Stochastic geometry and communication networks

Stochastic geometry and communication networks

OUTLINE I Panorama of some stochastic-geometry models in action,

Bartek Błaszczyszyn

II Basic geometric models,

tutorial lecture

III Signal-to-Interference-and-Noise ratio (SINR) coverage model,

Performance conference

IV Modeling ad-hoc networks,

Juan-les-Pins, France, October 3-7, 2005

V Power control in CDMA: from static to dynamic modeling — a spatial Erlang formula. INRIA-ENS

Stochastic geometry and communication networks, B. Błaszczyszyn; tutorial lecture, Performance’05, Juan-les-Pins, France, October 3-7, 2005

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PANORAMA...

I PANORAMA

• The mathematical principle of the Voronoi tessellation is widely used as a simple idealization of many complex “real” partitions of the plane (cells in cellular

• Boolean model is a first model of coverage of wireless network (it does not take into account interference). As the underlying model for the study of continuum

communication, Gupta & Kumar protocol model of ad-hoc networks; it takes into

percolation it can be used to address the questions of connectivity of ad-hoc

account only locations of antennas and ignores all other physical aspects of the

networks in the absence of interference.

communication technology as a.g. additive interference)

• The dual Delaunay graph can be used as a “protocol model” of neighbuorhood in networks ⇒ topology for routing


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• Mathematical representation of interferences based on (Poisson) shot noise processes ⇒ a variety of results on coverage, connectivity and capacity of large interference-limited networks.


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BASIC MODELS... Poisson Point Process Planar Poisson point process (p.p.)

II BASIC GEOMETRIC MODELS

• Number of Points Φ(B) of Φ in subset B of the plane is Poisson random variable with parameter λ|B|, where | · | is the Lebesgue measure on the plane; i.e.,

• Poisson point process, • Voronoi tessellation and Delaunay graph,

P{ Φ(B) = k } = e−λ|B|

• Boolean model, • Shot-Noise model,

(λ|B|)k , k!

• Numbers of points of Φ in disjoint sets are independent.

• References.

Laplace transform of the Poisson p.p.

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BASIC MODELS/Poisson p.p. ...

R

h(x) Φ(dx)

R

h(x)

] = e−λ (1−e ),dx , R P where h(·) is a real function on the plane and h(x) Φ(dx) = Xi ∈Φ h(Xi ). LΦ (h) = E[e


Φ of intensity λ:


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BASIC MODELS ... Voronoi Tessellation (VT) and Delaunay graph

Poisson p.p. is the basis of the stochastic-geometry modeling of communication

= {Xi } on the plane and a given point x, we define the Voronoi cell of this point Cx = Cx (Φ) as the subset of the plane of all locations that are closer to x than to any point of Φ; i.e., Given a collection of points Φ

networks. This modeling consist in treating the given architecture of the network as a snapshot of a (homogeneous) random model, and analyzing it in a statistical way. In this

Cx (Φ) = {y ∈ R2 : |y − x| ≤ |y − Xi | ∀Xi ∈ Φ} .

approach the physical meaning of the network elements is preserved and reflected in the model, but their geographical locations are no longer fixed but modeled by

= {Xi } is a Poisson p.p. we call the (random) collection of cells {CXi (Φ)} When Φ

random points of, typically, homogeneous planar Poisson point processes. Consequently, any particular detailed pattern of locations is no longer of interest.

the Poisson-Voronoi tessellation (PVT).

Instead, the method allows for catching the essential spatial characteristics of the network performance basically through the densities of these point processes (i.e.,

Edges of the Delaunay graph connect nu-

the densities of the network devices). Stochastic geometry and communication networks, B. Błaszczyszyn; tutorial lecture, Performance’05, Juan-les-Pins, France, October 3-7, 2005

Borders of Voronoi Cells

clei of the adjacent cells. 7


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BASIC MODELS/VT ...

BASIC MODELS... Boolean Model (BM)

˜ Let Φ

= {(Xi , Gi )} be a marked Poisson p.p., where {Xi } are points and {Gi }

are iid random closed stets (grains). We define the Boolean Model (BM) as the union

Ξ=

VT is a frequently used generic model of tessellation of the plane.

[ i

Points denote locations of various structural elements (devices) of the network (base station antennas and/or network controllers in cellular networks, concentrators in

Xi ⊕ G i

where x ⊕ G

Known: • Poisson distribution of the number of grains intersecting

fixed telephony, access nodes in ad hoc networks, etc.).

any given set.

