modelling and simulation of telecommunication networks ...hendrik/talks/prague06.pdf · s4g, prague...

Modelling and Simulation ofTelecommunication Networks:

Analysis of Mean Shortest Path Lengths

Hendrik Schmidt

Joint work with F. Fleischer, C. Gloaguen and V. Schmidt

University of Ulm

Department of Stochastics

France Telecom, Division R&D, Paris

S4G, Prague 2006 1

Outline

Stochastic–geometric Network Modelling

Real network data

Stochastic Subscriber Line Model

Geometry Model, Equipment Model, Topology Model

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Outline


Real network data



Model Selection

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Outline


Real network data



Model Selection

Mean Shortest Path Lengths

Simulation methods

Application of Neveu’s Exchange Formula for Palm Distributions

Mean Subscriber Line Lengths

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Stochastic–geometric Network ModellingReal network data

Infrastructure system of Paris

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Stochastic–geometric Network ModellingStochastic Subscriber Line Model

Main roads

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Main roads and side streets

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Location of subscribers

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Network nodes and serving zones

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Network Model Topology ModelGeometry Model

Placement of

Higher level comp.(HLC) and Lowerlevel comp. (LLC)

stations, repartiteur)and SAI (servicearea interface,sous−repartiteur)

SAI and subscriber

Topology of

(tree, ...)connections components

Serving zones:

of Voronoi−cells

SAI is connectedto WCS of its

Infrastructuresystem (roads, ...)

Simple Tessellations(PLT, PVT, PDT)

Nestings

Multi−type nestings

urban and rural

Stochastic Subscriber Line Model (SSLM)

2−level models:

2−level models:

WCS (wire center

serving zone

WCS are nuclei

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Model Selection

Based on minimization of distances betweencharacteristics of input data andcomputed values of these characteristics usingmean–value formulae

Decision in favor of optimal tessellation model within aclass of given random tessellation models

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Model Selection

Based on minimization of distances betweencharacteristics of input data andcomputed values of these characteristics usingmean–value formulae

Decision in favor of optimal tessellation model within aclass of given random tessellation models

Tessellations and their characteristics (per unit area)Mean number of vertices (γ1)Mean number of edges (γ2)Mean number of cells (γ3)Mean total length of edges (γ4)

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Model SelectionExamples of non-iterated tessellation models

3 non–iterated tessellation models (PLT, PVT, PDT)

(a)PLT, γP LT = 0.02 (b)PVT, γP V T = 0.0001 (c)PDT, γP DT = 0.000037

⇒ Mean total length of edges (per unit area) takes thesame value γ4 in all three cases (γ4 = 0.02)

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Model SelectionCharacteristics of non-iterated tessellations

Model γ1 γ2 γ3 γ4

PLT 1πγ2 2

πγ2 1πγ2 γ

PVT 2γ 3γ γ 2√

γ

PDT γ 3γ 2γ 323π

√γ

Values of γ1, . . . , γ4 for different tessellation models withparameter γ

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Model SelectionExamples of iterated tessellation models

9 iterated models (combination of PLT, PVT, PDT)

(a)PLT/PLT

γ(0)P LT

= 0.02 ,

γ(1)P LT

= 0.04

(b)PLT/PVT

γ(0)P LT

= 0.02 ,

γ(1)P V T

= 0.0004

(c)PLT/PDT

γ(0)P LT

= 0.02 ,

γ(1)P DT

= 0.0001388

⇒ Mean total length of edges (per unit area) γ4 takes thesame value (γ4 = 0.06)

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Model SelectionCharacteristics of iterated tessellations

