Stochastic Geometry and Wireless Networks

Stochastic Geometry and Wireless Networks Radha Krishna Ganti Department of Electrical Engineering Indian Institute of Technology, Madras Chennai, India 600036

Email:

[email protected] SPCOM 2012

July 22, 2012

What is stochastic geometry? Stochastic geometry is the study of random spatial patterns I

Point processes

I

Random tessellations

I

Stereology

Applications I

Astronomy

I

Communications

I

Material science

I

Image analysis and stereology

I

Forestry

I

Random matrix theory GRK (IITM)

Stochastic Geometry and Wireless Nets.

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Application to wireless networks

I I

Interference is a major limitation Networks are getting heterogeneous and decentralized GRK (IITM)


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Outline

* Primer on Point Processes * Ad hoc Networks * Cellular Networks * Heterogeneous Networks

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Primer on Point Processes

Primer on Point Processes *

Primer on Point Processes Poisson point process Transformations of PPP Reduced Palm probability

* Ad hoc Networks * Cellular Networks * Heterogeneous Networks GRK (IITM)


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What is a spatial point process? Let N be the set of all sequences φ ⊂ R2 satisfying 1. (Finite) Any bounded set A ⊂ R2 contains nite number of points. 2. (Simple) xi 6= xj if i 6= j .

Denition A point process1 in R2 is a random variable taking values in the space N. P A simple representation: Φ = i δXi

Notation:

1. Point process is denoted by Φ; An instance of the point process is denoted by φ 2. Number of points of the point process in a set A ⊂ R2 : Φ(A) 1 D.J. Daley, D. Vere-Jones , An introduction to the theory of point processes, Vol 1 and 2, Springer GRK (IITM)


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Example 1: An interesting but trivial point process 1. Contains only one point 2. The random point x is uniformly distributed in a bounded set A.

Uniform distribution

BPP: N=1, A= [−10,10]2 10 8 6 4

Let B ⊂ A

2 0

|B| P(x ∈ B) =

|A|

,

where |A| denotes the area of the set A.

Caveat: Dened only on bounded set, i.e., |A| < ∞. GRK (IITM)

−2

B

−4 −6 −8 −10 −10


−5

0

5

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Example 2: Binomial point process (BPP) BPP: N=20, A= [−10,10]2 10

A BPP on a set A is the superposition of N independent uniformly distributed points on the set A. Let B ⊂ A, then P(Φ(B) = k) =

N |B| k |B| N−k 1− k |A| |A|

8 6 4 2 0 −2 −4 −6 −8 −10 −10

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B


−5

0

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Other interesting examples 10

8

5 6

4 4

3 2

2 0

1 −2

0 −4

−1 −6

−2 −8

−3 −10 −15

−4

Hexagonal lattice

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−5 −5

−4

−3

−2

−1

0

1

2

3

4

5

Thomas cluster process


−10

−5

0

5

10

Determinantal point process (eigenvalues of a Gaussian matrix)

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Characterization of a point process

Given two point processes, is there a simple way to see if both of them are equivalent? A simple point process Φ is determined by its void probabilities over all compact sets, i.e., P(Φ(K ) = 0) for K ⊂ R2 and compact. I

This means that two point processes are equivalent if they have the same void probability distribution (for all sets).

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Stationary point processes Denition (Stationary point process) A point process is stationary if its distribution is invariant with respect to translations. I

The point process looks statistically similar from any point in space.

I

BPP is not a stationary point process.

I

A stationary point process cannot be dened on a subset of R2

The density of a stationary point process Φ is dened as

E[Φ(B)] , |B|

B ⊂ R2 .

The RHS does not depend on the particular choice of the set B . GRK (IITM)


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Stationary Poisson point process (PPP) 1. The most widely used model for spatial locations of nodes I I

Most amicable to analysis "Gaussian of point processes"

2. No dependence between node locations 3. Random number of nodes 4. Can be dened on the entire plane I

Limiting distribution of a BPP

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PPP of density 2 10 8 6 4 2 0 −2 −4 −6 −8 −10 −10

−5


0

5

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PPP: Formal denition A stationary Poisson point process Φ of density λ is characterized by 1. The number of points in a bounded set A ⊂ R2 has a Poisson distribution with mean λ|A|, i.e.,

(λ|A|)n P(Φ(A) = n) = exp(−λ|A|) n! 2. The number of points in disjoint sets are independent, i.e., for A ⊂ R2 , B ⊂ R2 and A ∩ B = ∅,

Φ(A) ⊥ Φ(B)

PPP of density 2 10 8 6 4

A

2 0 −2 −4

B

−6 −8 −10 −10

−5

0

5

10

A stationary PPP is completely characterized by a single number λ. GRK (IITM)


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Poisson point process

Properties of PPP Lemma The density of the PPP (as dened in previous slide) is λ. Proof: Let A ⊂ R2 . Then

E[Φ(A)] =

∞ X

n exp(−λ|A|)

n=0

(λ|A|)n n!

= λ|A|, which follows from the mean of a Poisson random variable. Hence

E[Φ(A)] = λ. |A| Observe that the above expression does not depend on the set A. GRK (IITM)


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Properties... Lemma Let A ⊂ R2 . Conditioned on the number of points Φ(A), the points are independently and uniformly distributed in the set A, i.e., the points form a BPP. Proof: We consider the void probability of a set K ⊂ A.

0

0

P(Φ(K ) = ∩ Φ(A) = n) P(Φ(A) = n) P(Φ(K ) = )P(Φ(A \ K ) = n) = P(Φ(A) = n)

P(Φ(K ) = |Φ(A) = n) =

0

e −λ|K | e λ|A\K | (λ|A \ K |)n /n! e λ|A| (λ|A|)n /n! n |W \ K |n |K | = − . = |A|n |A|

=

1

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A, Φ(A) = n

K

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Simulation of a PPP How to simulate a PPP of density λ on A = [−L, L]2 ? 1. The number of points in the set A is a Poisson random variable with mean λ|A|. 2. Conditioned on the number of points, the points are uniformly distributed as a BPP.

Matlab code N = poissrnd(λ|A|) Points = unifrnd(−L, L, N, 2)

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Distance properties of a PPP First contact distribution The CCDF of the distance of the nearest point of the process from the origin denoted by D is P(D ≥ r ) = exp(−λπr 2 ). Proof:

P(D ≥ r ) = P(B(o, r ) is empty ) = exp(−λ|B(o, r )|)

r

= exp(−λπr 2 ) Hence the PDF equals fN (r ) = 2λπr exp(−λπr 2 ). The average distance is Z ∞ 1 E[D] = r 2λπrfN (r )dr = √ . 2 λ 0 GRK (IITM)


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N -th closest point The CDF2 of the N -th closest point to the origin equals

P(Dn ≥ r ) =

n−1 X k=0

e −λπr

2 (λπr 2 )k

k!