Cells denote mutually disjoint regions of the plane served in some sense by these

• Asymptotic results (λ → ∞)

devices.

for the probability of complete

covering of a given set.

= {x + y : y ∈ G}.

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BM with spherical grains of random radii Stochastic geometry and communication networks, B. Błaszczyszyn; tutorial lecture, Performance’05, Juan-les-Pins, France, October 3-7, 2005

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BASIC MODELS/BM ...


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BASIC MODELS... Shot-Noise (SN) model

˜ Let Φ

= {(Xi , Si )} be a marked p.p., where {Xi } are points and {Si } are iid random variables. Given a real response function L(·) of the distance on the plane BM is a generic coverage model.

we define the Shot-Noise field

Points denote locations of various structural elements (devices) of the network.

IΦ˜ (y) =

Granis denote independent regions of the plane served these devices .

X i

In wireless networks it is a simplified model (it does not take into account

Si L(y − Xi ) .

˜ is a marked Poisson p.p. then we call I ˜ the Poisson SN. When Φ Φ

interference) for the study of coverage and connectivity.

For the Poisson SN, the Laplace transform of the vector (I Φ (y1 ), . . . , IΦ (yn )) is known for any y1 , . . . , yn


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∈ R2 (via Laplace transform of the Poisson p.p.).


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BASIC MODELS/SN ...

BASIC MODELS ...

References SN is a good model for interference in wireless networks.

1. Stoyan & Kendal & Mekke (1995) Stochastic Geometry and its Applications. Wiley, Chichester

Marks Si correspond to emitted powers.

2. Okabe & Boots & Sugihara (2001) Spatial Tessellations: Concepts and

Response function correspond to attenuation function.

Applications of Voronoi Diagrams. Willey, Chichester.


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Φ = {Xi , (Si , Ti )} marked point process (Poisson)

• Motivations,

{Xi } points of the p.p. on R2 — antenna locations,

(Si , Ti ) ∈ (R+ )2 possibly random mark of point Xi — (power,threshold) ¾ ½ Si l(y − Xi ) y: ≥ Ti cell attached to point Xi : Ci (Φ, W ) = W + κIΦ (y) P where Iφ (y) = i6=0 Si l(y − Xi ) shot noise process, κ interference factor, W ≥ 0 external noise, l(·) attenuation (response) function. Ci is the region where the SINR from Xi is bigger than the threshold Ti . Coverage PROCESS:

Ξ(Φ; W ) =

• Snapshots and qualitative results, • Typical cell study

(coverage probability for the point is some distance to the antenna, simultaneous coverage of several points, mean area of the cell),

• Handoff study

(overlapping of cells, coverage probability for a typical point, distance to different

handoff states),

• Macroeconomic optimization example,

Ci (Φ, W ).

i∈N Stochastic geometry and communication networks, B. Błaszczyszyn; tutorial lecture, Performance’05, Juan-les-Pins, France, October 3-7, 2005

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SINR COVERAGE MODEL ...

III SINR COVERAGE MODEL

[


• References. 15


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SINR COVERAGE MODEL / CDMA motivation ... Parameter values Intensity of Poisson process of base stations λBS ∼ 0.2 BS/km2 .

Motivation I: CDMA handoff cells

x0 — a point in R2 (location of an antenna), s0 ≥ 0 and t0 ≥ 0 — (pilot signal power of

the antenna and SINR threshold (bit energy-tonoise spectral power density

Pilot signal power s0 (x 1 , (s1 , t1 ))

pilot signal),

y

(x 3 , (s3 , t3 ))

(x 0 , (s0 , t0 )) (x 4 , (s4 , t4 ))

s0 l(y − x0 ) ≥ t0 w + Iφ (y)


∼ 30 mW

SINR threshold (bit energy-to-noise spectral power density

w

Eb /NO ) for the

φ = {xi , (si , ti )} — pattern of antennas, w ≥ 0 — external noise, 0 ≥ κ ≤ 1 — orthogonality factor, l(·) — attenuation function

(x 2 , (s2 , t 2))

−14 dB

Eb /NO ) for the pilot t0

External noise w

∼

∼ −105 dB

Interference factor for pilots from different BS’s

κ=1 Attenuation function

l(x) = A max(|x|, r0 )−α or l(x) = (1 + A|x|)−α with α ∼ 3 − 6. 17

SINR COVERAGE MODEL / CDMA motivation ...


