Networks and Stochastic Geometry - Models, Fitting, and ... · Networks and Stochastic Geometry - Models, Fitting, and Applications in Telecommunication Hendrik Schmidt France Telecom

IntroductionRandom Tessellations

Statistical Model Fitting of TessellationsCost Analysis through Network Trees

Networks and Stochastic Geometry - Models,

Fitting, and Applications in Telecommunication

Hendrik Schmidt

France Telecom NSM/R&D/RESA/NET

Reisensburg-Workshop February 2007

Hendrik Schmidt Networks and Stochastic Geometry



Outline

1 IntroductionReal Infrastructure Data of ParisStochastic Geometric Network Modeling

2 Random TessellationsNon–iterated Tessellations and their CharacteristicsIteration of Tessellations

3 Statistical Model Fitting of TessellationsMinimization Problem and Solution ApproachesApplication to Real Network Data

4 Cost Analysis through Network TreesGeometric Structural SupportShortest Path Analysis




Real Infrastructure Data of ParisStochastic Geometric Network Modeling

Real Infrastructure Data of Paris





Stochastic Geometric Network ModelingMain Roads





Stochastic Geometric Network ModelingMain Roads and Side Streets





Stochastic Geometric Network ModelingNetwork Devices and Serving Zones





Stochastic Geometric Network ModelingModeling Components

Geometric supportNetwork equipment Topology of placement

Random objects (infrastructure, equipment, topology)

provide a statistically equivalent image of reality

are defined by few parameters

allow to study separately the three parts of the network




Non–iterated Tessellations and their CharacteristicsIteration of Tessellations

Random TessellationsNon–iterated Tessellations and Their Characteristics

PLT, γPLT = 0.02 PVT, γPVT = 0.0001 PDT, γPDT = 0.000037





Random TessellationsNon–iterated Tessellations and Their Characteristics

Mean value relations for facet characteristics

Measured per unit area

λ1 mean number of verticesλ2 mean number of edgesλ3 mean number of cellsλ4 mean total length of edges

For the considered tessellation models with intensity γ

Model λ1 λ2 λ3 λ4

PLT 1

πγ2 2

πγ2 1

πγ2 γ

PVT 2γ 3γ γ 2√

γ

PDT γ 3γ 2γ 32

3π

√γ





Random TessellationsIteration of Tessellations

(a) PLT/PLT,

γ0 = 0.02, γ1 = 0.04

(b) PLT/PVT,

γ0 = 0.02, γ1 = 0.0004

(c) PLT/PDT, γ0 =

0.02, γ1 = 0.0001388

⇒ λ4 = 0.02 + 0.04 = 0.06






PLT with Bernoulli thinning PLT with multi–type nesting






Mean value relations for facet characteristics of X0/pX1-nestings

λ1 = λ(0)1

+ pλ(1)1

+4p

πλ

(0)4

λ(1)4

λ2 = λ(0)2

+ pλ(1)2

+6p

πλ

(0)4

λ(1)4

λ3 = λ(0)3

+ pλ(1)3

+2p

πλ

(0)4

λ(1)4

λ4 = λ(0)4

+ pλ(1)4

⇒ Mean-value formulae for nestings involving PLT, PDT and PVT




Minimization Problem and Solution ApproachesApplication to Real Network Data

Statistical Model Fitting of TessellationsMinimization Problem and Solution Approaches

Step 1 : Vector of estimates

λinp =

(λinp

1, λinp

2, λinp

3, λinp

4

)⊤

from input data in sampling window W using unbiasedestimators

λinp =

1

|W |(nv , ne , nc , le)⊤

nv number of vertices in W

ne number of edges, whose lexicographically smaller endpointlies in W

nc number of cells, whose lexicographically smallest vertex liesin W

le total length of edges in W






Step 2 : Calculate λ = λ(γ) = (λ1, λ2, λ3,λ4)⊤ (or λ(γ0, γ1))

