Applications to Traffic Analysis in Cellular Networks

 Assume that C₀ is a random set which has the same distribution as the typical cell of the Poisson-Voronoi tessellation $C=\{C_1,C_2,\ldots\}$.Using (3.8) and (3.9) we can construct this cell in the following way. It suffices to add a point at the origin to the underlying Poisson process Y. Then, we can identify C₀ with that cell of the resulting Voronoi tessellation whose nucleus lies at the origin. Now, besides Y, consider a further homogeneous Poisson process $X=\{X_1,X_2,\ldots\}$ with intensity $\lambda_X$ which is independent of Y and which can describe the locations of subscribers. Many interesting characteristics, e.g. the number N of subscribers being in the typical cell C₀, have the form $\left( \Phi_f = \right) \;\; \Phi(X,Y,f) \, = \, \sum_n \, f(X_n) \, \bfind (X_n \in C_0),$where $f \, : \, \RL^2 \to \RL_+$ is a given nonnegative function. In order to obtain N we simply put $f(x) \equiv 1$.Another example of such a characteristic is the sum L of the distances between all subscribers in C₀ and the nucleus of this cell (being at the origin). In this case we take f(x) = |x|. Note that $\Exp \Phi_f \, = \, \lambda_X \, \int \, f(x) \, \exp \left( -\lambda_Y \pi \vert x\vert^2\right) \; dx$.In the proof of this formula, we use the fact that a point x belongs to C₀ if and only if there is no point of Y in the circle with radius |x| and center at x; see Foss and Zuyev (1996). In particular,

$$\begin{array} {llllll} \Exp N & = & \frac{\D \lambda_X}{\D \... ...} + 0.147 \, \frac{\D \lambda_X^2}{\D \lambda_Y^3}.\end{array}$$

Furthermore it is shown in Foss and Zuyev (1996) that

and that similar inequalities hold for the tail function of L as well; see also Baccelli et al. (1996). Analogous formulas have been derived in Baccelli and Zuyev (1997) for a spatial road traffic model, where the random number of mobiles crossing the boundary of the typical cell C₀ in a fixed (small) period of time is considered. Some details of this model will be discussed in Frey and Schmidt (1997).

An extended point process model for communication networks with more than two levels of hierarchy has been studied in Baccelli et al. (1996); see also Baccelli and Zuyev (1998). Assume that there are k+1 different levels of hierarchy, where the subscribers are called 0-level stations, the stations directly connected to 0-level stations are called 1-level stations, and so on (see Figure 13).

 

Figure 13:   Communication network with three levels of hierarchy

The locations of the stations of level i are represented by a realization of a homogeneous Poisson process $X^{(i)} = \{X_n^{(i)}\}$ with intensity $\lambda_i$. Assume that the Poisson processes $X^{(0)}, X^{(1)}, \ldots,X^{(k)}$ are independent and $\lambda_0 \gt \lambda_1 \gt \ldots \gt \lambda_k$. Furthermore, assume that except for stations of level k, stations with the same level have no direct connection between them. As in the model with two hierarchy levels which we considered before, the stations of level i ($i=0,1,\ldots,k-1$) are connected to their closest station of level i+1. Thus, for each level $i=1,2,\ldots,k$,we consider the Poisson-Voronoi tessellation induced by X⁽ⁱ⁾ and the stations of level i-1 contained in the cell with nucleus X_n⁽ⁱ⁾ are directly connected to the latter.

This model has been used in Baccelli et al. (1996) to investigate the demand for service in multi-level hierarchical communication systems. In connection with this the probability H_ij(r,h) is considered to have a communication of a height h between two fixed stations of level i and j respectively which are located in distance r from each other. Here the height h is defined as the minimal level $h \geq \, i \vee j$ such that two stations belong to the same Voronoi cell induced by X^(h). If there is no such h, then h is set equal to the highest level, i.e,. h=k. In order to determine the probabilities H_ij(r,h), the tail function Q(r) of the linear contact distribution function of a normalized Poisson-Voronoi tessellation can be used. That is, let Q(r) be the probability that two fixed points in the plane, distant by r, belong to the same Voronoi cell induced by a homogeneous Poisson process with intensity 1. Then $Q(r) = 2 r^2 \int_0^{\pi} \int_0^{\infty} \, s \, \exp(-r^2 f(s,t)) \, ds \, dt$,where

$$f(s,t) = s \sin t + s^2 (\pi -t) + (s^2 +1 - 2 s \cos t) \le... ...arccos \frac{1-s \cos t}{\sqrt{ s^2 + 1 - 2 s \cos t}} \right);$$

see e.g. Baccelli et al. (1996), Meijering (1953), Muche and Stoyan (1992). On the other hand, H_ij(r,h) can be expressed by the tail function Q(r):

$$H_{ij}(r,h) = \left\{ \begin{array} {ll} \exp (-\lambda_{i\v... ...\lambda_m}))}\\ & \mbox{if } h=k, \, i \neq j\end{array}\right.$$

and analogously

$$H_{ii}(r,h)= \left\{ \begin{array} {ll} Q(r \sqrt{\lambda_h}... ...} (1-Q(r \sqrt{\lambda_m})) & \mbox{if } h=k.\end{array}\right.$$

The probabilities H_ij(r,h) can be used to determine the traffic matrix (n_ij(h), $i,j=0,1,\ldots,k,\, h \geq i\vee j)$ where n_ij(h) denotes the expected number of communications of height h per time unit between a fixed station of level i to stations of level j. Namely, $n_{ij}(h) = 2 \pi \lambda_j \, \int_0^{\infty} \, r f_{ij}(r) H_{ij}(r,h)\;dr$,where f_ij(r) is the expected number of communications per time unit which a station of level i asks for with a station of level j.