stochastic geometry & access telecommunication networks

1. stochastic geometry & access telecommunication networks Catherine GLOAGUEN – Orange Labs joint work with V. Schmidt and F. Voss – Institute of Stochastics, Ulm University, germany 7 Septembre 2010, Journées MAS, Bordeaux

2. summary partie 1 the complexity of telecommunication networks partie 2 the interest to "think stochastic geometry" partie 3 random models for roads partie 4 typical cell and estimation of shortest path length partie 5network modeling and validation on real data partie 6 conclusion

3. the complexity of telecommunication networks

4. the access telecommunication network • What is a network? A collection of equipements and links that aims to enable the customer to reach any possible service she subscribes. • This is realized by means of a suitable architecture defines how to aggregate links and to organize nodes in order to reduce costs while providing a good quality of service. • The fixed network is very important with new technologies like optical fiber; the existing Copper network remains a major cost point • The access network is the part closest to the customer It is very sensitive to the demography & geography and exhibits two major levels of complexity

5. complexity in cable pathes • the acces network merges in civil engineering • equipements are inside or in front of buidings • cables ly under the pavement or follow the road system • huge number and a variety of equipments Approximate scale 100 m x 200 m

6. complexity of the underlying road system major cities width 12km inner city and suburbs Lyon towns width 9 km Amiens and transition to rural areas nationwide width 950 km motorways, national and some secondary roads The morphology of the road system depends on the scale of analysis since it is designed for various purposes

7. some challenges for the network operator • for cost reduction or global planning purposesin adequacy with the topography and population density. • to analyze large scale networks in a short timefull reconstruction of realistic optimized networks is impossible, partial reconstruction is limited in size. • to use external public data as inputto compensate for too voluminous databasis, that are not always complete nor reliable and often need dedicated software • to address rupture situations in technology and architectureby definition no databasis are available and extrapolation from actual situation may be dubious

8. first positive point even such complex systems as access networks can be described in a global way by simple and logical principles due to the underlying careful building. • they can be decomposed in 2 levels sub-networks connecting L(ow) nodes to H(igh) nodes • a serving zone is associated to each H node with respect to L nodes • the physical connexion L -> H is achieved according to a "shortest path" rule, which meaning depends on the technology

9. second positive point the interest to build a global vision • it is questionable to work on detailed analysis with the aim to deduce for the purpose of detailed reconstructions when possible are sometimes used to estimate global behaviour • allows to simplify the reality only keeping strcturing features • allows to turn the observed variability and complexity as an advantage • considering the network areas as a statistical set of realizations of a random network

10. the interest to "think stochastic geometry"

11. the "translation" of the problem is easy global vision relationship between the process parameters contains all structuring geometrical features In fine instantaneous results spatial variability random spatial processes node location choice of point process avarage number as global parameter stochastic geometry geometrical characteristics estimated via the right functionnals connexion rules geometrical considerations serving zone apply logical connexion rule to process for node

12. the simplest 2 levels network as an example • L and H nodes location as independant Poisson point process in R2 , 2 intensities • logical connexion rule from L the nearest H euclidian distance defines the serving zone a Voronoï cell • the physical connexion follows the straight line • analytical global results for distributions of geometrical features • distances L -> H as Exp (intensity H) • action area characteristics : area, perimeter.. simplest network Fully described by 2 intensities "Géométrie aléatoire et architecture de réseaux", F. Baccelli, M. Klein, M. Lebourges, S. Zuyev, Ann. Téléc. 51 n°3-4, 1996

13. a key object : the typical serving zone Poisson Voronoï tessellation Point process of H nodes probability distribution typical cell Conditioned with a H node in the origin The typical serving zone is representative for all the serving zones that can be observed (ergodicity). Efficient simulation algorithms are derived. Empirical distribution of all cells Distribution of the typical cell perimeter

14. simulation algorithm for PVT typical cell "Spatial stochactic network models" F. Voss Doctoral dissertation, Dec. 2009, Ulm

