the theorem formalizes.

An Understanding of Clustering

R(x,�1) � R(x,�2)

x

y1 y2

y3y4

x

y1 y2

y3

�1 = {(x, Tx(x)), (y1, Tx(y1)), (y2, Tx(y2)),(y3, Tx(y3)), (y4, Tx(y4))}

�1 = {(x, Tx(x)), (y1, Tx(y1)), (y2, Tx(y2)),(y3, Tx(y3))}

Special CaseCase when T=0.!( Link lengths are very small compared to network size.)

(Configuration at time t).�t = {x1, · · · , xNt} , xi 2 S

The qualitative features (mathematically) retained.

R(x,�) = C log2

1 +

1

N0 +P

y2�t\{x} l(||y � x||)

!

The rate function

SBD Model - Special Case

Spatial Domain S

Increasing Time

(Time = t) The red lines represent space-time !position of points that are either dead !by time t or not yet born at time t.

The “ “ appear as a PPP on S⇥ R

The green represent �t

abishek sankararaman phd qualifying exam supervisor

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