=

Γ(n, λπr 2 ) (n − 1)!

The average distance to the N -th closest point equals r Γ(n + 21 ) n ∼ E[Dn ] = √ πλ πλΓ(n)

2 M. Haenggi, "On Distances in Uniformly Random Networks," IEEE Transactions on Information Theory, vol. 51, pp. 3584-3586, Oct. 2005 GRK (IITM)


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Sums over PPP Lemma (Campbells theorem) Let Φ be a PPP of density λ and f (x) : R2 → R+ . E[

X

Z f (x)] = λ

R2

x∈Φ

f (x)dx

Proof: We have

E[

X

f (x)] = lim E[ R→∞

x∈Φ

X

f (x)].

x∈Φ∩B(o,R)

Let n = Φ(B(o, R)). Conditioning on the number of points n,      X X E f (x) = En E  f (x) n x∈Φ∩B(o,R)

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x∈Φ∩B(o,R)


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Since conditioned on the number of points, the points are i.i.d uniform Z X f (x) E[ f (x)|n] = n dx. |B(o, R)| B(o,R) x∈Φ∩B(o,R)

Averaging over n

Z

X

E[

f (x)] = E[n] B(o,R)

x∈Φ∩B(o,R)

f (x) dx |B(o, R)|

As E[n] = λ|B(o, R)|, and tending R → ∞ we obtain the result. Let g (x, y ) : R2 × R2 → R+ . Then3

E

6= X x,y ∈Φ

g (x, y ) = λ

2

Z R2

Z R2

g (x, y )dx dy .

D. Stoyan, W. Kendall, and J. Mecke, "Stochastic Geometry and Its Applications", 2nd ed. John Wiley and Sons, 1996 3

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Example: Mean and variance of interfernece

Let the transmitters be distributed as a PPP Φ of density λ.

Denition (Interfernce) 5

The interfernce (sum power) at location y ∈ R2 is X I (y ) = `(x − y ),

4

3

2

1

0

−1

x∈Φ −2

where `(x) is the path-loss function.

−3

−4

−5 −5

−4

−3

−2

−1

0

1

2

3

4

5

Mean of interference: By Campbell theorem, E[I (y )] = E[

X x∈Φ

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Z `(x − y )] = λ

R2

`(x − y )dx = λ


Z R2

`(x)dx July 2012

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Variance of interference: !2 

 X

2

E[I (y ) ] = E 

`(x − y )

"

=E

X

`(x − y )2 ] + E[

x∈Φ

Z =λ

2

=λ

R2

6= X

!# X

`(z − y )

z∈Φ

`(x − y )`(z − y )]

x,z∈Φ

Z

Z

`(x − y ) dx + λ `(x − y )`(z − y )dx dz R2 R2 Z `(x)2 dx + (λ `(x)dx)2 2

ZR

`(x − y )

x∈Φ

x∈Φ

= E[

! X

2

R2

Hence variance equals, var(I (y )) = λ

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Z R2

`(x)2 dx.


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Products over PPP Lemma (Probability generating functional (PGFL)) Let Φ be a PPP of density λ and f (x) : R2 → [0, 1] be a real valued function. Then " E

#

Y

Z f (x) = exp −λ (1 − f (x))dx . R2

x∈Φ

Proof: We prove the result for Ψr = Φ ∩ B(o, r ). Observe that Ψr is a PPP with number of points n distributed as a Poisson random variable with mean λπr 2 .     Y Y E f (x) = En E  f (x) n x∈Ψr

x∈Ψr

= En E[f (x)]n GRK (IITM)


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But E[f (x)] =

1

πr 2

B(o,r ) f (x)dx .

R

" E

# Y


Hence

1

"

f (x) = En

Z

πr 2

x∈Ψr

B(o,r )

f (x)dx

!n # .

Let z > 0. Let n be a Poisson random variable with mean a. Then

E[z n ] = exp(−a(1 − z)).



 E

Y

f (x) = exp −λπr 2

x∈Ψr

= exp −λ

1 1− 2 πr

Z

!!

Z

f (x)dx

B(o,r )

! (1 − f (x))dx

.

B(o,r )

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Application of PGFL: Laplace transform of interference "

!#

E[exp(−sI (y ))] = E exp −s

X

`(x − y )

.

x∈Φ

This can be rewritten as,

" E[exp(−sI (y ))] = E

# Y

exp (−s`(x − y )) .

x∈Φ

Using the PGFL,

E[exp(−sI (y ))] = exp −λ Substituting x − y → x , we have Z LI (s) = exp −λ

R2

GRK (IITM)

Z R2

1−e

1−e

−s`(x−y )

−s`(x)


dx .

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Transformations of PPP

Transformations of PPP: Independent Thinning Let Φ be a PPP of density λ 1. A node x ∈ Φ is coloured red with probability p and blue with probability 1 − p . 2. Let Φr denote the red point process and Φb denote the blue point process. So we have Φ = Φr ∪ Φb . Can be used to model ALOHA MAC protocol.

Lemma (Thinning) 1. Φr is a PPP of density λp 2. Φb is a PPP of density λ(1 − p) 3. Φr is independent of Φb . Proof: Look at void probabilities of Φr and Φb . GRK (IITM)


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Matern hard-core process: Dependent thinning 1. Begin with a PPP Φ of density λ. 2. To each x ∈ Φ, associate a mark mx ∼ U[0, 1] independent of every other point. 3. A node x ∈ Φ selected if it has the lowest mark among all the points in the ball B(x, R).

Ψ = {y : y ∈ Φ, my ≤ mx , ∀x ∈ B(y , R)∩Φ}

A minimum distance process for modelling CSMA MAC.

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0.2 0.3

0.03

0.25 0.8

0.4

0.35

0.62

0.9

0.12

Ψ = {y : y ∈ Φ, my ≤ mx , ∀x ∈ B(y , R)∩Φ}


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0.2 0.3

0.03

0.25 0.8

0.4

0.35

0.62

0.9

0.12

Ψ = {y : y ∈ Φ, my ≤ mx , ∀x ∈ B(y , R)∩Φ}


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0.2 0.3

0.03

0.25 0.8

0.4

0.35

0.62

0.9

0.12

Ψ = {y : y ∈ Φ, my ≤ mx , ∀x ∈ B(y , R)∩Φ}


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Density of Matern hard-core process A node x ∈ φ is retained with probability

p = P(mx ≤ my , ∀y ∈ B(x, R) ∩ Φ). I I

Let the mark of x be equal to t ∈ [0, 1]. Let n represent the number of points of Φ in B(x, R). I

I I

.

n ∼ Poi(λπR 2 )

Conditioned on the mark t , the probability that x is selected equals E[(1 − t)n ] = exp(−λπR 2 t). Averaging over t , Z 1 1 − exp(−λπR 2 ) p= exp(−λπR 2 t)dt = . λπR 2 0

So the nal density of the process equals λm = pλ. λm = GRK (IITM)

1 − exp(−λπR 2 ) πR 2


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Reduced Palm probability

Conditioning: Reduced Palm probability I

Notion of a typical point, i.e., conditioning on the existence of a node at a particular location

Nearest-neighbour distribution function The distance of the nearest neighbour from a "typical" point.

