SINR coverage process

Example 2: Wireless communication SINR based coverage process - physical model

The model developed for mobile and wireless communications includes mobile user trajectories, on-off switching, hand-over policies, etc. The principle remains the same: all elements are represented by stochastic processes with few parameters.

The first models used a simplified representation of cells based on Voronoi tessellations. Such models are sometimes referred to as protocol models (see, e.g., articles by P.Kumar below ). Our main research direction on the matter now bears on so called physical models , where cells are defined by the Signal to Interference (and Noise) Ratio (SIR) (SINR) principle. This type of cells plays an important role e.g in the Code Division Multiply Access (CDMA). In this technology SINR is a key function in that

It is used for the definition of cell ( handoff zones ),

It determines whether uplink communication (resp. coverage/outage of a user) is possible in a given cell,

It is a key parameter in the calculation of capacities: number of users and their bit rates (via Shannon capacity).

CDMA handoff zones - a typical pattern CDMA handoff zones under strong attenuation

Let us illustrate how one can model CDMA handoff zones. The model is based on the following assumptions:

Irregular patterns of locations of Base Stations (BS's) are represented by spatial Poisson point process;
The total emitted power, the fraction of total power devoted to pilot signal, the shadow fading, the SINR threshold and other characteristics of each BS are represented by random marks;
External noise is represented by an independent spatial process;
Path loss is modeled by a continuous attenuation function of the distance;
The handoff zone of a given BS is defined as the region of the space where the ratio of the pilot power received from this BS to the cumulative power received from all other BS (possibly multiplied by an interference / orthogonality coefficient ) exceeds some threshold. The cumulative effect is the shot noise process associated with the Poisson point process and response function given by the attenuation.

Level sets for the Shannon capacity; in contrast with a Boolean model, two adjacent antennas fight each other for space.

What kind of problems can be addressed by such a model?

- The coverage probability; i.e., the probability that a typical antenna contains in its handoff zone a mobile located at a given distance.

- The mean handoff level of typical location; i.e., expected number of antenna containing this location in its handoff zone.

- Qualitative results on the shape of the cells in function of such parameters as the strength of the attenuation or the interference coefficient;

- (Exact) simulation of the contact disttributions of the model gives other network QoS, as e.g. the proportion of the plane where the handoff level is at least k, or the probability that a mobile moving along a random line remains with such a handoff level for more than time t;

- Parametric optimization issues, such as maximization of coverage under cost constraints.

Linear and spherical contact d.f. are available by simulation	Mean handoff as a functions of mean antenna power under budget constraints. Each maximum gives the optimal solution to the densification vs magnification trade-off problem

Percolation in the SINR based coverage processes

An interesting question is the impact of interferences on the percolation of the SINR based coverage process. By the percolation we mean existence of the infinite cluster of connected zones.

Percolation domain in the SINR coverage process.

One finds that if the interference coefficient is non zero but small enough, there exist spatial densities of antennas for which the percolation exists. But in contrast to the Boolean model, an increase of this density can spoil the percolation.

For a given density of antennas there is a critical value of the interference factor above which the network is made of disconnected clusters of nodes.

The above results have impact on connectivity of large-scale ad-hoc wireless networks, where distant nodes communicate in multiple hops.

Bibliography

: Baccelli, F. and Blaszczyszyn, B. (2001). On a coverage process ranging from the boolean model to the poisson voronoi tessellation, with applications to wireless communications. Adv. Appl. Prob. SGSA 33, 293-323.
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: Baccelli, F., Blaszczyszyn, B. and Tournois, F. (2002). Spatial averages of coverage characteristics in large CDMA networks. Wireless Networks 8, 569-586.
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: Dousse, O., Baccelli, F. and Thiran, P. (2003). Impact of interferences on the connectivity of ad hoc networks. In Proc. of IEEE INFOCOM. San Francisco. to appear in IEEE Transactions on Networking.
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