Step 3 : Minimize d(λinp, λ) . Example :

d(λinp, λ) =4∑

i=1

((λinp

i − λi )/λinpi

)2

→ min

Inserting the mean value relation (PLT)

f (γ) =

((λinp

1− 1

πγ2)/λinp1

)2

+

((λinp

2− 2

πγ2)/λinp2

)2

+

((λinp

3− 1

πγ2)/λinp3

)2

+

((λinp

4− γ)/λinp

4

)2

→ min

Step 4 : Determine d(λinp, λmin) and γmin (or γmin0 and γmin

1 )








d(λinp, λ) =4∑

i=1

((λinp

i − λi )/λinpi

)2

→ min


f (γ) =

((λinp

1− 1

πγ2)/λinp1

)2

+

((λinp

2− 2

πγ2)/λinp2

)2

+

((λinp

3− 1

πγ2)/λinp3

)2

+

((λinp

4− γ)/λinp

4

)2

→ min


1 )








d(λinp, λ) =4∑

i=1

((λinp

i − λi )/λinpi

)2

→ min


f (γ) =

((λinp

1− 1

πγ2)/λinp1

)2

+

((λinp

2− 2

πγ2)/λinp2

)2

+

((λinp

3− 1

πγ2)/λinp3

)2

+

((λinp

4− γ)/λinp

4

)2

→ min


1 )





Statistical Model Fitting of TessellationsApplication to Real Network Data

Raw data Preprocessed data

|W | = 3000m × 3000m = 9km2






Fitting strategy : Exploit hierarchical data structure

Main roads : ? ? ?Model Distance γmin

PLT 0.21101 0.002384PVT 0.29749 0.000001PDT 0.73378 0.000001

Whole network : PLT/ ? ? ?

Model Distance γmin0 γmin

1

PLT/PLT 0.15224 0.002384 0.013906

PLT/PVT 0.20455 0.002384 0.000044

PLT/PDT 0.36649 0.002384 0.000028






Fitting strategy : Exploit hierarchical data structure

Main roads : ? ? ?Model Distance γmin

PLT 0.21101 0.002384PVT 0.29749 0.000001PDT 0.73378 0.000001

Whole network : PLT/ ? ? ?

Model Distance γmin0 γmin

1

PLT/PLT 0.15224 0.002384 0.013906

PLT/PVT 0.20455 0.002384 0.000044

PLT/PDT 0.36649 0.002384 0.000028






Preprocessed road system Realization of optimal PLT/PLT




Geometric Structural SupportShortest Path Analysis

Cost Analysis through Network TreesGeometric Structural Support

Network devices on real

infrastructure

0 0.5 1 1.5 2 2.5

00.

20.

40.

60.

8 sans voirievoirie réelle

Histogram of shortest

paths Spatial placement of

network devices






Network devices on real

infrastructure

0 0.5 1 1.5 2 2.5

00.

20.

40.

60.

8

voirie réellePVT fit

Histogram of shortest

paths Network devices on

fitted infrastructure






Randomtessellation models

can replace realinfrastructuredataafter havingbeen fit

Simulation studiesare very expensive

=⇒ Analytical formulae are needed





Cost Analysis through Network TreesShortest Path Analysis

Poisson linetessellation Xℓ

Intensity γ > 0 :Mean total lengthof lines per unitarea






Stationary ergodicpoint processXH = {Xn}n≥1 onthe lines

Special case :Poisson process XH

Linear intensityλ1 > 0

Planar intensityλH = γ λ1






Sequence of servingzones {Ξ(Xn)}n≥1

Later on :

Typical servingzone Ξ∗

Typical linesystem L(Ξ∗)






Cox-Voronoi tessellation (CVT) based on a Poisson line process






Linear placement on lines

Stationary Poisson process{Xn}n≥1 on the lines

Linear intensity λ2 > 0Planar intensity λL = γ λ2

Stationary marked process XL ={[Xn, c(P(Xn,N(Xn)))]}n≥1

N(Xn) location of nearest

HLC of Xn

P(Xn,N(Xn)) shortest

path from Xn to N(Xn)

c(P(Xn,N(Xn))) length

(costs) of P(Xn,N(Xn))