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SINR COVERAGE MODEL / Motivations II

This is a relatively simple model, which takes into account only locations of the Base Stations, their pilot signal powers and SIR’s for the pilots.

Gupta & Kummar physical model for ad-hoc networks

In particular there is no any pattern of mobiles assumed yet and it does not take into

(see Modeling of ad-hoc networks)

account power control issues.


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SINR COVERAGE MODEL / Snapshots ...

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≡ 1, T = 0.4, W = 0, l(r) = (Ar)−β and attenuation exponent β → ∞.

Constant emitted powers Si , ei

Small interference factor allows one to approximate SINR cells by a Boolean model SIR cells tend to Voronoi cells whenever attenuation is stronger, e.g. in urban areas.

(via perturbation methods) Stochastic geometry and communication networks, B. Błaszczyszyn; tutorial lecture, Performance’05, Juan-les-Pins, France, October 3-7, 2005

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SINR COVERAGE MODEL / Typical cell study ...


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Probability for a typical cell to cover a point

Φ — marked Poisson point process representing antennas in R2 , (0, (S, T )) — additional antenna located at fixed point 0 with random (S, T ) distributed as any mark of Φ, independent of it (thus Φ ∪ {(0, (S, T )} has 2 Poisson Palm distribution), y — location (of a mobile) in R . Given:

IΦ (y) that is

− 1), W

and the Laplace transform of

· Z ³ ´ i 1 − LS (ξl(y − z)) µ(dz) , E[exp(−ξIΦ (y))] = exp −

where LS (ξ)

´

pR = P y ∈ C 0 ³ ´ = P S(1/T − 1)l(R) − W − IΦ (y) > 0 . Stochastic geometry and communication networks, B. Błaszczyszyn; tutorial lecture, Performance’05, Juan-les-Pins, France, October 3-7, 2005

be given via Laplace transforms of S(1/T

Rd

Probability for C0 to cover a given point y located at the distance R to the origin:

³

Res. For M/G case (general distribution of (S, T )) the coverage probability p R can

= E[e−ξS ] is the Laplace transform of S .

Cor. Fourier transform of the Poisson shot-noise variable I φ (y) → Rieman Boundary Problem

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→ probability of coverage by the typical cell.


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SINR COVERAGE MODEL / Typical cell study ... Special M/M case

Example Fourier transform FIΦ (ξ) of the homogeneous Poisson (intensity λ) shot noise with

exponential S (parameter m) and attenuation l(x)

£

−iξIΦ

FIΦ (ξ) = E e " r

for ξ

= A max(|x|, r 0 )

−4

Res. [Baccelli&BB&Muhlethaler (2004)]: Assume that {Si } are exponential r.vs. with

par.

µ and Ti = T are constant. Then the probability for C0 to cover a given point

located at the distance R: is equal

¤

pR = exp

r ¶ µ r λ iAξ m iAξ − π2 arctan r02 = exp λπ m iAξ 2 m ¸ 4 4 r − iAξ − r m 0 + λπr02 0 , 4 iAξ + r0 m

− 2πλ

Z

∞ 0

¾ u du . 1 + l(R)/(T l(u))

proof: Say the emitter is at the origin and consider the corresp. Palm distribution P;

pR = P(S ≥ T IΦ1 /l(R)) Z ∞ e−µsT /l(R) dP(IΦ ≤ s) = 0

= ψIΦ (µT /l(R)) ,

∈ R, where the branch of the complex square root function is chosen with

where LIΦ (·) is the Laplace transform of the value of the hom. Poisson SN I Φ .

positive real part.


½

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SINR COVERAGE MODEL / Typical cell study / M/M case ...


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SINR COVERAGE MODEL / Typical cell study / M/M case ...

Some optimizations One can study the following optimization problems for the expected effective transmission range r Cor. For the attenuation function l(u)

= (Au)−β

pR (λ) = e−λR where C

2 T 2/β C

• given the density of stations λ find the targeted range r that optimizes the expected effective transmission range

,

³ ´ = C(β) = 2πΓ(2/β)Γ(1 − 2/β) /β .

