Fitting and Simulation of Models forTelecommunication Access Networks
Frank FleischerJoint work with C. Gloaguen, H. Schmidt, V. Schmidt
University of Ulm
Department of Applied Information Processing
Department of Stochastics
France Telecom R & D, Paris
Workshop Freudenstadt 2005, Frank Fleischer 1
Outline
Stochastic–geometric network modellingAims
The Stochastic Subscriber Line Model
Models for the road systemRandom tessellations
Nestings of tessellations
Model choice procedureModel choice based on distance measures
Results for real input data
Typical Cox-Voronoi cellsSimulation algorithm
Results
Workshop Freudenstadt 2005, Frank Fleischer 2
Stochastic–geometric network modellingAims
Aims of modellingCost analysis and risk evaluationAnalysis of performance indicatorsSimulation of present and future network designscenariosDescription of the network by a minimum number ofstructural parameters
Models are necessary both for the road system and forthe telecommunication equipment
The model choice has to be based on the statisticalanalysis of real network data as well as of simulatednetwork data
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Stochastic–geometric network modellingInfrastructure system of Paris
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Stochastic–geometric network modellingStochastic Subscriber Line Model (SSLM)
System of main roads
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Stochastic–geometric network modellingStochastic Subscriber Line Model (SSLM)
System of main roads and side streets
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Stochastic–geometric network modellingStochastic Subscriber Line Model (SSLM)
Placement of network nodes and their serving zones
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Models for the road systemRandom tessellations
A sequence {Pn}n≥1 of convex polytopes Pn ∈ IR2 iscalled (deterministic) tessellation of IR2 if
int Pn 6= ∅ for all n ≥ 1
int Pn ∩ int Pm = ∅ for all n 6= m
S∞n=1
Pn = IR2
P
n≥1
1{Pn∩K 6=∅} < ∞ for all compact sets K ∈ IR2
The Pn’s are called cells of the tessellation
A sequence X0 = {Ξn}n≥1 of random convex polytopesis called random tessellation of IR2 if
IP(X0 ∈ T ) = 1 ,
where T denotes the family of all tessellations in IR2
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Models for the road systemPoisson line tessellation (PLT)
Realization of PLT
Induced by Poissonline process
The intensity γPLT isthe mean total length oflines per unit area
Notice: 2γPLT is themean number of linesintersecting the unit ball
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Models for the road systemPoisson–Voronoi tessellation (PVT)
Realization of PVT
Cells are formed withrespect to a set ofnuclei
The intensity γPV T isthe mean number of nu-clei per unit area
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Models for the road systemPoisson–Delaunay tessellation (PDT)
Realization of PDT
Cells are triangles
Vertices are nuclei of aPVT
The intensity γPDT isthe mean number ofvertices per unit area
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Models for the road systemSome characteristics of random tessellations
Realization of PLT
Global characteristicsλ1 mean number of vertices
λ2 mean number of edges
λ3 mean number of cells
λ4 mean total length of
edges
Local characteristicsmean perimeter
mean area
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Models for the road systemSome characteristics of random tessellations
Tessellation λ1 λ2 λ3 λ4
PLT 1πγ2 2
πγ2 1
πγ2 γ
PVT 2γ 3γ γ 2√
γ
PDT γ 3γ 2γ 323π
√γ
Values of λ1, . . . , λ4 for a tessellation with intensity γ
Workshop Freudenstadt 2005, Frank Fleischer 11
Models for the road systemRandom nestings of tessellations
A (deterministic) iterated tessellations of IR2 is given by
{Pn ∩ Pnν : int Pn ∩ int Pnν 6= ∅ ; n, ν ∈ N}an initial tessellation {Pn}n≥1
a sequence ({Pnν}ν≥1)n≥1 of component tessellations
A (random) nesting of tessellations in IR2 is given by
{Ξn ∩ Ξnν : int Ξn ∩ int Ξnν 6= ∅ ; n, ν ∈ N}X0 = {Ξn}n≥1 is an arbitrary random tessellation in IR2
{Xn}n≥1 = ({Ξnν}ν≥1)n≥1 is an independent sequence of independent
and identically distributed random tessellations in IR2
Notation: X0/X1–nesting
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Models for the road systemRandom nestings of tessellations
PLT/PVT–nesting PDT/PLT–nesting
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Models for the road systemRandom nestings of tessellations
PVT/PVT–nesting PLT/PVT-nesting
Nestings with Bernoulli thinning (p = 0.75)
