For example, interference is higher if users are close to the cell border; also, traffic in neighboring cells adds to interference. Capacity planning thus has to consider the number, properties and positions of users in a cell and also in the neighboring cells. The power levels resulting from the power control mechanism need to be calculated in order to determine if there is sufficient capacity for serving a given amount of traffic.
Figure 3: Soft capacity: even if the cell's resources suffice to serve a set of users a if all are located at the cell center, capacity may be exceeded if b the same users are located at the cell border, or c if there is too much interference from users in neighboring cells. Instead of considering the dynamic influences such as short-term channel variation and user mobility, snapshots of users' demand in time are evaluated.
A schematic view of a user snapshot served by a radio network is depicted in Figure 1a. Two crucial assumptions for static modeling are: first, the average interference can be considered constant over short time spans; second, power control adjusts the signal quality at the receiver to the minimum feasible value that ensures proper decoding at all times. The bottom graph in Figure 2 is thus approximated by a straight horizontal line.
This is called perfect power control. The assumption of constant average signal quality is captured by the carrier-to-interference equation illustrated in Figure 4. Note: We state the CIR equation for the uplink only, as it is simpler than that for the downlink. Moreover, some technicalities are omitted for the sake of a smoother presentation. A full account of the model can, for example, be found in  An equation is formulated for each radio link.
The fraction at the left-hand side is the link's carrier-to-interference ratio CIR. The numerator contains the received signal power, which is the product of the transmit power and the attenuation factor.
The denominator is the total strength of all other interfering signals and noise. Again, the transmitted powers are multiplied by the respective attenuation factors. Figure 4: Static model: a system of CIR equations is used to calculate transmit powers and check capacity. The identification of sensible coefficients requires the collaboration with different fields in the engineering sciences.
In addition, an experimentally determined activity factor is introduced, indicating the fraction of the time a link is actually active. As an example, a link carrying voice telephony is typically active for 67 percent of the time in one direction including protocol overhead. An average interference is thus considered. Because perfect power control is assumed, the CIR is equal to the connection-specific constant on the right hand side of the equation, the CIR target. The set of all CIR equalities constitutes a linear equation system, whose solutions represent the equilibrium of transmit powers toward which power control drifts.
In order to assess whether the network provides sufficient capacity to serve a given snapshot, we have to determine whether the CIR equation system admits a solution. Furthermore, the solution has to be positive in all components and within the cell power limits. The concept of systems of CIR equations is simple to grasp, but using it in practice poses some challenges. For supplying the model with appropriate coefficient values, mathematicians rely upon the results of different disciplines in the engineering sciences.
The number of users, their services and their positions in a snapshot is determined by traffic modeling. Traffic modeling usually identifies a stochastic distribution of usage intensities in the busy hour the hour of the day during which the network is maximally loaded ; an example traffic intensity function is depicted in Figure 5a. The user-specific parameters activity factor and CIR target are selected according to service and link models.
Live measurements or detailed link-level simulations determine the CIR targets for the different services. Figure 5: Elements of planning data: a average normalized traffic intensity in Berlin, b received signal strength and cell areas. The attenuation factors between antennas and users are predicted by radio wave propagation models . This is difficult in urban areas, because signals are reflected and diffracted by several obstacles and reach the receiver on multiple paths.
Radio wave propagation prediction uses terrain height data and, ideally, building data. Also the influence of the antenna needs to be considered. Figure 5b illustrates a use of propagation data; for the indicated network configuration, the strongest received signal at all points in the planning area and the resulting cell structure are indicated. There are two computational difficulties with using the static model for network planning. First, planning for a single snapshot is pointless. Many snapshots should be considered. Second, solving the CIR equation system is numerically difficult as the quantities involved often span a range of orders of magnitude.
There is a mathematical model of the entire system that pinpoints the interference coupling among cells in a strikingly simple fashion. Note: We focus on the uplink here.
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The same ideas apply to the downlink. The model was first conceived as a technique for reducing the dimension of the CIR equation system from the number of users typically several thousands to the number of cells typically a few hundred . To this end, cell power variables summing up in the uplink all average received powers at the base station antennas are introduced.
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By eliminating the link power variables from the system, we obtain an equation system describing the relation of the different cell powers:. The vector of average cell powers is here denoted by p. The coefficients of the cell power variables after the transformation form a nonnegative square matrix, the interference-coupling matrix, denoted by C. The interference-coupling matrix is constructed for a specific user configuration and network design.
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See [10, 3] for details. For the user configuration and network design depicted in Figure 1a, the line color intensity in Figure 1b indicates the magnitude of the corresponding off-diagonal elements representing the cell coupling.
Originally, the dimension reduction technique provided merely a computational speedup for solving the CIR equation system in a two-step approach. First, the cell powers are computed as the solution of the coupling equation system. This is easy because the system isand well-conditioned. Second, the individual link powers are derived. Beyond the alleviation of computational problems, however, the coupling equation system enables a new view on the system at cell level.
After all, individual link powers in a snapshot are of little interest for network planning. The cell powers are essential for capacity evaluation, and they can be computed without explicitly considering individual users. The coupling equation system thus allows one to understand the properties of a network through the coupling matrix alone.
We next outline two approaches to UMTS network modeling and performance evaluation that rely on this principle. The expected network performance for a random model of snapshots is usually more interesting than the network's performance on a single snapshot. The expected network performance can be estimated by solving the coupling equation with the expected coupling matrix.
This corresponds to considering an "average snapshot" ; for typical random models, the corresponding calculations are straightforward. It can be shown empirically  that the expected coupling matrix is a good representative for the network design and its performance. International Electrotechnical Commission. Justis Pub. Lund University Libraries M. Nucleus Medical Art, Inc. Bowker Readex Readex, Karger s. Sage Publications Sage Publications, Inc. Wharton School.
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