1 √ r≥0 T 1/β 2λpC 1 √ = rmax (p) = argmaxr≥0 {rpr (λ)} = T 1/β 2λC ρ = ρ(p) = max{rpr (p)} =

rmax Stochastic geometry and communication networks, B. Błaszczyszyn; tutorial lecture, Performance’05, Juan-les-Pins, France, October 3-7, 2005

× pr :

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SINR COVERAGE MODEL / Typical cell study / M/M case optimization


Probability for a typical cell to cover two points

(y1 , y2 ) — two point to be covered by a given cell C0 (Φ, W ) under Palm distribution of Φ ∪ {(0, (S, T )}

• given the targeted range R find the density of emitters λ that optimize the expected effective transmission range R × pR :

We need the joint Laplace transform of

(IΦ (y1 ), IΦ (y2 )) that is given by h ³ ´¤ E exp −ξ1 IΦ (y1 ) − ξ2 IΦ (y2 ) · Z ³ ´ i 1 − LS (ξ1 l(y1 − z) + ξ2 l(y2 − z)) µ(dz) . = exp −

1 R2 T 2/β C 1 max{RpR (λ)} = 2 2/β λ≥0 R T eC

λmax = λmax (R) =argmaxλ≥0 {RpR (λ)} =

Rd


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SINR COVERAGE MODEL / Typical cell study / Perturbation of Boolean model ...

Coverage probability via perturbation of Boolean model valid for small interference factor κ

Res.

(κ) (κ) Denote pR = P(x ∈ C0 ), where |x| = R and n o (κ) 2 C0 = y ∈ R : Sl(y) ≥ κIΦ (y) + W .

Assume F∗ (u)

= P((Sl(x) − W ) ≤ u) admits Taylor approximation at 0:

F∗ (u) = F∗ (0) + and R∗ (u)

(κ)

pR =

h (k) X F∗ (0) k=1

k!

k

u + R∗ (u)

z³

}| ´{ P Sl(x) ≥ W provided E[(IΦ (x))

h

= o(u ) u & 0.


correcting terms value for the Boolean model

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2h

−

z h X k=1

κ

}|

(k) k F∗ (0)

k!

] < ∞.


E

£

(IΦ (y))

{ ¤ k

error

z }| { + o(κh ) ,

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SINR COVERAGE MODEL / Typical cell study ... Numerical examples

p(κ) (1)

p(0.1) (y)

1

Mean cell area formula

v (κ) (1)

1

2 1.8

0.8

= E[|C0 |]. Recall that pR is the coverage probability for location at distance R. Denote the mean area of the cell of the BS located at 0 by v 0

0.8

1.6 1.4

0.6

0.6

1.2

0.4

0.4

0.8

0.2

0.2

0.4

1

0.6

Then

v0 =

Z

0.2

Rd

p|y| dy.

0

0.2

0.4

0.6

1

0.8

κ

0 1

1.2

1.4

1.6

1.8

2

function of the distance.

coefficient Stochastic geometry and communication networks, B. Błaszczyszyn; tutorial lecture, Performance’05, Juan-les-Pins, France, October 3-7, 2005

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SINR COVERAGE MODEL / Handoff study ...

0.02

0.04

0.06

0.08

0.1

κ

y

Probability of coverage as a Probability of coverage as a function of the interference

0

Mean cell area as a function of the interference coefficient.


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SINR COVERAGE MODEL / Handoff study / Overlapping of cells...

Overlapping of cells Deterministic scenario: given n cells C(xi , si , ti ; φ, w), i = 1, . . . , n Pn Res. The inequality i=1 ti /(1 + ti ) < 1 is a necessary condition for Tn i=1 C(xi , si , ti ; φ, w) 6= ∅ Random scenario:

Cor. If the distribution of the ratio T is such that T

≥τ

for some τ

> 0, then the number Ky of cells of the coverage process Ξ covering any given point y is a.s. bounded

Example: For the maximal pilot’s bit energy-to-noise spectral power density

τ = Eb /NO = −14 dB the pole handoff number (theoretical maximal handoff number) K ≤ 26.

1+τ . τ (Given point cannot be covered by (1 + τ )/τ or more cells, no matter how close Ky
1/(β/2−δ)

< C/|x − y| β for some

0, β > 0 and if the distribution of S has finite moment E[S ] < ∞ for some δ ∈ [1, β/2], then one can show that for any R, ε, α > 0³, there exists R0 > 0 such that ´ P P sup|y|R0 Si l(y, Xi ) < ε > 1 − α . Stochastic geometry and communication networks, B. Błaszczyszyn; tutorial lecture, Performance’05, Juan-les-Pins, France, October 3-7, 2005

˜t ) is obtained using The exact stationary distribution of the Markov process ( Z backward simulation (coupling from the past) similar to that proposed by Kendall. Stochastic geometry and communication networks, B. Błaszczyszyn; tutorial lecture, Performance’05, Juan-les-Pins, France, October 3-7, 2005

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Macroeconomic optimization example:

densification / magnification

Increase the mean power m of existing antennas or increase the density λ of antennas?