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Models for the road systemSome characteristics of random nestings
PVT/PVT-nesting
Characteristics of X0
(λ(0)1 , λ
(0)2 , λ
(0)3 , λ
(0)4 )
Characteristics of X1
(λ(1)1 , λ
(1)2 , λ
(1)3 , λ
(1)4 )
Measured per unit area
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Models for the road systemSome characteristics of random nestings
Joint characteristics of X0/pX1–nesting
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
⇒ Formulae for nestings involving PLT, PDT and PVT
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Idea of the model choiceReal network data
Data possess certainhierarchical structure
Examples of datacharacteristics
Number of road intersections
(λ1)
Number of edges (λ2)
Number of cells (λ3)
Total length of roads (λ4)
Real network data and their characteristics
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Idea of the model choiceSummary
Observe input data in sampling window W
Get vector of estimates
λinp = (λinp
1 , λinp2 , λinp
3 , λinp4 )
from input data by using unbiased estimators
For certain values of γ (or γ0 and γ1) compute theentries λ1, λ2, λ3, and λ4 of λ
Minimize distance d(λinp,λ) ⇒ dmin(λinp,λ)
Repeat for all competing models: λopt and γopt
Validation by Monte-Carlo tests
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Model choice procedureRelative distance measures
Let x = (x1, . . . , xn) and y = (y1, . . . , yn) denote twovectors
Euclidean distance
de(x,y) =
√
√
√
√
n∑
i=1
(
xi − yi
xi
)2
Absolute value distance
da(x,y) =n
∑
i=1
∣
∣
∣
∣
xi − yi
xi
∣
∣
∣
∣
Maximum norm distance
dm(x,y) = maxi=1,...,n
∣
∣
∣
∣
xi − yi
xi
∣
∣
∣
∣
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Model choice procedureOptimization of distances
Estimate λinp = (λinp
1 , λinp2 , λinp
3 , λinp4 ) from input data
Consider PLT (γPLT ), PVT (γPV T ), and PDT (γPDT )Go through a range of values for γPLT , γPV T , andγPDT
Each time compute values λ1, λ2, λ3, and λ4 of λ
Minimize distance d = d(λinp,λ)
Obtain dminPLT , dmin
PV T , and dminPDT
Obtain γopt through dopt = min{dminPLT , dmin
PV T , dminPDT }
Analogously in case of nestings
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Numerical examplesSimulated input data
Consider PLT with γ = 0.1
Quadratic sampling window (side length 10000)
Estimated Theoretical
λ1 0.00318 0.00318
λ2 0.00630 0.00637
λ3 0.00318 0.00318
λ4 0.09995 0.10000
Estimated and theoretical characteristics of the PLT
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Numerical examplesSimulated input data
Let γ ∈ [0.0001, 0.5], step width 10−5
de,min γ da,min γ dm,min γ
PLT 0.0075 0.0998 0.0102 0.0999 0.0049 0.0997
PVT 0.4672 0.0020 0.7450 0.0021 0.3349 0.0021
PDT 0.6455 0.0017 1.0968 0.0016 0.4372 0.0018
Estimated and theoretical characteristics of the PLT
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Numerical examplesSimulated input data
X0 = PLT (γ0 = 0.08), X1 = PDT (γ1 = 0.0008)
Quadratic sampling window (side length 10000)
Estimated Theoretical
λ1 0.01231 0.012619
λ2 0.02165 0.021147
λ3 0.00829 0.008528
λ4 0.17433 0.176034
Estimated and theoretical characteristics of thePLT/PDT–nesting
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Numerical examplesSimulated input data
Let γ0, γ1 ∈ [0.00001, 0.15], step width 10−5
de,min γ0 γ1
PLT/PLT 0.07422 0.05727 0.10703
PLT/PVT 0.06844 0.11961 0.00052
PLT/PDT 0.04008 0.08235 0.00073
PVT/PLT 0.06844 0.00052 0.11961
PVT/PVT 0.21489 0.00001 0.00641
PVT/PDT 0.06424 0.00101 0.00119
PDT/PLT 0.04008 0.00073 0.08235
PDT/PVT 0.06424 0.00119 0.00101
PDT/PDT 0.07416 0.00074 0.00074
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Numerical examplesSimulated input data
Solutions for problem of ambiguous decisions
If thinning factor p < 1 decisions uniqueIntroduce p as additional optimization parameterIncreases runtimeSometimes p known
Iterative fitting procedureFitting procedure for X0
Fitting of nesting with X0 givenHierarchical data structure needed
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Numerical examplesReal input data
(a) Raw data (b) Preprocessed data
Data of a local region within Paris
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Numerical examplesReal input data
Decide between PLT, PVT, and PDT
Let γ ∈ [10−6, 0.03], step width 10−8
Tessellation de,min γ
PLT 0.29555 0.016922
PVT 0.20147 0.000065
PDT 0.77293 0.000046
Decision in favor of PVT with γ = 0.000065
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Numerical examplesReal input data
Decision for a PVT is obvious?