D(r ) = P(Φ(B(o, r ) = 1|o ∈ Φ) = Po (Φ(B(o, r ) = 1) = P!o (Φ(B(o, r ) = 0) Probability conditioned on there being a point at the origin (but not counting it).

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A spatial average interpretation 6

The reduced Palm probability can also be interpreted as a spatial average

4

2

0

P (Y ) = lim E !o

R→∞

X x∈Φ∩B(o,R)

P(Φ−x \ {x} ∈ Y ) . λπR 2

−2

−4

−6 −6

−4

−2

0

2

4

6

Palm distribution of PPP: Slivnyak theorem P!o = P,

i.e., reduced Palm distribution of a PPP equals the original distribution. Hence for a PPP, a new point can be added to the process without disturbing other points of the process. GRK (IITM)


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Campbell Mecke Theorem

Let f (x, φ) : R2 × N → [0, ∞] be a real valued function, Z X E[ f (x, Φ \ {x})] = λ E!o [f (x, Φ)]dx. R2

x∈Φ

Hence for a PPP,

E[

X x∈Φ

GRK (IITM)

Z f (x, Φ \ {x})] = λ

R2

E[f (x, Φ)]dx.


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Ad hoc Networks

Analysis of Ad Hoc Networks * Primer on Point Processes * Ad hoc Networks SINR analysis Interfernece correlation * Cellular Networks * Heterogeneous Networks GRK (IITM)


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Ad hoc Networks

SINR

analysis

Dipole model I

I

The transmitters are distributed as a PPP Φ of density λ Each transmitter has a receiver at a distance d in a random direction I

I

Not part of the process Φ.

Path loss function is denoted by `(x)

1. Examples: `(x) = kxk−α , `(x) = min{1, kxk−α }. 2. α > 2 3. Path loss between nodes x and y is `(x − y ).

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5

4

3

2

1

0

−1

−2

−3

−4

−5 −5

−4

−3


−2

−1

0

1

2

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Ad hoc Networks

SINR

analysis

Dipole model I

I


I




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6

4

2

0

−2

−4

−6 −6

−4


−2

0

2

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Ad hoc Networks

SINR

analysis

Dipole model I

I


I




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6

4

2

0

−2

−4

−6 −6

−4


−2

0

2

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Ad hoc Networks

SINR

analysis

System model I

All nodes transmit at the same power

I

TX has Nt transmit antenna

I

RX has Nr receive antenna Fading between any two nodes is i.i.d Rayleigh

I

1. The fading power is exponentially distributed with unit mean. 2. The fading power between nodes is denoted by hxy I

P(hxy ≥ z) = exp(−z)

What is the performance of a "typical" link?

3 F. Baccelli, B. Blaszczyszyn, P. Muhlethaler, "An ALOHA protocol for multihop

mobile wireless networks," Information Theory, IEEE Transactions on , vol.52, no.2, pp. 421- 436, Feb. 2006 GRK (IITM)


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Ad hoc Networks

SINR

analysis

SISO ad hoc network: Nt = Nr = 1 Typical link: By Slivnyaks theorem, we can add a new reference transmitter with its receiver at the origin The Signal-to-interference-noise ratio (after processing) between the receiver at the origin and its corresponding transmitter is h d −α SINR = 2 , σ + I (o) 6

4

where I (o) is the interference at receiver at origin X I (o) = hyo ky k−α . y ∈Φ\{x}

2

0

−2

−4

How to compute P(SINR ≥ θ) for the reference link? GRK (IITM)

−6 −6

−4


−2

0

2

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Ad hoc Networks

SINR

analysis

SINR distribution ps (θ, λ) = P(SINR(o) > θ) = P

h d −α ≥θ σ 2 + I (o)

= P h ≥ d α θ(σ 2 + I (o)) = E exp(−d α θ(σ 2 + I (o))) = exp(−d α θσ 2 ) E exp(−dα θI (o)) | {z } T1 =LI (o) (d α θ)

Observe that T1 is the Laplace transform of I (o) evaluated at s = d α θ. We now evaluate the Laplace transform of interference X LI (o) (s) = E exp(−s hyo ky k−α ) y ∈Φ

=E

Y

exp(−s hyo ky k−α )

y ∈Φ

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Ad hoc Networks

SINR

analysis

Laplace transform of interference Y

LI (o) (s) = E

exp(−s hyo ky k−α )

y ∈Φ (a)

=E

Y

Ehyo exp(−s hyo ky k−α )

y ∈Φ (b)

= E

Y y ∈Φ

1 1 + sky k−α

(a) follows from the independence of the fading random variables and (b) follows from the Laplace transform of an exponential random variable.

Recall PGFL of a PPP E

Y y ∈Φ

GRK (IITM)

Z f (x) = exp −λ

R2

1 − f (x)dx


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Ad hoc Networks

SINR

analysis

Using the PGFL of a PPP,

Z 1 LI (o) (s) = exp −λ 1− dx 1 + sky k−α R2 Z 1 = exp −λ dx − 1 ky kα R2 1 + s = exp(−λs 2/α C (α)), where C (α) =

2 2π α sin(2π/α) .

