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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
S4G, Prague 2006 2
Outline
Stochastic–geometric Network Modelling
Real network data
Stochastic Subscriber Line Model
Geometry Model, Equipment Model, Topology Model
Model Selection
S4G, Prague 2006 2
Outline
Stochastic–geometric Network Modelling
Real network data
Stochastic Subscriber Line Model
Geometry Model, Equipment Model, Topology Model
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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Stochastic–geometric Network ModellingStochastic Subscriber Line Model
Main roads and side streets
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Stochastic–geometric Network ModellingStochastic Subscriber Line Model
Location of subscribers
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Stochastic–geometric Network ModellingStochastic Subscriber Line Model
Network nodes and serving zones
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Stochastic–geometric Network ModellingStochastic Subscriber Line Model
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
S4G, Prague 2006 6
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)))
S4G, Prague 2006 30
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
S4G, Prague 2006 30
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
S4G, Prague 2006 31
Mean Subscriber Line LengthsComputational algorithm
Samples of typical Cox–Voronoi serving zones
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Mean Subscriber Line LengthsComputational algorithm
(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
Mean Subscriber Line LengthsComputational algorithm
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
S4G, Prague 2006 34
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
S4G, Prague 2006 36
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
S4G, Prague 2006 37
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
S4G, Prague 2006 38
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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