service zones- stations- links.
|
|
| One-Level Access Network:
service zones- stations- links. |
|---|
Define the elements of the simplest possible access network as a set of stations (a countable set of points in the plane) connected by certain links (segments). If two users wish to establish a connection via the network, they connect themselves to the stations serving their geographical locations. Thus each station has its service zone (subset of the plane), and the set of all such zones covers the whole service area.
The model is based on the following assumptions:
Two stations are linked if they have neighboring zones. The set of links thus defined is called Delaunay graph. This graph is dual to the Voronoi tessellation.
Stations form a random pattern on the plane which has the distribution of a Poisson point process. The term Poisson random field would also be appropriate since there is no notion of time so far.
Thus the triple Poisson point process - Voronoi tessellation - Delaunay graph provides a basic network model. If a homogeneous point process is chosen, then the only parameter is the intensity of the point process.
|
| Multi-Level Hierarchical Network |
|---|
A hierarchical network with multiple types of stations can be represented
by a combination of basic cell models. For example, each station of
level 1 serves users in its zone and is linked to the closest
station of level 2. Stations of level 2 communicate via
Delaunay-type links.
- Cost Analysis. For example, given the intensities of two point processes, what is the average number of the stations of level 1 connected directly to a station of level 2?
- Parametric optimization of architecture. For example, in a three-level system, how many stations of level 2 should be deployed at the second level to minimize the total cost of the infrastructure?
- Optimization of architecture in the space of intensity measures. For example, given the intensity measure representing subscriber density, what is the intensity measure of stations minimizing the average cost of connection?
- Analysis of routing algorithms yielding short paths on the graph of links. For example: what is the distribution of the number of hops (or of links) of the Delaunay graph which are required for connecting two stations located at the distance bR from each other.
- Other problems were considered lately in collaboration with ENST (Paris) in relation with multicast trees; the following questions can be addressed within this setting: what is the number of levels and what are the degrees of the nodes of the multicast trees which minimize network resource consumption?
Toward inhomogeneous stochastic architecture The homogeneous-Poisson location of the network devices reflects various irregularities of a real network architecture. This irregularity is however homogeneous, meaning e.g. that the respective mean densities are constant on the plane. This assumption is often not very realistic. It is enough to give a look at a map of the density of population of a given region to realize that an optimal network that is supposed to reflect the traffic demand, should be inhomogeneous too.
|
| Density of the population
in France in 1990, in habitants/km2 |
|---|
|
|
a simulation of the homogeneous PVT tessellation |
|
|
simulation of a PVT modulated by a
Boolean model; the density of nuclei in 20 circular, randomly chosen regions is 10 times bigger than in the remaining part |
Modeling of inhomogeneity is not an easy task and more adequate, non-homogeneous models rapidly become too difficult to analyze. A possible attitude to take if we want to improve upon this situation is to find a general framework, in which available results concerning homogeneous cases could be used as approximations in inhomogeneous cases.
In to order preserve the advantages of the methodology one needs to propose simple parametric models of inhomogeneous Poisson point processes. Moreover, for these models the respective mean functionals should be, at least approximately, expressed in explicit way in terms of the model parameters.
A possible attitude to take is an approximation technique for the distribution of characteristics of typical cell of the Voronoi tessellation (VT) generated by some class of modulated-Poisson point processes. The idea is to approximate the unknown distribution in the non-homogeneous case by a mixture of the known distributions for homogeneous Poisson cases. This approach makes possible the analysis of a wide class of inhomogeneous Poisson-Voronoi tessellations (PVT)'s by means of the formulae and estimates already established for homogeneous cases.
Specifically, one considers the Poisson point process whose intensity takes different constant values on sets of some finite partition of the space. Note that the cell of the VT about a given point is fully shaped by the neighbors of that point in the system of generating points. Thus, provided the partition of the space is not very ``fine'' with respect to the intensities of the points, the resulting modulated-Poisson Voronoi tessellation (mPVT) is ``locally homogeneous'' PVT. Consequently, the ``typical cell of a given partitioning set'' is highly probably identical to the typical cell of the homogeneous scenario and a ``randomly chosen cell from the whole mPVT'' should have a distribution close to the mixture of the homogeneous cases. The error of such approximation comes from existence of cells whose fundamental domain cross the boundaries between the partitioning sets. This analysis applies also to the Poisson process modulated by an independent stationary random partition, in which case the error of the approximation of the double-stochastic Poisson VT depends on the so-called covariance function of the modulating partition.