The CCDF of SINR is ps (θ, λ) = P(SINR(o) > θ) = exp(−d α θσ 2 )LI (o) (d α θ)

= exp(−d α θσ 2 ) exp(−λd 2 θ2/α C (α)) | {z }| {z } Noise

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Interference

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Ad hoc Networks

SINR

analysis

Multiple antenna systems:

Post-processing SIR depends on the processing at the receiver I Maximal-ratio combining (MRC), Nt = 1, Nr = n : Let hx denote 1 × n channel vector from a node x to the receiver at the origin. The received signal is

Y =

X hx kxk−α/2 ho d −α/2 √ √ ao + ax , n n x∈Φ

where ax are the transmitted symbols. Hence the received SIR after multiplying with hoH is SIR =

I

kho k2

1 H 2 −α n |ho ho | d P 1 H 2 −α x∈Φ |ho hx | kxk n

=P

kho k2 d −α

hoH 2 −α x∈Φ | kho k hx | kxk

is χ2 distributed with 2n degrees of freedom. is exponentially distributed with unit mean.

hH I | o h |2 kho k x

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Ad hoc Networks

I

SINR

analysis

Zero forcing receiver (ZF), Nt = n, Nt = n, # streams =n. Looking at the k -th stream the received vector is

√

nYk = ho (k)d −α/2 aok +

n X

ho (i)d −α/2 aoi +

n XX

hx (n)kxk−α/2 ax

x∈Φ i=1

i=1,i6=k

hx (i) is the i -th column of the channel matrix between the node x and the tagged receiver. Multiplying with the v that is orthogonal to the vectors ho (i), i 6= k the received SINR is Sd −α −α x∈Φ Sx kxk

SIR = P

1. S is exponentially distributed. 2. Sx is also exponentially distributed. GRK (IITM)


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Ad hoc Networks

SINR

analysis

The post process signal-to-interference ratio (after processing) is generally of the form Sd −α SIR = 2 , σ + I (o) P where I (o) = y ∈Φ\{x} hˆyo ky k−α is the interference at receiver at origin. Let the CCDF of S be

F (x) =

n X

ak x k exp(−bk x)

k=0

For example 1. When S is exponentially distributed, n = 1, a1 = 1 and b1 = 1 2. When S is χ2 with 2m degrees of freedom, n = m − 1, ak = bk = 1.

GRK (IITM)


1

k!

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Ad hoc Networks

SINR

analysis

Laplace trick ps (θ, λ) = P

Sd −α ≥θ I (o)

α

= P (S ≥ θd α I (o))

= EF (S ≥ θd I (o)) =

n X

h i ak E (θd α I (o))k exp(−bk θd α I (o))

k=0

=

n X

h

i ak (θd α )k E I (o)k exp(−bk θd α I (o))

k=0

k −x

E[x e

k k k d −xs k d ] = E (−1) e L (s) = (− 1 ) X k k ds ds s=1 s=1

Hence, ps (θ, λ) =

n X k=0

GRK (IITM)

ak θk d kα (−1)k

dk L (s) I (o) k ds s=bk θd α


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Ad hoc Networks

SINR

analysis

Summary: Key steps in the SINR evaluation (a) ps (θ, λ) = P h ≥ d α θ(σ 2 + I (o)) = E exp(−d α θ(σ 2 + I (o))) = exp(−d α θσ 2 )E exp(−dα θI (o)) Y = exp(−d α θσ 2 )E Lh (d α θky k−α ) y ∈Φ

= exp(−d α θσ 2 )E

Y y ∈Φ

1 1 + d α θky k−α

(b)

= exp(−d α θσ 2 )exp(−λd 2 θ2/α C (α))

1. The distribution of h being exponential in (a). 2. The distribution of the fading between the interferers and the tagged receiver is not crucial. 3. Using PGFL in (b). GRK (IITM)


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Ad hoc Networks

SINR

analysis

Interference distribution The Laplace transform of interference I =

−α y ∈Φ hyo ky k

P

LI (s) = exp(−λs 2/α C (α)), I

is given by

α > 2.

The Laplace transform of an alpha stable distribution with parameter 2/α.

1. Heavy tailed distribution. Not Gaussian4 . 2. No closed form expression for CDF 3. Integer moments don't exist. R I

I

I

I =

E[I ] = λ R2 kxk−α dx = ∞ An artefact of the singularity of path loss model `(x) = kxk−α at x = o.

−α y ∈Φ hyo ky k

P

is also referred as shot noise (SN) process.

4 R.K. Ganti and M. Haenggi. "Interference in ad hoc networks with general motion-invariant node distributions", ISIT, 2008 GRK (IITM)


July 2012

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Ad hoc Networks

SINR

analysis

Tail bounds on interference5 Evaluate the CCDF of P(I ≥ y ), where I = I

y ∈Φ hyo ky k

P

−α .

Divide the point process into two sets

Φy = {x ∈ Φ, hxo kxk−α > y } Φcy = {x ∈ Φ, hxo kxk−α ≤ y } I

Lower bound:

P(I ≥ y ) = P(IΦy + IΦcy ≥ y ) ≥ P(IΦy ≥ y ) = 1 − P(IΦy ≤ y ) = 1 − P(Φy = ∅) 4 S. Weber, X. Yang, J. G. Andrews and G. de Veciana, "Transmission Capacity of Wireless Ad Hoc Networks with Outage Constraints", IEEE Transactions on Information Theory, Vol. 51, No. 12, Dec. 2005 GRK (IITM) Stochastic Geometry and Wireless Nets. July 2012 45 / 89

Ad hoc Networks

SINR

analysis

I

P(Φy = ∅) = E

Y

1(hxo kxk−α < y )

x∈Φ

=E

Y

Ehxo 1(hxo kxk−α < y ) = E

x∈Φ

α

1 − e −y kxk

x∈Φ

Z = exp −λ

R2

I

Y

e

−y kxkα

dx

= exp −λy −2/α πΓ(1 + 2/α)

Upper bound

P(I ≥ y ) = P(I ≥ y |IΦy > y )P(IΦy > y ) + P(I ≥ y |IΦy ≤ y )P(IΦy ≤ y ) = P(IΦy > y ) + P(I ≥ y |IΦy ≤ y )P(IΦy ≤ y ) = 1 − P(Φy = ∅) + P(I ≥ y |IΦy ≤ y )P(Φy = ∅) = 1 − (1 − P(I ≥ y |IΦy ≤ y ))P(Φy = ∅) GRK (IITM)


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Ad hoc Networks I

SINR

analysis

Using Markov inequality

P(I ≥ y |IΦy ≤ y ) = P(I ≥ y |Φy = ∅) E[I ≥ y |Φy = ∅] ≤ y 1 X = E hxo kxk−α 1(hxo kxk−α ≤ y ) y x∈Φ Z λ kxk−α E[hxo 1(hxo kxk−α ≤ y )]dx = y R2 Z Z y kxkα λ −α = kxk he −h dhdx y R2 0 2πΓ (1 + 2/α) −2/α = y 2−α

1−e

−λy −2/α E[h2/α ]

GRK (IITM)

2πE[h2/α ] −2/α ≤ P(I ≥ y ) ≤ 1− 1 − y 2−α Stochastic Geometry and Wireless Nets.

