Hendrik Schmidt France Telecom NSM/RD/RESA/NET hendrik.schmidt@orange-ftgroup

1. Comparison of Network Trees in Deterministic and Random Settings using Different Connection Rules Hendrik Schmidt France Telecom NSM/RD/RESA/NEThendrik.schmidt@orange-ftgroup.com SpasWin07, Limassol, Cyprus 16 April 2007

2. Overview 1 • Introduction and motivation • Geometric support: Models and their fitting • Comparison of network trees • Infrastructure and costs • Outlook and conclusion 2 3 4 5

3. 1 Introduction and motivation

4. Introduction Study areas in Paris Real data A single study area

5. Introduction Network devices in the plane, Euclidean distance connection Serving zone (two levels of network devices), connection along infrastructure • Place lower level devices (LLDs) in a serving zone • Each LLD is connected to the corresponding higher level device (HLD) • Length distribution LLD → HLD influences costs and technical possibilities LLD HLD Distribution of distances LLD → HLD HLD LLD

6. IntroductionStochastic Subscriber Line Model • Geometric considerations are essential: The access network … • … runs along the infrastructure • … contributes mainly to total network costs • Telecom providers are confronted with new challenges • Network analysis of competing providers / in different countries • New technologies / data • Need for simple and global modeling tools • Fast comparison of scenarios • Fast technical and cost evaluations • Minimal number of parameters, maximal information about reality • One solution: Stochastic-geometric modeling • Disregard too detailed information for the sake of clarity • Study random objects and their distribution • Take into account the spatial geometric structure of networks

7. IntroductionSSLM: Main roads Main roads Cells: Subscribers are situated there

8. IntroductionSSLM: Main roads and side streets Main roads Two level hierarchy of side streets

9. IntroductionSSLM: Infrastructure, subscriber, serving zones A serving zone HLD

10. R SR CP CS IntroductionSSLM: Summary • The SSLM consists of 3 parts • 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 Topology of connections Network equipment (nodes, devices) Geometric Support (infrastructure)

11. 2 Geometric support: Models and their fitting

12. Non-iterated tessellations PLT PVT PDT • Stationary non-iterated Poisson tessellations • Characterized by one parameter, called intensity g (measured per unit area) • PLT (Poisson Line Tessellation): g … mean total length of edges • PVT (Poisson Voronoï Tessellation): g … mean total number of cells • PDT (Poisson Delaunay Tessellation): g … mean total number of vertices

13. Non-iterated tessellationsMean value relationships • Consider facet characteristics • They can be expressed in terms of the intensity g

14. Nesting of tessellations PLT/PVT PLT/PDT PLT/PLT g0= 0.02 g1= 0.0004 g0= 0.02 g1= 0.0001388 g0= 0.02 g1= 0.04 The mean total length of edges is always

15. Nesting of tessellationsGeneralizations • Bernoulli thinning: Nesting in cell with probability p • Multi-type nesting: Different nestings in different cells PLT multi-type nesting PLT / PVT with Bernoulli thinning

16. Nestings of tessellations • Mean value relationships X0 / pX1 with and hence • Immediate application to PVT/(PLT, PVT, PDT), PDT/(PLT, PVT, PDT) and PLT/(PLT, PVT, PDT)

17. Model fitting Preprocessed data Raw data

18. Model fitting Realisation of the optimal tessellation: PLT g0 /PLT g1 Preprocessed data • Estimation of characteristics • Choice of a distance function • Class of tessellation models • Minimization of distance function

19. Model fittingUnbiased Estimation

20. Model fittingNumerical Minimisation • Solution of minimization problem • analytically for non-iterated models • numerical methods for nested models, e.g. Nelder-Mead algorithm • fast • easy to implement • minimum depends on initial point → random variation • Example: Simulated PLT/PLT model ( )

21. Model fittingExample • Fitting strategy: Exploit hierarchical data structure • Side streets • Main roads • Monte Carlo test • Null hypothesis H0 : The optimal model is PLT g0= 0.02384 / PLT g1= 0.013906 • Decision: H0is not rejected

22. 3 Comparison of network trees

23. Comparison of network trees Geometric support • Two levels of network devices: • Lower level devices (LLD) • Higher level devices (HLD) • Two connection rules: • Euclidean distance • Connection along geometric support

