Abishek Sankararaman PhD Qualifying Exam Supervisor ...
Transcript of 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
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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
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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
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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
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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 ?
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!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.
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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
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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 �
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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
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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
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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
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�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
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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
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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.
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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
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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
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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
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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
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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
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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.
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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
Queues at every integer.
Discrete Time
xi(t), i 2 Z, t 2 NLength of queue i at time t
SINR at a queue.
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0 321-2 -1Z
Time
xi(t), i 2 Z, t 2 NLength of queue i at time t
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
Queues at every integer.
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)
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!!
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)
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• 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]
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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.
, PPP of intensity � �with an additional point at the origin.
39
Random Graph Model
Color uniformly and independently
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.
, PPP of intensity � �with an additional point at the origin.
Conditional on the location and !colors, draw edges independently. !
40
Random Graph Model
Conditional on the location and !colors, draw edges independently. !
, PPP of intensity � �
Color uniformly and independently
8r � 0, fin
(r) � fout
(r)
Thus edge structure can have some information about the colors.
Two points at distance are connected with probability!• - if they have same colors.!• - if they have opposite 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
Color uniformly and independently
pn
pn
Place points independently and !uniformly
�n�pn
2,
pn
2
�
Place an extra point at the origin.
42
Random Graph Model - Alternatively
Conditional on the location and !colors, draw edges independently. !
Two points at distance are connected with probability!• - if they have same colors.!• - if they have opposite colors.
rfin(r)fout
(r)
pn
pn
Place points independently and !uniformly �
pn
2,
pn
2
�
Ignore Edge Effects
Color uniformly and independently
Place an extra point at the origin.
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
Community Detection Problem
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
Community Detection Problem
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.
Community Detection Problem
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
Community Detection on an Euclidean Random Graph
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)
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.”
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)
Intuition for Phase Transition
�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
Intuition for Phase Transition
�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
Proof Idea - Stability
Consider points interacting with function !instead of
l✏(·, ·)l(·)
✏
✏
Discretize continuum space.
Assume link length = 0.
Proof Idea - Stability
Consider points interacting with function !instead of
l✏(·, ·)l(·)
✏
✏
Discretize continuum space.
Assume link length = 0.
l✏(x, y) � l(||x� y||) 8x, y 2 S
l✏(·, ·) Constant inside a cell
•
•
Proof Idea - Stability
Consider points interacting with function !instead of
l✏(·, ·)l(·)
✏
✏
Discretize continuum space.
Assume link length = 0.
l✏(x, y) � l(||x� y||) 8x, y 2 S
l✏(·, ·) Constant inside a cell
•
•
# of points in the cells is MarkovX(✏)t 2 Nn✏
+
Proof Idea - Stability
Consider points interacting with function !instead of
l✏(·, ·)l(·)
✏
✏
Discretize continuum space.
Assume link length = 0.
l✏(x, y) � l(||x� y||) 8x, y 2 S
l✏(·, ·) Constant inside a cell
•
•
# of points in the cells is MarkovX(✏)t 2 Nn✏
+
X(✏)t < �t=)Monotonicity
Proof Idea - Stability
✏
✏
• # of points in the cells is MarkovX(✏)t 2 Nn✏
Proof Idea - Stability
✏
✏
• # of points in the cells is MarkovX(✏)t 2 Nn✏
Trick -!
Max-Lyapunov function in the fluid scale
Proof Idea - Stability
✏
✏
• # of points in the cells is MarkovX(✏)t 2 Nn✏
Trick -!
Max-Lyapunov function in the fluid scale
X(✏)t < �t
Thus stable stable.X(✏)t =)
Have
�t
Proof Idea - Stability
✏
✏
• # of points in the cells is MarkovX(✏)t 2 Nn✏
Trick -!
Max-Lyapunov function in the fluid scale
X(✏)t < �t
Thus stable stable.X(✏)t =)
✏ ! 0Let
Have
�t
Community Detection on an Euclidean Random Graph
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
Other Possible Definition
Ignore edge effects.n
Cluster just the nodes in a !ring of radius .r
a.s. clustering finite sets.
Other Possible Definition
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.
Performance MetricsNon-overlapping bandwidths
pi(·) : R+ ! R+For technology denote the reward function.i
SINRi,j0 SINR from the jth nearest BS of technology i.
Performance MetricsNon-overlapping bandwidths
pi(·) : R+ ! R+For technology denote the reward function.i
SINRi,j0 SINR from the jth nearest BS of technology 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
How Information affects Optimal Association
When averaged across fading
Sudden very good signal which the UE can sense.
SINR < SINR
How Information affects Optimal Association
When averaged across fading
Sudden very good signal which the UE can sense.
In this case, UE should associate to
SINR < SINR
How Information affects Optimal Association
When averaged across fading
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
(⌦,F ,P) The Probability space containing the random elements ! and {�i}T1=1 {Hi
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
(⌦,F ,P) The Probability space containing the random elements ! and {�i}T1=1 {Hi
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
For any policy , the expected reward ⇡ R⇡ = E[p⇡(0)(SINR⇡0 )]
Optimal Association
Optimal Policy - ⇡⇤I = arg sup
⇡R⇡
I
(sup over all measurable functions )⇡ : ⌦ ! [1, T ]⇥ NFI
For any policy , the expected reward ⇡ R⇡ = E[p⇡(0)(SINR⇡0 )]
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