! e −λy

−2/α E[h2/α ]

July 2012

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Ad hoc Networks

SINR

analysis

Interference is heavy tailed Lemma When path loss is given by `(x) = kxk−α , interference is heavy tailed P(I ≥ y ) ∼ λy −2/α E[h2/α ], y → ∞ Proof: We have 1−e

−λy −2/α E[h2/α ]

2πE[h2/α ] −2/α ≤ P(I ≥ y ) ≤ 1− 1 − y 2−α

! e −λy

−2/α E[h2/α ]

.

Use exp(−x) ∼ 1 − x , x → 0.

GRK (IITM)


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Ad hoc Networks

SINR

analysis

Is Gaussian modelling of interference appropriate? PDF of Interference α =4

PDF of interference , α=4, Log Normal fading σ=6dB 0.2

Interference Monte−Carlo simulation Non−singular Exponential fading

0.2

Monte carlo

0.18

Inverse Gaussian, µ=6.3,k=6.7

Inverse Gaussian,ν=4.9,κ=20.8

Gamma, k=1.57, θ=4

0.16

Gamma, k=5.1, θ=0.96

Normal

Normal Inverse Gamma a=3, β =9.8

0.15

Inverse Gamma α=2.7, β=11.01

0.14

Density

Density

0.12

0.1

0.1

0.08

0.06 0.05

0.04

0.02 0

2

4

6

8

10 Data

12

14

16

18

0

0

5

10

15

20

25

30

Data

PDF of interference for Rayleigh and log normal fading with path loss `(x) = (1 + kxkα )−1 .

GRK (IITM)


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Ad hoc Networks

SINR

analysis

Transmission capacity(TC) Let ∈ (0, 1). TC is dened as

TC () = (1 − ) arg maxλ>0 {ps (θ, λ) > 1 − } 1. arg maxλ>0 {ps (θ, λ) > 1 − } is the maximum density of transmitting nodes that can be supported for an outage constraint . 2. 1 − fraction of these nodes are successful. 3. Hence TC measures the maximum spatial intensity of successful transmissions per unit area for a given outage capacity. 4. Can be related to area spectral eciency (ASE) by multiplying with log2 (1 + θ).

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Ad hoc Networks

SINR

analysis

Transmission capacity of the PPP dipole network Lemma When σ 2 ≈ 0, the TC of a PPP dipole network with Nt = Nr = 1 is (1 − ) TC () = 2 ln d C (α)θ2/α Proof: We have,

1 1−

.

p(θ, λ) = exp(−λd 2 θ2/α C (α))

Observe that p(θ, λ) increases with λ. Hence solving for p(θ, λ) > 1 − , we obtain the result. GRK (IITM)


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Ad hoc Networks

SINR

analysis

Sphere packing interpretation of TC When ≈ 0, by Taylor series expansion, ln

(1 − ) TC () = 2 ln d C (α)θ2/α

1 1−

∼

1

−

1

= + o(). Hence

= d 2 C (α)θ2/α

1 q 2 . π d α sin2(π2π/α)

Interpretation (heuristic) Hence each transmission approximately requires an interference free disc of radius 1/2 2π R=d . α sin(2π/α) I

The disc radius increases as

I

The disc radius decreases with increasing α I

√1 .

Higher path loss exponent → better packing.

GRK (IITM)


July 2012

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Ad hoc Networks

SINR

analysis

Transmission capacity for other schemes when ≈ 0 I

SISO: TC () =

I

MIMO: MRC with n receive antenna

d 2 C (α)θ2/α

n2/α n2/α Γ(1 − 2/α) ≤ TC () ≤ d 2 C (α)θ2/α d 2 C (α)θ2/α I

MIMO eigen-beamforming: m transmit and n receive max{n, m}2/α (nm)2/α Γ(1 − 2/α) ≤ TC () ≤ d 2 C (α)θ2/α d 2 C (α)θ2/α Can be used to analyse a multitude of systems with interference.

A. M. Hunter, J. G. Andrews and S. P. Weber, "Transmission Capacity of Ad Hoc Networks with Spatial Diversity", IEEE Transactions on Wireless Communications, Vol. 7, No. 12, pp. 5058-71, Dec. 2008 GRK (IITM)


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Ad hoc Networks

SINR

analysis

References I

MIMO R.H.Y. Louie, M. R. McKay, I. B. Collings , "Open-Loop Spatial Multiplexing and Diversity Communications in Ad Hoc Networks",/ IEEE Transactions on Information Theory, Vol. 57, No.1, pp. 317-344, Jan. 2011

I

Power control N. Jindal, S. Weber, and J. Andrews, "Fractional Power Control for Decentralized Wireless Networks, IEEE Trans. Wireless Communications, Vol. 7, No. 12, pp. 5482-5492, Dec. 2008 X. Zhang and M. Haenggi, "Random Power Control in Poisson Networks," IEEE Transactions on Communications, 2012.

I

Multihop R. Vaze,"Throughput-Delay-Reliability Tradeo with ARQ in Wireless Ad Hoc Networks", Wireless Communications, IEEE Transactions on , vol.10, no.7, pp.2142-2149, July 2011

Monographs F. Baccelli and B. Blaszczyszyn "Stochastic Geometry and Wireless Networks" NOW Publishers S. Weber and J. G. Andrews, "Transmission Capacity of Wireless Networks", NOW Publishers M. Haenggi and R. K. Ganti, "Interference in Large Wireless Networks", NOW Publishers GRK (IITM)


July 2012

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Ad hoc Networks

Interfernece correlation

Spatial and temporal correlation of interference in PPPs with ALOHA I

I

At each time instant each node transmits with probability p Let Φk denote the set of active transmitters at time k . i.e.,

5

4

3

2

1

Φk = {x; x ∈ Φ, x is on at time k}

0

−1

I

The interference is X I (Φk , z) = hxz [k]`(x − z) x

I

∈Φk

−2

−3

−4

−5 −5

−4

−3

We assume that the fading is indpendent across space and time.

GRK (IITM)


−2

−1

0

1

2

3

4

5

Time instant 1. July 2012

55 / 89

Ad hoc Networks



I


5

4

3

2

1


0

−1

I


I

∈Φk

−2

−3

−4

−5 −5

−4

−3


GRK (IITM)


−2

−1

0

1

2

3

4

5


55 / 89

Ad hoc Networks



I


5

4

3

2

1


0

−1

I


I

∈Φk

−2

−3

−4

−5 −5

−4

−3


GRK (IITM)


−2

−1

0

1

2

3

4

5


55 / 89

Ad hoc Networks


Spatio-temporal correlation coecient Correlation coecient, ρX ,Y =

cov(X , Y ) . σX σY

Lemma The spatio-temporal correlation coecient of the interferences I (Φm , u) and R I (Φn , v ), m 6= n for ALOHA and path loss functions `(x) satisfying R2 `(x)dx < ∞, is ζ(u, v ) =

p

R

R2

`(x)`(x − ku − v k)dx R . E[h2 ] R2 `2 (x)dx

Proof: Follows6 from Campbell's theorem. 6 R. K. Ganti and M. Haenggi. "Spatial and temporal correlation of the interference in ALOHA ad hoc networks", IEEE Communications Letters, 13(9):631 -633, September 2009 GRK (IITM) Stochastic Geometry and Wireless Nets. July 2012

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Ad hoc Networks

Interfernece correlation p=0.5

0.5

0.45

No Fading

0.4 ξ(u,v)

When u = v , the temporal correlation is equal to p . E[h2 ]

0.35

Hence for Nakagami-m fading, it is equal to 1pm +m .