References 2

C total budget of an operator per km , Cλ cost of one antenna, Cm cost of increasing the power of one antenna by 1W. constraint: λCλ + Cm λ/m = C.

1. Baccelli & Błaszczyszyn (2001), On a coverage process ranging from the Boolean model to the Poisson Voronoi tessellation, with applications to wireless communications Adv. Appl. Probab. 33 mean handoff level EN0

2. Tournois (2002), Perfect Simulation of a Stochastic Model for CDMA coverage

10

INRIA report 4348,

8

Plots of mean handoff as a functions of mean antenna power

1/m 6

under budget constraint with top: Cm

3. Baccelli, Błaszczyszyn & Tournois (2002), Spatial averages of downlink coverage

C = 1000, Cλ = 500 and from the

= 1, 2, 5.

Solution: Plot maximum = Optimal configuration.

4

0


characteristics in CDMA networks (INFOCOM ).

2

2

4

6

8

10

12

mean antenna power 1/m

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AD-HOC NETWORKS ...

• A random set of users distributed in

IV MODELING AD-HOC NETWORKS

space and sharing a common Hertzian

• Ad-hoc networks,

medium.

• A few sg “interference aware” models for ad-hoc networks,

• Users constitute ad-hoc network that is

• Some optimization problem in capacity and medium access control,

emitter receiver

in charge of transmitting information far away via several hops.

• References.

• Users switch between emitter and receiver modes.


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AD-HOC NETWORKS ...


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AD-HOC NETWORKS ...

Interference and successful reception

Multi-hop transmissions

• Emitter sends a packet in some given di-

• Emitter sends a packet emitting some

• The packet is received by some number

• Transmission distance attenuates the

power.

rection far away via several hops.

emitted power.

(possibly 0) of neigbouring receivers.

• An optimal receiver among them is in

emitter

charge of forwarding this packet in (one receiver

towards destination optimal receiver

emitter used sign

receivers.

of) his next emission time-slots.

al receiver

interferen

ce

• Reception is successful if the Signal to In-

• In the case of no reception, emitter re-

terference Ratio (SIR) at the receiver is large enough.

emits the packet next authorized time.


• Emitted power causes interference at all

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AD-HOC NETWORKS ...

AD-HOC NETWORKS / A few models / General settings ...

A few sg interference aware models

General settings

A way of choosing marks ei for nodes is called medium access control. Two

• Nodes distributed in (a subset of) the plane according to a Poisson p.p. Φ = {Xi },

possibilities can be considered:

• Nodes Xi are marked in some way (not necessarily independently) by ei = 1 when node is emitting or 0 when not (it is a potential receiver); • Call – –

• Some “central authority” assigns marks ei (in some optimal way) for each given configuration of nodes

⇒ {ei } are dependent in some way. • Each node independently switches between modes ei = 1 and ei = 0 ⇒ {ei } are independent, given configuration of nodes.

Φ1 = {Xi ∈ Φ : ei = 1} — emitters and

Φ0 = {Xi ∈ Φ : ei = 0} — potential receivers.


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AD-HOC NETWORKS / A few models ...


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AD-HOC NETWORKS / A few models / Protocol model ... Examples of medium access centralized

independent

Voronoi tessellation principle: Protocol Model [Gupta & Kumar (2000)]

• When the node Xi ∈ Φ1 transmits it can be successfully received by node Xj ∈ Φ0 if |Xj − Xi | ≤ (1 + ∆)|Xj − Xk | ∀Xk ∈ Φ1 , where ∆

≥ 0 is some constant. ⇔ Xj ∈ modified Voronoi Cell CXi (Φ1 ) The Protocol Model with centralized medium access was studied asymptotically


(when number of nodes goes to ∞) by G & K (2000); to be commented. 51


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AD-HOC NETWORKS / A few models / Carrier-Sense model ... Examples of medium access

Hard-core principle: Carrier-Sense Model

´ hard-core model: • Matern ⇒ Nodes are first independently marked by some auxiliary marks in [0, 1]. A node Xi is selected as emitter (ei = 1) if its auxiliary mark is larger than all auxiliary marks within its neighbourhood of range R cs . Otherwise the node is

Close to some currently used protocols implemented e.g. in IEEE 802.11

• When the node Xi ∈ Φ1 transmits it can be successfully received by node Xj ∈ Φ0 if

marked receiver.

|Xj − Xi | ≤ R and |Xk − Xi | > Rcs ∀Xk ∈ Φ1 , where Rcs

> R ≥ 0 are some constants.