Characteristics of X0/pX1-nesting

γ1 = γ(0)1 + pγ

(1)1 +

4p

πγ

(0)4 γ

(1)4

γ2 = γ(0)2 + pγ

(1)2 +

6p

πγ

(0)4 γ

(1)4

γ3 = γ(0)3 + pγ

(1)3 +

2p

πγ

(0)4 γ

(1)4

γ4 = γ(0)4 + pγ

(1)4

⇒ Mean-value formulae for nestings involving PLT, PDTand PVT

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Model SelectionAlgorithm

Step 1 : Observe input data in sampling window W

Step 2 : Get vector of estimates (from input data)

γinp = (γinp1 , γinp

2 , γinp3 , γinp

4 ) =1

ν2(W )(nv, ne, nc, le) ,

nv number of vertices in W

ne number of edges, whose lexicographically smallerendpoint lies in W

nc number of cells, whose lexicographically smallestvertex lies in W

le total length of edges in W

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Model SelectionAlgorithm

Step 3 : Minimization problem (rel. Eucl. distance)

dist(γinp,γ) =

√√√√4∑

i=1

((γinp

i − γi)/γinpi

)2

→ min

Insertion of mean-value formulae (example PLT):

f(γ) =

((γinp

1 − 1πγ2)/γinp

1

)2

+

((γinp

2 − 2πγ2)/γinp

2

)2

+

((γinp

3 − 1πγ2)/γinp

3

)2

+

((γinp

4 − γ)/γinp4

)2

→ min

Step 4 : Repeat for all competing models: Obtain γopt

Step 5 : Model with overall minimum distance = optimaltessellation model

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Model SelectionRoad system of Paris

Line segments with marks(type of road):

highway

national route

main road

side street

...Real infrastructure data of Paris

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Model SelectionPreprocessing of data

Main roads and side streets under consideration

Construction of a tessellation with polygonal cellsRemove dead end streetsDiscard points where only two line segmentsemanate from

(a) Raw data (b) Preprocessed data

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Model SelectionSampling window

Analysed sampling window W

(a) Raw data (b) Preprocessed data

Location and size of W : lower left vertex [5000,3000], upper right vertex [8000,6000];

ν2(W ) = 3000m × 3000m = 9km2

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Model SelectionResults: Main roads

Model Distance γopt

PLT 0.21101 0.002384

PVT 0.29749 0.000001

PDT 0.73378 0.000001Distance and corresponding optimized parameter γopt

Main roads modelled by a PLT with intensityγPLT = 0.002384

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Model SelectionResults: All roads

Road system - decision between PLT/PLT, PLT/PVT andPLT/PDT

Model Distance γ(0)opt γ

(1)opt

PLT/PLT 0.15224 0.002384 0.013906

PLT/PVT 0.20455 0.002384 0.000044

PLT/PDT 0.36649 0.002384 0.000028

Distance and corresponding optimized intensity parameters γ(0)opt and γ

(1)opt

Road system modelled by a PLT/PLT-nesting withγ

(0)PLT = 0.002384 and γ

(1)PLT = 0.013906

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Model DescriptionGeometry model

Poisson linetessellation Xℓ

Intensity γ > 0:Mean total lengthof lines per unitarea

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Model DescriptionPlacement of higher level components

Stationary ergodicpoint processXH = Xnn≥1 onthe lines

Special case:Poisson processXH

Linear intensityλ1 > 0

Planar intensityλH = γ λ1

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Model DescriptionServing zones

Sequence ofserving zonesΞ(Xn)n≥1

Later on:Typical servingzone Ξ∗

Typical linesystem L(Ξ∗)

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Model DescriptionServing zones

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

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Mean Shortest Path LengthsPlacement of lower level components

HLC

LLC

LLC

LLC

Linear placement on lines

Stationary Poisson process

eXnn≥1 on the lines

Linear intensity λ2 > 0

Planar intensity λL = γ λ2

Stationary marked point

process XL =

[ eXn, c(P ( eXn, N( eXn)))]n≥1

N( eXn) location of nearest

HLC of eXn

P ( eXn, N( eXn)) shortest path

from eXn to N( eXn)

c(P ( eXn, N( eXn))) length

(costs) of P ( eXn, N( eXn))