0.3 Rayleigh Fading

0.25 0

20

40

60

80

100

m Spatial Correlation versus |u−v|, α=4, λ =1 1 0.9 ε=0.01

Spatial Correlation

When the path loss is given by `(x) = 1/( + kxkα ) and u 6= v the correlation is equal to

0.8

ε=0.1

0.7

ε=0.3 ε=0.5

0.6

ε=0.7

0.5 0.4 0.3

lim ξ(u, v ) = 0.

→0

0.2 0.1 0 0

0.5

1 c=|u−v|

1.5

2

Interference is spatio-temporally correlated. GRK (IITM)


July 2012

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Ad hoc Networks


Link formation delay A TX at x can connect to a RX at y if

I

SIR(x, y ) =

hxy [k]`(d) ≥θ I (Φk , y )

I

ALOHA MAC with access probability p

I

D : No of attempts required for a connection to form.

5

4

idle

3

interferers 2

TX

First attempt

RX

1

No connection

0

−1

−2

−3

−4

−5 −5

−4

−3

−2

−1

GRK (IITM)

0

1

2

3

4

5


July 2012

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Ad hoc Networks



I

SIR(x, y ) =

hxy [k]`(d) ≥θ I (Φk , y )

I


I


5

4

idle

3

interferers 2

TX

Second attempt

RX

1

No connection

0

−1

−2

−3

−4

−5 −5

−4

−3

−2

−1

GRK (IITM)

0

1

2

3

4

5


July 2012

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Ad hoc Networks



I

SIR(x, y ) =

hxy [k]`(d) ≥θ I (Φk , y )

I


I


5

4

idle

3

interferers 2

TX

Third attempt

RX

1

Connected

0

−1

−2

D=3

−3

−4

−5 −5

−4

−3

−2

−1

GRK (IITM)

0

1

2

3

4

5


July 2012

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Ad hoc Networks



I

SIR(x, y ) =

hxy [k]`(d) ≥θ I (Φk , y )

I


I


5

4

idle

3

interferers 2

TX

What is the average delay E[D] ?

RX

1

0

−1

−2

−3

−4

−5 −5

−4

−3

−2

−1

GRK (IITM)

0

1

2

3

4

5


July 2012

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Ad hoc Networks


Average delay E[D]: Neglecting interference correlation Recall I

ALOHA corresponds to independent thinning I

I

Φm

(the set of transmitters at time m) is a PPP of density λp.

The probability of link formation in a PPP network with density pλ is ps (θ, pλ) = exp(−pλd 2 θ2/α C (α))

1. At each time instant a link is formed with probability ps (θ, λ) independent of every other time 2. So the delay D is a geometric random variable with mean

(θ,λ) . 1

ps

E[D] = exp(pλd 2 θ2/α C (α)) 3. Observe that the delay increases with p . GRK (IITM)


July 2012

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Ad hoc Networks


Average delay E[D]: With interference correlation [k]`(d) I (Φk ,o) P(E c ∩ E c ∩

I

Let Ek denote the event

I

Then P(D > k) = ... ∩ Ekc ) Fail for k times 1 2 P∞ E[D] = k=0 P(D > k) Average of a positive random variable

I

I

h

≥θ

P∞ E[D] = EΦ E[D|Φ]] = EΦ [ k=0 P(D > k|Φ)]

P(E1c |Φ)P(E2c |Φ) . . . P(Ekc |Φ)

Conditioning on Φ

Conditional

I

P(D > k|Φ) = Independence

I

Probability that a link is not formed at time m is h [m]d −α c ≤θ Φ P(Em |Φ) = P I (Φm , y )

= 1 − E[exp (−d α θI (Φm , o)) |Φ] | {z } T1

GRK (IITM)


July 2012

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Ad hoc Networks I


Two sources on randomness: Fading and ALOHA MAC

T1 = Ee −d θ x∈Φ hxo [k]kxk 1(x is Tx at time m) Y α −α =E e −d θhxo [k]kxk 1(x is Tx) α

−α

P

x∈Φ

=E

Yh

e −d

α θh

−α xo [k]kxk

1(x is Tx) + 1 − 1(x is Tx)

i

x∈Φ I

First averaging over ALOHA, i Yh α −α e −d θhxo [k]kxk p + 1 − p T1 = E x∈Φ

I

Averaging over fading,

P(Emc |Φ)

= 1 − T1 = 1 −

Y x∈Φ

Observe that GRK (IITM)

P(Emc |Φ)

p 1− α 1 + d θkxkα

does not depend on the time index m.


July 2012

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Ad hoc Networks


I

P(D > k|Φ) = P(E1c |Φ)P(E2c |Φ) . . . P(Ekc |Φ) !k Y p = 1− 1− 1 + d α θkxkα x∈Φ

I

Hence the conditional average of delay is

E[D|Φ] =

∞ X

P(D > k|Φ) =

k=0

=Q =

Y 1− 1−

k=0

x∈Φ

p 1 + d α θkxkα

!k

1 x∈Φ

Y x∈Φ

GRK (IITM)

∞ X

h 1−

p

i

1+d α θkxkα

p 1− α 1 + d θkxkα

−1


July 2012

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Using PGFL of a PPP,

Average delay versus ALOHA parameter p 400

−1 p − + d α θkxkα −1 ! Z p −λ − − dx + d α θkxkα R2

350

1 1

Y x∈Φ

=


exp

1

1 1

300

|x|−α with correlation −α

|x| 250

without correlation α −1

(1+|x| )

α −1

(1+|x| ) E[D]

E

Ad hoc Networks

with correlation without correlation

200 150 100 50

With correlation E[D] = exp

pλd 2 θ2/α C (α) (1 − p)1−2/α

Observe E[D] = ∞ for p = 1. I I

0 0

0.2

0.4

0.6

0.8

1

P

!

Recall: Without considering correlation E[D] = exp pλd 2 θ2/α C (α)

Relying on fading is not sucient for ARQ to succeed. Correlation of interference cannot be neglected.