⇔ Each communication within the transmission range R is protected by the exclusion disc centered at the transmitter with radius R cs > R. (One can also

• Gibbs hard-core model: ⇒ The steady state distribution of a spatial birth-and-death process of nodes. When a node is born it is marked as emitter if in its neighbourhood of range R cs there is no any other emitter. Otherwise it is marked receiver.

think of the models with exclusion discs centered at the receiver and at both

The Carrier Sense Model, has not been studied yet by means of stochastic geometry

transmitter and receiver.)

tools (to the best of our knowledge).


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AD-HOC NETWORKS ...

SINR principle: Physical model

• Points Xi are independently identically marked by some Si ≥ 0 representing powers used when emitting signals.

• l(r) = (Ar)−β signal attenuation function, • T signal-to-interference ratio threshold, Connectivity for the physical model

• W > 0 constant external noise, κ > 0 interference factor.

• When the node Xi ∈ Φ1 transmits it can be successfully received by node Si l(|Xj − Xi |) Xj ∈ Φ0 if ≥T, W + κIΦ1 \Xi (Xj ) P where IΦ1 is the shot-noise process of Φ1 IΦ1 (y) = Xk ∈Φ1 Sk l(|y − Xk |). ⇔ Communication is successful if the signal-to-interference ratio at the receiver is bigger than the threshold T . Stochastic geometry and communication networks, B. Błaszczyszyn; tutorial lecture, Performance’05, Juan-les-Pins, France, October 3-7, 2005

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See Performance’05 tutorial by P. Thiran.


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AD-HOC NETWORKS ...

AD-HOC NETWORKS / Some optimization problem ...

Some optimization problem in capacity and medium access for the physical model Suppose emitter X0

= 0 has to send information to infinity along the real axis. Define also a modified progress

Define effective distance traversed in one hop, called also progress effective distance

effective distance

probab. of z ³ }| µsuccessful ´+{¶ z }| rec.{ ˜ = max p|X | (λp) |Xj | cos(arg(Xj )) . D j 0

rec. ind.z }| µsuccessful ³ ´+{¶ z }| { , D = max0 δ(Xj , 0, Φ1 )|Xj | cos(arg(Xj ))

Xj ∈Φ

Xj ∈Φ

Denote d(λ, p)

effective distance emitter optimal receiver

˜ p) = E[D] ˜. = E[D], d(λ,

Search for receivers that realize the progresses, D and


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˜ can be implemented. D


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Fact The mean total distance traversed in one hop by all transmissions initialized in some unit area (called density of (modified) progress) is equal to λp

d(λ, p) (resp.

˜ p)). λp d(λ,

³ zet ´ G(z) = 2 arccos √ dt . √ 2et {t:et / 2et≤1/z}

⊂ R of unit area, by the Campbell’s formula · X ¸ Z E Di = λp 1(x ∈ B)E[D0 ] dx Xi ∈Φ1 ∩B

Fact For all λ, p

∈ [0, 1], define an auxiliary function Z

2

proof: For B

For z

R2

Res. For the M/M model with W

= 0 The distribution function of the modified

progress is given by

= λp d(λ, p) .

˜ ≤ z) = e−λ(1−p)(rmax (λp))2 G(z/ρ(λp)) . FD˜ (z) = P(D

˜ p). > 0 we have d(λ, p) ≥ d(λ,

proof follows from Jensen’s inequality.


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60


˜ is an extremal shot-noise maxX ∈Φ0 proof: D i


g(Xi ) with the response function

g(x) = |x|p|x| (cos(arg(x)))+ . Its distribution function can be expressed by

the Laplace transform of the (additive) shot noise

P( max

Xi ∈Φ0

·

g(Xi ) ≤ z) = E exp

·X Xi

∈Φ0

ln(1(g(Xi ) ≤ z))

Cor. For the M/M model with W

¸¸

˜ is equal to = 0 the expectation of D

˜ p) = E[D] ˜ d(λ, Z 1 h³ 1 1 ´ G(z) i = 1/β √ 1 − exp 1 − dz , p 2T 2/β C T 2λpeC 0

Φ0 with intensity λ(1 − p) · ¸ Z ˜ ≤ z) = exp −λ(1 − p) P( D 1(g(x) > z) dx .

and thus, for Poisson p.p.