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Mean Shortest Path LengthsSimulation methods

Natural ApproachSimulate network in (large) sampling windowW ⊂ IR2

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)))

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

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Mean Shortest Path LengthsSimulation methods

Alternative ApproachWii≥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(IR2))

1

λLν2(B)IE

X

n≥1

1IB( eXn)c(P ( eXn, N( eXn))) = IEXLc(P (o, N(o)))

DisadvantagesSimulation not clearNot very efficient

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Neveu’s Exchange Formula for Palm Distributions

Flow θx , x ∈ IR2 with θx : Ω → Ω

Palm distribution IP∗XD

of XD : Ω → M(IR2 × D),

IP∗XD

(A × G) =1

λν2(B)

Z

Ω

Z

IR2×G

1IB(x) 1IA(θxω) XD(ω, d(x, g))IP(dω)

for B ∈ B(IR2) and A ∈ A, G ∈ B(D), 0 < λ < ∞

f : IR2 × D × D × Ω → [0,∞) measurable function

λ

∫

Ω×D

∫

IR2× eD

f(x, g, g, θxω)X eD(ω, d(x, g))IP∗XD

(d(ω, g))

= λ

∫

Ω× eD

∫

IR2×D

f(−x, g, g, ω)XD(ω, d(x, g))IP∗eX eD

(d(ω, g))

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Mean Shortest Path Lengths

Application of Neveu

IEXLc(P (o,N(o))) =

1

IEXHν1(L(Ξ∗))

IEXH

∫

L(Ξ∗)

c(P (u, o)) du

With1

IEXHν1(L(Ξ∗))

= λ1

Independence from λ2

Simulation algorithm for typical cell of CVT needed

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Mean Shortest Path LengthsComputational algorithm

Estimators for c∗LH (k ≥ 1),

cLH(k) =1

k∑i=1

ν1(L(Ξ∗i ))

k∑

i=1

Mi∑

j=1

∫

S(j)i

c(P (u, o)) du

cLH(k) = λ11

k

k∑

i=1

Mi∑

j=1

∫

S(j)i

c(P (u, o)) du

∫S

(j)i

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

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Mean Shortest Path LengthsComputational algorithm

S

S

SS

S

SS

S

SS

S

S

SS

S

1

23

4

56 7

89

10

1112 13

14

15

16

o

S

S

S1

17

8

*Ξ

Partitioning of L(Ξ∗) into segments

A(S) B(S)D(S)

S

c (S) c (S)A B

o

Mean shortest path length for S

RS

c(P (u, o)) du = 14(ν1(S))2 + 1

2(cA(S) + cB(S))ν1(S) + 1

4(cB(S) − cA(S))2

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Mean Subscriber Line LengthsPlacement of lower level components

Last meter

HLC

LLC

Projected point

Spatial placement andprojection to nearest line

Stationary Poisson point

process X ′nn≥1 (int. λL )

Stationary marked point

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

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Mean Subscriber Line LengthsSimulation methods

Mean subscriber line length

dLH(W ) =1

#n : X ′n ∈ W

∑

n≥1

1IW (X ′n) c(P (X ′

n, N(X ′n)))

Ergodicity of X ′L

limi→∞

dLH(Wi) = d∗LH = IEX ′Lc(P (o,N(o)))

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Mean Subscriber Line LengthsSimulation methods

Mean subscriber line length

dLH(W ) =1

#n : X ′n ∈ W

∑

n≥1

1IW (X ′n) c(P (X ′

n, N(X ′n)))

Ergodicity of X ′L

limi→∞

dLH(Wi) = d∗LH = IEX ′Lc(P (o,N(o)))

Application of Neveu

IEX ′L

c(P (o,N(o))) =1

IEXHν2(Ξ∗)

IEXH

∫

Ξ∗

c(P (u, o)) du

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Mean Subscriber Line LengthsComputational algorithm