15. a real network involves the road system • as a support for nodes location • as a support for physical connections following a shortest path principle Road system H node Serving zone L node connection "Comparison of network trees in deterministic and random settings using different connection rules. " Gloaguen C, Schmidt H, Thiedmann R, Lanquetin JP, Schmidt V SpaSWiN, Limassol, 2007

16. stochastic modelling in realistic settings with the following methodology • stochastic models for road systems • typical cell for nodes located on the road systems • dedicated simulation algorithm for typical cells • geometric characteristics are expressed as functionals of the processes and estimated from the content of the typical cell We focus on the estimation of the distribution length of the shortest path connexions as an example

17. Random models for roads

18. simple Poissonian models for road systems Line Throw lines Delaunay Throw points and relate them to their neighbours Voronoï Throw points, draw Voronoi tesselation, erase the points throw points or line in the plane in a random way to generate a "tessellation" that can be used as a road system. More sophisticated models (iterated, aggetagted) are available

19. models are discriminated by mean values A stationary model is fully described by its intensity g "Stationary iterated random tessellations" Maier R, Schmidt V ,Adv Appl Prob (SGSA) 35:337-353, 2003

20. partition of urban area • fitting algorithm to find the "best" model to represent real data • automatized segmentation • morphogeneis of urban street systems --> new stationary models Bordeaux built up area PVT 163 km-2 PVT 52 km-2 PVT 37 km-2 PVT 18 km-2 "Mathematics and morphogenesis of the city" T. Courtat,Workshop Transportation networks in nature and technology, 24 juin 2010 Paris "Fitting of stochastic telecommunication network models, via distance measures and Monte-Carlo tests" Gloaguen C, Fleischer F, Schmidt H, Schmidt V, Telecommun Syst 31:353—377, 2006

21. databasis for road systems in a single Excel sheet

22. why road models ? • a model captures the structurant features of the real data set • a "good" choice takes into account the history that created the observed data (ex PDT roads system between towns) • statistical characteristics of random models only depend on a few parameters • the real location of roads, crossings, parks is not reproduced …but the relevant (for our purpose) geometrical features of the road system are reproduced in a global way. • models allow to proceed with a mathematical analysis • final results take into account all possible realizations of the model • no simulation is required

23. typical cell and estimation of shortest path length

24. the serving zone revisited to incorporate streets PLCVT Poisson-Line-Cox-Voronoï-tessellation • H nodes are randomly located on random tessellations (PVT, PDT, or PLT) and not in the plane • the serving zone has the same formal definition as a Voronoï cell • the serving zones define a Cox-Voronoï tessellation (PLCVT, PDCVT or PVCVT) Road system (PLT) H node Serving zone

25. simulation algorithm for PLCVT typical cell Distance are Exp distributed Nearest points to 0 P1 and P2. Radial simulation of line l2 and P3 and P4 Initial line l1 through 0, orientation angle ~ U[0,2p) Add one point at the origin d0 Further simulated points on l2 and radial simulation of other lines Further simulated points on l2 and radial simulation of other lines Construction of first initial cell and radius =2 max (|Opi|)

26. shortest path on streets Shortest path with PLT model for streets • H nodes are located on a random tessellation (ex PLT) • L nodes are located on the same system independantely from H-nodes • L node belongs to one serving zone and is connected to its nucleus • the connexion is the shortest path on the road system : edge set of the tessellation road system serving zone H node L node Euclidian along the edge set

27. shortest path length C* • the length of the shortest path to its H node is associated to every L as a marked point process • "natural" computation simulate the network in a sequence of increasing sampling windows Wn and compute some function of the length of all paths and average process for H nodes process for H nodes marked process with path length

28. representation of the distribution of length C* • consider the distribution of the path length from a L node conditionned in O • use Neveu exchange formula for marked point processes in the plane applied to XC and XH • write the distribution in terms of a H node conditionned in O • the result • depends on the inside line system • does not depend on L nodes process Length from y to 0 H nodes typical PLCVT cell and its line segment content L*H