R2

R

Passing to polar coordinates in the integral R2

. . . dx completes the proof.


61



62


˜ p) is attained for p Res. The maximal density of modified progress λp d(λ, satisfyingZ 1 ³

0

The optimization

h³ 1 ´ G(z) i G(z) ´ exp 1 − 1+ dz = 1 . p 2T 2/β C pT 2/β C ˜ p) λpd(λ,

Goal: optimize the density of progress λp d(λ, p) in MAP p, for fixed λ.

0.006

Note that replacing d by d˜ in the optimization problem will result in conservative

0.005

0.004

bound on the density of progress. 0.003

Example: Modified transport capacities for 0.002

the special case with T

= {13, 15, 17}dB

0.001

(curves from top to bottom).

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

p Stochastic geometry and communication networks, B. Błaszczyszyn; tutorial lecture, Performance’05, Juan-les-Pins, France, October 3-7, 2005

63


64


AD-HOC NETWORKS ...

Relation to Gupta & Kumar’s (2000) result

• The transport capacity of a bounded network with Protocol √and SIR (Physical) Model with centralized medium access is proportional to

λ (as λ → ∞).

This law holds true even if the network architecture and operation is optimally organized in a centralized manner.

• Gupta & Kumar (2000), The Capacity of Wireless Networks, IEEE Inf. Theory, • Dousse & Baccelli & Thiran (2003) Impact of Interferences on the Connectivity of

• Our density of progress can be seen as an instantaneous transport capacity.

Ad Hoc Networks, IEEE INFOCOM,

• We show that the independent medium access gives an instantaneous transport √ capacity of K(p)

References

λ.

• Closed form of K(p) allows for optimization of K(p) in p.

• Baccelli & Blaszczyszyn & Muhlethaler (2004), An Aloha protocol for multihop

mobile ad-hoc wireless networks, Proc of 16th ITC Specialist Seminar to appear

in IEEE Inf. Theory

• Question: Is this performance implementable and stable in the strong sense (existence of positive recurrent Markov process describing the system)?


65


66

POWER CONTROL ... CDMA: Interference limited radio channel

V POWER CONTROL IN CDMA: FROM STATIC TO DYNAMIC MODELING

• CDMA Power allocation algebra,

(Y2 , S 2 )

(Y5 , S5 )

• Decentralized Power Allocation Principle DPAP,

W

• Network architecture models, • Maximal load estimates,

X

(Y1 ,S1 )

• Feasibility probabilites, (Y3 ,S3 )

• Blocking rates via a spatial Erlang formula, • References. Stochastic geometry and communication networks, B. Błaszczyszyn; tutorial lecture, Performance’05, Juan-les-Pins, France, October 3-7, 2005

(Y4 ,S4 )

S1 l(X − Y1 ) ≥C W + I(X) 67


68

POWER CONTROL ...

POWER CONTROL / Power allocation algebra...

CDMA Power allocation algebra NBS = {Yj }j : locations of base-stations (BS’s) in R2 , NM = {Xij }i : locations of mobiles served by BS No. j , Cij : SINR required for mobile Xij , Wij : total non-traffic noise at Xij (from common overhead channels, thermal noise), κj , γ : orthogonality factors,

Local and global problem

l(x, y): path-loss from y to x.

Power allocation feasible

Power allocation feasible if exist antenna powers 0

≤

Sij

< ∞ such that

Sij l(Yj − Xij ) j X X X ≥ C i Wij + κj l(Yj , Xij ) Sij0 + γ l(Yk , Xij ) Sik0 |

{z

i0 6=i

own-cell interference

|

all

i, j .

for each BS

{z

}

other-cell interference


69


Global problem

Fix total powers emitted on traffic channels by other BS’s:

Sk =

Power allocation is locally feasible in cell j if exist powers 0

P

Suppose for each BS local power allocation is feasible.

k i0 Si0 (k

≤ Sij < ∞ s.t.

6= j ).