Estimators for d∗LH (k ≥ 1),

dLH(k) =1

k∑i=1

ν2(Ξ∗i )

k∑

i=1

Ki∑

j=1

∫

Ψ(j)i

c(P (u, o)) du

dLH(k) = λ1γ1

k

k∑

i=1

Ki∑

j=1

∫

Ψ(j)i

c(P (u, o) du

∫Ψ

(j)i

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

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Samples of typical Cox–Voronoi serving zones

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(a) Typical cell

Ψ

Ψ

Ψ

Ψ

1

2

3

4

(b) A micro–cell

Ψ1

(c) Part of this micro–cell

Ψ1

S

(d) Further decompositionS4G, Prague 2006 33


S B(S)

Ψ

ΨΨ

A(S)=ΨA

C

B

RΨ

c(P (u, o)) du = ca(S) ν2(ΨA) + cB(S) ν2(ΨB) +RΨC

c(P (u, o)) du

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Numerical ResultsScaling property

Let γ(1)/λ(1)1 = γ(2)/λ

(2)1

γ(1) c∗LH(γ(1), λ(1)1 )

= γ(2) c∗LH(γ(2), λ(2)1 )

γ(1) d∗LH(γ(1), λ

(1)1 )

= γ(2) d∗LH(γ(2), λ

(2)1 )

Different intensities but same κ = γ/λ1

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Numerical ResultsMean shortest path lengths

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∗LH forκ = 10

κ = 50

κ = 120

c∗LH(γ, λ1) = m(κ)/γ

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Numerical ResultsMean shortest path lengths

0

2

4

6

8

10

12

m( )^

20 40 60 80 100 120 140

κ

κ

m(κ) of c∗LH for increasing κ

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Numerical ResultsMean subscriber line lengths

d*^LH

10

20

30

40

50

60

70

0 2 4 6 8

1/γ

50000 realizations of Ξ∗

(and L(Ξ∗))

Estimated results of d∗LH forκ = 10

κ = 50

κ = 120

d∗LH(γ, λ1) = m′(κ)/γ

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Numerical ResultsMean subscriber line lengths

0

2

4

6

8

m’( )^

20 40 60 80 100 120 140

κ

κ

m′(κ) of d∗LH for increasing κ

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Numerical ResultsComparison and summary

c∗LH > d∗LH for any parameter pair (λ1, γ)

Good approximations m(κ) ≈ aκb and m′(κ) ≈ a′κb′

Thereby estimation of c∗LH and d∗LH without simulationpossible

κ = γ/λ1

m(κ) = aκb

m′(κ) = a′κb′

c∗LH = m(κ)γ−1

d∗LH = m′(κ)γ−1

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OutlookExtensions to other geometry models

CVT based on Poisson-Voronoi

CVT based on Poisson-Delaunay

CVT based on iterated tessellations

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Literature

C. Gloaguen, F. Fleischer, H. Schmidt, V. Schmidt.Fitting of Stochastic Telecommunication Networks via DistanceMeasures and Monte Carlo TestsTelecommunication Systems 31, 353-378 (2006).

M. Beil, S. Eckel, F. Fleischer, H. Schmidt, V. Schmidt,P. Walter.Fitting of Random Tessellation Models to Cytosceleton NetworkDataJournal of Theoretical Biology 241, 62–72 (2006).

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Literature

C. Gloaguen, F. Fleischer, H. Schmidt, V. Schmidt.Simulation of typical Cox-Voronoi cells, with a special regard toimplementation tests.Mathematical Methods of Operations Research 62(3),357-373 (2005).

C. Gloaguen, F. Fleischer, H. Schmidt, V. Schmidt.Analysis of shortest path and subscriber line lengths intelecommunication access networks.Preprint, (2006).

For more information www.geostoch.de

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modelling and simulation of telecommunication networks ...hendrik/talks/prague06.pdf · s4g, prague...

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