F. Baccelli, B. Baszczyszyn, "A New Phase Transitions for Local Delays in MANETs," INFOCOM, 2010 Proceedings IEEE , vol., no., pp.1-9, 14-19 March 2010 GRK (IITM) Stochastic Geometry and Wireless Nets.

July 2012

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Cellular Networks

Analysis of Cellular Networks * Primer on Point Processes * Ad hoc Networks * Cellular Networks SINR distribution Frequency reuse * Heterogeneous Networks GRK (IITM)


July 2012

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Cellular Networks

Cellular Trends

I

Interference is a main challenge in cellular systems

1. SINR more important than SNR I I

Universal frequency reuse Denser and denser deployments

2. BS cooperation and other interference-suppression techniques require good models for other-cell interference

I

Networks are becoming unplanned, decentralized and heterogeneous I I I

Picocells placed strategically in high-trac areas Femtocells/relays being placed randomly BS deployments increasingly driven by capacity needs rather than coverage needs

GRK (IITM)


July 2012

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Cellular Networks

Emerging cellular networks

What we think of

4G+femto/pico

Cells getting smaller, more random and chaotic GRK (IITM)


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Cellular Networks

Some current models

Wyner model 1. Fixed background interference 2. Highly inaccurate (averaging) Hexagonal model 1. Is it accurate? 2. Not very tractable. GRK (IITM)


July 2012

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Cellular Networks

SINR distribution

Proposed Model: PPP Base Stations I Base stations: big dots. Mobile users: little dots.

I

BS locations are drawn from a PPP of density λ Each mobile associates with the closest BS I

Cells are Voronoi tessellations

Advantages I

Non uniform cell sizes

I

Tractable?

Disadvantages I

GRK (IITM)

BSs might get very close


July 2012

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Cellular Networks

SINR distribution

Comparison with real deployment

Base stations: big dots. Mobile users: little dots.

Actual BS locations in a 4G Urban Network 20

15

Y coordinate (km)

10

5

0

−5

−10

−15

−20 −20

GRK (IITM)

−15

−10

−5

0 X coordinate (km)

5

10

15

20


July 2012

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Cellular Networks

SINR distribution

System model: Downlink

I

The BSs are spatially distributed as a PPP of density λ

I

Mobile users connect to the nearest (geographical) BS

I

The path loss is given by `(x) = kxk−α , α > 2.

I

All BSs transmit at the same power P

I

The fading between a BS x and a mobile y is denoted by hxy What is the SINR distribution of a typical mobile user?

Without loss of generality, we can assume the typical mobile user to be at the origin o .

GRK (IITM)


July 2012

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Cellular Networks

SINR distribution

Analysis of SINR Let x ∈ Φ be the BS that is closest to the reference MS at the origin.


The downlink SINR of the MS at the origin is hxo r −α −α y ∈Φ\{x} hyo ky k

SINR = P where r = kxk.

Compute P(SINR > θ)

GRK (IITM)


July 2012

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Cellular Networks

SINR distribution






GRK (IITM)


July 2012

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Cellular Networks

SINR distribution






GRK (IITM)


July 2012

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Cellular Networks

SINR distribution

Analysis of SINR Recall that the distribution of the nearest BS equals the rst contact distribution f (r ) = λ2πr exp(−λπr 2 ) I

We rst condition on the distance to the nearest BS r .

P(SINR > θ) = Er P(SINR > θ|r ) Z ∞ = P(SINR > θ|r )λ2πr exp(−λπr 2 )dr 0

I

Focusing on P(SINR > θ|r ) which we denote by pr ! hxo r −α pr = P P ≥ θ r −α y ∈Φ\{x} hyo ky k     X = E exp −θr α hyo ky k−α  r  y ∈Φ\{x}

GRK (IITM)


July 2012

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Cellular Networks

SINR distribution

I Base stations: big dots. Mobile users: little dots.

 pr = E 

Y

 exp −θr α hyo ky k−α r 

y ∈Φ\{x} I

Since the fades are independent and exponentially distributed with unit mean,   Y 1  pr = E  r 1 + θr α ky k−α y ∈Φ\{x}

I

Using PGFL on Φ ∩

B(o, r )c

pr = e I

−λ

R

B(o,r )c

1 dy 1+θ−1 r −α ky kα .

Un-conditioning on r ,

P(SINR > θ) =

Z

∞

e

−λ

R

B(o,r )c

1 dy 1+θ−1 r −α ky kα f (r )dr .

0

GRK (IITM)


July 2012

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Cellular Networks

SINR distribution

SINR distribution The SIR distribution in a PPP BS network where a MS connects to the nearest BS is given by7

P(SIR ≥ θ) = where ρ(θ, α) = θ2/α I

I

1

+u α/2

1

du .

Increasing the density of BSs does not increase the coverage probability.

Simple expression I

I

θ−2/α

Does not depend on the density of BSs λ. I

I

R∞

1 1 + ρ(θ, α)

For α = 4 reduces to (1 +

√

√

2 arctan(1/

θ(π/ −

θ)))−1

.

Extensions to general fading distributions and noise possible. Since the PDF of SIR is known the average ergodic can be computed I

For α = 4, the computed ergodic rate is 1.49nats/sec/Hz

7 J. G. Andrews, F. Baccelli, and R. K. Ganti. A tractable approach to coverage and rate in cellular networks. IEEE Trans. on Communications, Nov. 2011 GRK (IITM) Stochastic Geometry and Wireless Nets. July 2012 74 / 89

Cellular Networks

SINR distribution

Numerical results Coverage probability for α = 3, 6dB LN shadowing, no noise 1 Grid N=24 Real Data Random (PPP)

Coverage probability for α = 4 1

0.9

Grid N=8, SNR=10 Grid N=24, SNR=10 Grid N=24, No Noise PPP BSs, SNR=10 PPP BSs, No Noise

0.9

0.8

0.8

Probability of Coverage


0.7 0.7

PPP

0.6

0.5

0.4 Square Grid

0.3

0.5

0.4

0.3

0.2

0.2

0.1

0.1

0 −10

0.6

−5

GRK (IITM)

0

5 SINR Threshold (dB)

10

15

20

0 −10

−5


0

5 SINR Threshold (dB)

10

July 2012

15

20

75 / 89

Cellular Networks

Frequency reuse

Frequency reuse I

I

Frequency planning necessary for increasing the coverage. Assume that there are δ frequency bands I I

PPP: random allocation of bands Corresponds to thinning of a PPP Φ I

I

I

1

Density of Φm is λ/δ .

We rst consider the BS x ∈ Φ to which the MS at the origin connects. I

Frequency bands: 4 2

2 exp(−λπr 2 ) The interferers density is now λ/δ.