Sij l(Yj − Xij ) X X ≥ Cij j j P j j Wi + κj l(Yj , Xi ) i0 6=i Si0 + γ l(Yk , Xi ) Sik0

κj

X i

P

{z

j k6=j l(Yk ,Xi )Sk

Cij

1+

κj Cij

bj =

i

i

l(Yj , Xij )

X HjW j i i i

}

X H j l(Yk , X j ) i

i

all .

l(Yj , Xij )

,

for j

6= k

and

j where Hi

ajj =

X

κj Hij ,

i

=

Cij

1 + κj Cij

.

Denote the matrix (ajk )

= A, the vector (bj ) = b and (Sj ) = S. Global power allocation is feasible if exist antenna powers 0 ≤ S j < ∞ (total

< 1.

powers emitted on traffic channels) such that

S ≥ b + AS

(local) pole capacity condition


Define:

ajk = γ

i0

k6=j

fixed=γ

70


Fix BS j .

Res. Local power allocation is feasible iff

Global power allocation feasible


Local problem

|

and

i0

k6=j

}

Local power allocation feasible

m

71


72


POWER CONTROL ...

Res.

Decentralized Power Allocation Principle (DPAP)

• Global power allocation feasible iff the spectral radius of the matrix A is less

j Each BS j verifies for the pattern NM of the mobiles it controls if

than 1.

• The minimal solution S is equal to

X

n

κj

A b.

n

|

• The minimal solution can be obtained as the limit of the iteration

equivalently

• A sufficient condition for the spectral radius to be less than one is that A is Principle

⇒ Decentralized Power Allocation


73

POWER CONTROL ...

}

|

{z

< 1,

}

other-cell-interference correcting term

total path loss of user i Hij < 1. own-BS path loss of user i j | {z } Xij ∈NM

X

user’s i weight


74

{Yj } located according to Poisson p.p, with intensity λBS .

• BS’s

located according to hexagonal

• All antenna parameters are i.i.d. marks.

grid, p.p, with spatial density λBS .

• All mobiles form independent Poisson p.p.

• All antenna parameters are i.i.d. marks.

NM

• All mobiles form independent Poisson p.p.

NM

{z

k6=j X j ∈N j i M

j j l(Yk , Xi ) Hi l(Yj , Xij )

Poisson-Voronoi (P-V) model (“too random”)

Hexagonal (Hex) model (“too regular”)

{Yj }

j Xij ∈NM

+γ

X X

POWER CONTROL / Network architectures ...

Network architecture models

• BS’s

Hij

pole capacity term

Ab, A2 b, . . .

substochastic (has row-sums less then 1).

X

• Each mobile is served by the nearest BS.

with spatial density λM .

j (Equivalently: Each BS j serves mobiles NM

• Each mobile is served by the nearest BS.


with intensity λM .

in its Voronoi cell.)

75


76

POWER CONTROL ...

POWER CONTROL / Maximal load estimations ... Explicit mean formulas

(Wrong) Idea: Given density of BS’s λBS find maximal density of mobiles λM ,

¯ – mean number of users per cell, R ∼ – mean distance between BS’s; M λBS = 1/(πR2 ), L(r) = (Kr)α – path-loss, α – path-loss exponent, ¯ ∼ – bit rate (SINR threshold) κ – (downlink) orthogonality factor, H Pmax – power limit

such that power allocation is feasible with probability 1.

downlink:

Maximal load estimations of P-V and Hex model

Res. Given density of BS’s λBS , for any λM > 0 in both P-V and Hex model, the spectral radius of A is equal ∞ with probability 1, and thus power allocation in not feasible!

1 − Pcch /Pmax ¯ ≤ M ¯ + f¯ + L(R)g(M ¯ , α)/Pmax ) H(κ f¯H = 1/(α − 2), f¯P V = 2/(α − 2),

g¯H () ≈ . . . , g¯P V () = . . .

uplink:

Conclusion: A reduction of mobiles (admission control) is necessary for any λM > 0. Calculate blocking probabilities.

1 ¯ ≤ M ¯ + f¯ + L(R)h(M ¯ , α)/Pmax ) H(1 ¯ P V () = . . . ¯ H () ≈ . . . , h h


77

POWER CONTROL / Maximal load estimations ...


78

POWER CONTROL ...

¯ M 35

30

Feasibility probabilities for DPAP

25 20 15

P

10 5 0

1

2

3

4

5

µ

j Xij ∈NM

|

R

Hexagonal model Poisson-Voronoi model

X

Hij

total path loss of user i own-BS path loss of user i

¶