24. LLD and HLD in the plane Connection according to Euclidean distance Comparison of network trees LLD and HLD on the roads Connection along infrastructure LLD and HLD on optimal geometric support Connection along infrastructure Note: Run time of simulations is very long! Distribution of distances LLD → HLD

25. LLD and HLD on other geometric support Connection along infrastructure Comparison of network treesExample 1: Influence of fitting procedure LLD and HLD on optimal geometric support Connection along infrastructure

26. Comparison of network treesExample 1: Different models – different distributions 50 km Comparisons: Different … … intensities … geometric supports 20 km

27. Comparison of network trees LLD and HLD in the plane Connection according to Euclidean distance LLD and HLD on the roads Connection along infrastructure Note: Run time of simulations is very long! LLD and HLD on optimal geometric support Connection along infrastructure Distribution of distances LLD → HLD

28. LLD and HLD on other geometric support Connection along infrastructure Comparison of network treesExample 2: Influence of fitting procedure LLD and HLD on optimal geometric support Connection along infrastructure

29. Comparison of network trees Example 2: Different models – different distributions 50 km Comparisons: Different … … intensities … geometric supports 20 km

30. Optimal geometric support: Non-iterated model Comparison of network treesExample 2: Non-iterated vs. iterated models LLD and HLD on the roads Connection along infrastructure Optimal geometric support: Iterated model

31. 4 Infrastructure and costs

32. R SR CP CS The model Geometric support (infrastructure) Topology Network equipment (devices) • An example of the SSLM • Geometric support: Stationary PLT • Network devices: 2 layer model of stationary Poisson point processes • Lower level devices (LLD) • Higher level devices (HLD) • Topology of connection • Logical connection: LLD connected to closest HLC • Physical connection: Shortest path along the infrastructure • Questions • What are the mean shortest path costs from LLD to HLD? • Is a parametric description of the distribution possible?

33. Infrastructure and costsGeometric support … • Geometric support: Assume stationary PLT Xlwith intensity g (> 0) Geometric support: PLT

34. Infrastructure and costs… and network devices • Road system: Assume stationary PLT Xlwith intensity g • Higher level devices (HLD) • Stationary point process (independent of Xl) • Poisson process on Xl(Cox process) with linear intensityl1 • Stationar planar point process XH with planar intensity HLD

35. Infrastructure and costs… and network devices • Road system: Assume stationary PLT Xlwith intensity g • Higher level devices (HLD) • Stationary point process (independent of Xl) • Poisson process onXl(Cox process) with linear intensityl1 • Stationar planar point process XH with planar intensity • Lower level devices (LLD) • Stationary point process (indep. ofXland XH) • Poisson process onXl(Cox process) with linear intensityl2 • Stationar planar point process with planar intensity LLD

36. Infrastructure and costsLogical connection • Random placement of HLD along the lines • Each LLD is connected to the closest HLD • Serving zones induce a Cox-Voronoi tessellation (CVT)

37. Infrastructure and costsPhysical connection (1)

38. Infrastructure and costsPhysical connection (2)

39. Infrastructure and costsMean shortest path length (1) • Natural approach • Disadvantages

40. Infrastructure and costsMean shortest path length (2) • Alternative approach • Disadvantages • Simulation not clear • Not very efficient

41. Infrastructure and costsMean shortest path length (3) • Application of Neveu • Independent from l2 • The typical serving zone (the typical cell of a CVT) has to be simulated

42. Infrastructure and costsMean shortest path length (4)

43. Infrastructure and costsMean shortest path length (5) • Estimation of • Note: The integrals can be calculated analytically

44. Infrastructure and costsMean shortest path length (6)

45. Infrastructure and costsMean shortest path length (7)

46. Infrastructure and costsMean shortest path length (8)

47. Infrastructure and costsMean shortest subscriber line length

48. Infrastructure and costsApplication Mean length from LLD to HLD [km] in case of spatial placement

49. 5 Outlook and conclusion

50. Outlook • Analysis of shortest paths • Formulas for other types of geometric support • Not only mean values but (parametric) distributions of cost functions • Typology of infrastructure • Within the cities • Nationwide extension Main roads: Optimal intensity of nested tessellation (within PDT) Deterministic PDT in France (level préfectures and sous-préfectures)