0

kxk = r ∼ λ πr

The coverage probability is

−1

−2 −2

−1

0

1

2

1 P(SIR ≥ θ) = 1 1 + δ ρ(θ, α) GRK (IITM)


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Cellular Networks

Frequency reuse

Coverage versus rate 1. The coverage probability is

P(SIR ≥ θ) =

1 1 1 + δ ρ(θ, α)

~ w wδ

2. The ergodic rate equals 1δ E[ln(1 + SIR)], which equals Z Z 1 ∞ 1 ∞ 1 P(ln(1 + SIR) > θ)dθ = dθ, 1 δ 0 δ 0 1 + δ ρ(e θ − 1, α) which equals

Z R= 0

∞

1

δ + ρ(e θ − 1, α)

w

dθ w δ

Coverage increases with δ while the average rate decreases with δ . GRK (IITM)


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Cellular Networks

Frequency reuse

Numerical results Avergare rate: SNR=10dB, λ=0.25 (PPP)

Coverage probability, no noise, α = 4, δ=4 1

2

0.9

1.8

0.8

1.6

0.7

1.4

Random (PPP), α=2.2

Average rate: τ(λ,α,δ)


Random (PPP), α=4

0.6 0.5

Random (PPP) Square Grid

0.4

Actual

0.3

1 0.8 0.6 0.4

0.1

0.2

−5

0

5 10 SINR Threshold (dB)

GRK (IITM)

15

20

Actual,α= 2.2

1.2

0.2

0 −10

Actual, α=4

0 1

2


3

4 5 Frequency bands δ

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Heterogeneous Networks

Analysis of Heterogeneous Networks * Primer on Point Processes * Ad hoc Networks * Cellular Networks * Heterogeneous Networks

GRK (IITM)


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System Model I I

K tier network The BS locations in tier i are modelled by a PPP Φi I I I

I

I I

Density of Φi is λi BSs in tier i transmit with power Pi A mobile can connect to a BS in tier i if the received SIR is greater than θi

All tiers transmit in the same frequency band and hence contribute to interference Fading is i.i.d Rayleigh Path loss is given by kxk−α , α > 2

Assumption The SIR thresholds θi > 1. GRK (IITM)

10 8 6 4 2 0 −2 −4 −6 −8 −10 −10

−5

0

5

10

An example of k = 3 network. Red: FemtoBS, blue: PicoBS, black: MacroBS


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Max SIR model Connectivity model A mobile user can connect to a BS of any tier provided that the SIR constraint is satised, i.e., to connect to a BS of tier i , the SIR should be greater than θi .

Lemma If θi > 1, a mobile user can connect to at most one BS

Proof. Let ai denote the received power from a BS i , then only one of the following terms can be greater than 1

ai P

j6=i

GRK (IITM)

aj

,

i = 1, 2, . . .


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Coverage probability The receive SIR of a mobile at origin and a BS x ∈ Φm is SIR(x) = PK P i=1

y ∈Φi

hxo kxk−α hyo ky k−α − hxo kxk−α

Let pc denote the coverage probability.   K Y Y 1(SIR(x) < θm ) 1 − pc = E  m=1 x∈Φm

 = E

K Y Y

 1 − 1(SIR(x) > θm )

m=1 x∈Φm

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Contd... Expanding the inner product, 1 − pc =1 − E

K X X

1(SIR(x) > θm )

m=1 x∈Φm

+E

X

1(SIR(x) > θm )1(SIR(y ) > θn ) −( three terms ) . . . |

{z

}

T2

The term T2 and higher order terms are zero since the MS can connect to at most 1 BS. Hence pc =

K X m=1

GRK (IITM)

E

X

1(SIR(x) > θm )

x∈Φm


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Contd... How to evaluate E

P

x∈Φm

1(SIR(x) > θm ).

Recall Campbell Mecke theorem For a PPP of density λ

X

E

Z f (x, Φ \ {x}) = λ

R2

x∈Φ

E

X

1(SIR(x) > θm ) = λm

x∈Φm

As before,

P

GRK (IITM)

Pm hxo kxk−α > θm I

Z R2

E[f (x, Φ)]dx

P(SIR(x) > θm )dx

−1 = LI (kxkα θm Pm ).


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Contd...

Observe that the total interference is the sum of the interference from each tier which are independent. Hence −1 LI (kxkα θm Pm )=

K Y

−1 LIj (kxkα θm Pm )

j=1

Recall that the Laplace transform of interference Ij = /α

LIj (s) = exp(−λj Pj

2

where C (α) =

P

x∈Φj

Pj hxo kxk−α is

s 2/α C (α)),

2 2π α sin(2π/α)

Hence /α

−2/α

−1 LI (kxkα θm Pm ) = exp(−kxk2 θm Pm 2

K X

/α

2

λi Pi

C (α))

i=1

GRK (IITM)


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Coverage results8

Combining everything, pc =

k X m=1

Z λm

R2

/α

−2/α

exp(−kxk2 θm Pm 2

K X

/α

2

λ i Pi

C (α))dx

i=1

Lemma The coverage probability in a K tier heterogeneous network is π pc = C (α)

2/α −2/α m=1 λm Pm θm , PK 2/α m=1 λm Pm

PK

θi > 1 −2/α

1. A simple expression. Convex combination of θm 2. pc does not depend on the densities if all the thresholds θm are equal I

Can add more tiers without changing the coverage

7 H. Dhillon, R. K. Ganti, F. Baccelli, and J. G. Andrews. "Modelling and analysis of k-tier downlink heterogeneous cellular networks ". IEEE JSAC, April 2012 GRK (IITM) Stochastic Geometry and Wireless Nets. July 2012

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Fraction of users connected to j -th tier is /α −2/α θj PK 2/α −2/α m=1 λm Pm θm 2

βj =

λj Pj

Lemma (Closed access) When the user is allowed to connect to only a subset of tiers B ⊂ {1, 2, 3, .., K }, the coverage probability is π pc = C (α)

GRK (IITM)

2/α −2/α m∈B λm Pm θm , PK 2/α λ P m m m=1

P


θi > 1

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Numerical results 1 Simulation: PPP Simulation: Actual BSs Simulation: Grid Analysis

0.9

Coverage Probability (Pc)

0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 −6

−4

−2

0

2 4 SIR Threshold (β1)

6

8

10

A two-tier HCN, K = 2, α = 3, P1 = 100P2 , λ2 = 2λ1 , β2 = 1dB GRK (IITM)


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Conclusions

I

Networks are getting more random and chaotic

I

Random spatial models are necessary for modelling current networks.

I

A rich set of mathematical tools are provided by stochastic geometry

GRK (IITM)


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