Idea: Consider main roads first
Tessellation de,min γ
PLT 0.21101 0.002384
PVT 0.29749 0.000001
PDT 0.73378 0.000001
Main roads would be modeled by PLT with γ = 0.002384
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Numerical examplesReal input data
Decide between PLT/PLT, PLT/PVT, and PLT/PDT
X de,min γ0 γ1
PLT/PLT 0.15224 0.002384 0.013906
PLT/PVT 0.20455 0.002384 0.000044
PLT/PDT 0.36649 0.002384 0.000028
Road system would be modeled by PLT/PLT–nesting withγ0 = 0.002384 and γ1 = 0.013906
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Typical Cox-Voronoi cells
For main roads decision mostly in favor of PLT
Higher-level telecommuncation equipment placed onmain roads only
Placement modeled by linear Poisson processesCox processes induced by Poisson line processesTypical serving zones of interestTypical cells of Cox-Voronoi tessellation (CVT)’Typical’ means drawing uniformly from all cells
Simulation algorithm based on Slyvniak’s theorem
Usage of properties of Poisson point processes andPoisson line processes
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Typical Cox-Voronoi cells
Random placement of points on PLTWorkshop Freudenstadt 2005, Frank Fleischer 32
Typical Cox-Voronoi cellsSimulation algorithm
Starting line with initial nucleusWorkshop Freudenstadt 2005, Frank Fleischer 33
Typical Cox-Voronoi cellsSimulation algorithm
Placement of neighboring nucleiWorkshop Freudenstadt 2005, Frank Fleischer 33
Typical Cox-Voronoi cellsSimulation algorithm
Second line with points on itWorkshop Freudenstadt 2005, Frank Fleischer 33
Typical Cox-Voronoi cellsSimulation algorithm
Initial cellWorkshop Freudenstadt 2005, Frank Fleischer 33
Typical Cox-Voronoi cellsSimulation algorithm
Stopping criterionWorkshop Freudenstadt 2005, Frank Fleischer 33
Typical Cox-Voronoi cellsSimulation algorithm
Cutting the initial cellWorkshop Freudenstadt 2005, Frank Fleischer 33
Typical Cox-Voronoi cellsSimulation algorithm
Typical Cox-Voronoi cellWorkshop Freudenstadt 2005, Frank Fleischer 33
Typical Cox-Voronoi cellsScaling property
Typical cell can be analyzed regarding geometriccharacteristics (e.g. area, perimeter, number ofvertices, shape,..)
Model parametersγ (intensity of line tessellation)λC (intensity of components on the lines)λ = γλC (intensity of the point process with respectto unit area)
Scaling propertySame structure but on a different scaleκ = γ/λC important parameter1/γ good measure for scale
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Typical Cox-Voronoi cellsScaling property
Scaling property, different intensities but same κ
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Typical Cox-Voronoi cellsResults
Estimations for first order and second order momentsof geometric characteristics for any given pair (γ∗, λ∗
C)
Information about distribution of geometriccharacteristics (=> risk analysis)
Similarities to typical cell of Poisson-Voronoitessellation, especially for large κ
Useful for simulation of network characteristicsMean shortest path lengthsMean subscriber line lengths
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References
C. Gloaguen, F. Fleischer, H. Schmidt and V. SchmidtFitting of stochastic telecommunication network models viadistance measures and Monte-Carlo tests.Preprint (submitted to Telecommunication Systems),2004.
C. Gloaguen, F. Fleischer, H. Schmidt and V. SchmidtSimulation of typical Cox-Voronoi cells, with a special regard toimplementation tests.Preprint (submitted to Mathematical Methods ofOperations Research), 2005.
See also www.geostoch.de.
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