29. S2 S1 0 Si density estimation the distribution of length C* simulate the typical cell and the (Palm) line segment system it contains explicit the line segments compute the estimator of the density as a step function simulates exact distributions, no runtime or memory problems, unbiased and consistent estimator, convergence theorems for maximal error, but needs to develop the simulation algorithm for the correponding serving zone

30. available algorithms • indirect simulation algorithms • simulate random cells and weigth it • PVCVT and PDCVT • other processes for nodes location • Cox on iterated tessellations • thinned vertex sets Nodes location on iterated tessellations or as thinned vertex set "Simulation of typical Poisson-Voronoi-Cox-Voronoi cells, F. Fleischer, C. Gloaguen, H. Schmidt, V. Schmidt and F. Voss. " Journal of Statistical Computation and Simulation, 79, pp. 939-957 ,2009

31. scaling invariance for simple tesselations, the statistical properties of functionals of the typical cell only depend on a scale factor k PLCVT cell k = 1 PLCVT cell k = 1000

32. PDCVT PLCVT PVCVT library of fitted formulas for densities k = 50, g = 1 n = 50 000 • empirical densities are computed from n simulations • large range of k values • all available road models

33. selection of parametric families to fit empirical densities • ensuring theoretical convergence to known distributions & limit values • not too many parameters • best if one family for all models • truncated Weibull distribution PDCVT k = 250 k = 750 k = 2000 empirical fitted

34. 2 level subnetwork case is solved Area to be equipped parameters for road model • instantaneous results for 2 level networks • analytical parametric formulas for the repartition function, majoration of the length, averages and moments • explicit dependancy on the morphology of the road system number of H nodes -> k Length distribution (road model, k) bloc de texte

35. network modeling and validation on real data

36. Large scale WCS Middle scale SAI ND SAIs ND real networks A synthetic spatial view of real networks is obtained from the identification of 2-level subnetworks and the partitionning of the area in serving zones for every subnetwork. It maps the architecture on the territory (here on Paris). "Parametric Distance Distributions for Fixed Access Network Analysis and Planning". Gloaguen C, Voss F, Schmidt V, ITC 21, Paris, 2009

37. the family of parametric densities at work large scale subnetwork WCS-SAI the mean area of a typical serving zone = total area /(mean number of WCS); containing an average of 50 km road. k ~1000 = (total length of road /area) x (total length of road / number H nodes)

38. medium scale subnetwork SAI-SAIs or SAI-ND the mean area of a typical serving zone = total area /(mean number of SAI); containing an average of 2 km road. k ~35 = (total length of road /area) x (total length of road / number H nodes)

39. small scale subnetwork SAIs-ND the mean area of a typical serving zone = total area /(mean number of SAI); containing an average of 300 m road. k ~5 = (total length of road /area) x (total length of road / number H nodes)

40. global analysis of a network middle size French town Partitioned in homogeneous road models customer-WCS connexion obtained by convolutions and ponderated average of 2 level subnetworks no computational time : the time investment comes form the mapping of the architecture on the area, i.e. describing the interweaving of 2 level networks. The models and parameters for the road systems (Excel sheet) are determined once and do not vary in time.

41. impact of new technologies on QoS middle size French town optical gain of the end to end connexions for optical network the optical fiber gain depends on the number of nodes and technology obvious application to optical networks. Given the architecture, the technology (coupling devices, optical losses) and the number of nodes, the probability distribution of the optical gain of the end to end connexion is easily deduced.

42. conclusion

43. key points • global analysis of fixed access networks explicitely accounting for regional specificities, without runtime problems • analytical formulas for network geometric characteristics • analytical models for road systems • with potential use in mobility problems • can't be ignored to model cabling systems • open methodology : choice of functionals • mathematical results for convergence, limit theorems, fitting & simulation tools

44. merci