Natural Approach

Simulate network in (large) sampling window W ⊂ R2

Compute c(P(Xn,N(Xn))) for each Xn

Compute mean shortest path length

cLH(W ) =1

#{n : Xn ∈ W }∑

n≥1

1IW (Xn)c(P(Xn,N(Xn)))

Disadvantages

W small ⇒ edge effects significantW large ⇒ memory and runtime problems






Alternative Approach

{Wi}i≥1 averaging sequence of sampling windowsErgodicity of XL yields with probability 1 that

limi→∞

cLH(Wi ) = c∗LH

The limit c∗LH

is given by (B ∈ B0(IRd))

1

λLν2(B)E

∑

n≥1

1IB(Xn)c(P(Xn, N(Xn))) = EXLc(P(o, N(o)))

Disadvantages

Simulation not clearNot very efficient






Application of Neveu

EXLc(P(o, N(o))) =

1

EXHν1(L(Ξ∗))

EXH

∫

L(Ξ∗)

c(P(u, o)) du ,

where1

EXHν1(L(Ξ∗))

= λ1

Independence from λ2

Simulation algorithm for typical cell of CVT needed






Estimators for c∗LH (k ≥ 1) are given by

cLH(k) =1

k∑i=1

ν1(L(Ξ∗i ))

k∑

i=1

Mi∑

j=1

∫

S(j)i

c(P(u, o)) du

and

cLH(k) = λ1

1

k

k∑

i=1

Mi∑

j=1

∫

S(j)i

c(P(u, o)) du

Note :∫S

(j)i

c(P(u, o)) du can be analytically calculated






Let γ(1)/λ(1)1

= γ(2)/λ(2)1

.

Then γ(1) c∗LH

(γ(1), λ(1)1

)

= γ(2) c∗LH

(γ(2), λ(2)1

)

Different intensities but same κ = γ/λ1






c*^LH

20

40

60

80

100

0 2 4 6 8

1/γ

50000 realizations of Ξ∗

(and L(Ξ∗))

Estimated results of c∗LHfor

κ = 10κ = 50κ = 120

c∗LH(γ, λ1) = m(κ)/γ






0

1

2

3

4

5

6

7

m( )^

20 40 60 80 100 120κ

κ

Good approximationm(κ) ≈ aκb

Computation of c∗LHwithout simulation

κ = γ/λ1

m(κ) = aκb

c∗LH

= m(κ)γ−1






Last meter

HLC

LLC

Projected point

Spatial placement and

projection to nearest line

Stationary Poisson point

process {X ′n}n≥1 (int. λL )

Stationary marked process X ′L

={[X ′

n, c(P(X ′n,N(X ′

n)))]}n≥1

N(X ′n) location of nearest

HLC of X ′n

Project X ′n onto X ′′

n

c(P(X ′n,N(X ′

n))) =c ′(X ′

n,X′′n ) +

c(P(X ′′n ,N(X ′

n)))c ′(X ′

n,X′′n ) cost value of

last meter





Cost Analysis through Network TreesOutlook

CVT based on

PVT PDT Nesting





Cost Analysis through Network TreesOutlook

Intensity map of Paris (suppose underlying PLT)





Literature can be found onwww.geostoch.de

This talk is based on joint work withF. Fleischer, Institute of Stochastics, Ulm University

C. Gloaguen, France Telecom NSM/R&D/RESA/NETV. Schmidt, Institute of Stochastics, Ulm University

Thank you for your attention !


Documents

Networks and Stochastic Geometry - Models, Fitting, and ... · Networks and Stochastic Geometry - Models, Fitting, and Applications in Telecommunication Hendrik Schmidt France Telecom