**5. Numerical results**

This section evaluates the performance of the proposed algorithms in terms of convergence and solution accuracy under different conditions. Radio base stations are considered to be located at cell centers and the traffic is assumed to be inhomogeneous, with each cell having a different and *a priori* known traffic demand. Following the ideas shown in (Lai, 1996), the initial population is constructed using the available *a priori* information, i.e., the algorithm assigns a valid string of frequencies to all the cells following a simple approach: first, the algorithm attempts to assign a set of valid frequencies to as many base stations as possible. In the event that valid frequencies cannot be located to some of the cells, they are then randomly assigned.

Simulated Quenching Algorithm for Frequency Planning in Cellular Systems 61

**Figure 6.** Cellular geometry for the Philadelphia benchmark problem with *n*=21 cells.

In this section, the convergence properties of the proposed methods are studied. Results shown in Table 2 are average values over 25 trials for each problem. The parameters to be set in the GA are: the number of iterations *ng*, the initial mutation and crossover probabilities, the population size *np*, and the parameters of functions *pc(k)* and *pm(k)*. After several trials that helped to fine tune the parameters ensuring that the computation is

• Number of fitness evaluations: 100 (for problem 14), 25,000 (problems 5−8 and 11), 50,000 (problems 12, 15 and 16), 75,000 (problem 13), 100,000 (problems 1−4), 150,000 (problem 10) and 300,000 (problem 9). If the values corresponding to problems 10, 12, 14 and 16 are compared to those shown in (Ngo, 1998) it can be seen that these values mean a reduction of 75% (in problems 15, 12 and 16) and 62.5% (in problem 10) with respect to the number of iterations needed in (Ngo, 1998); in problem 14 both approaches require 100 iterations. Notice that, since not every offspring needs to be evaluated in each generation, the number of fitness evaluations is a more representative

• Initial crossover probability, *pc(0)*: this parameter is set to 0.35 in problems 5−8 and

• Population size, *np*: 10 individuals, except in problems 1−4, 9 and 10, which required 20. • Simulations show that λCSC=1 and λACC=1.3 lead to faster convergence as compared to λACC=1. This result is in accordance to (Lai, 1996), where λACC-optimal=1.1/2 was

On the other hand, the SQ algorithm has been implemented with a mixture of standard and modified *flip-flops* (described in section 4.2). Problems 5−8, 11−12, 14−16 are solved with a configuration of 50-70% of modified flip-flops, while problem instances 1−4, 9, 10 and 13 used 20-40% of modified flip-flops. The remaining cases, in all problem instances, were implemented with standard flip-flops. To explain this experimental adjustment, just notice that the more complex is the problem instance, the more explorative must be the global

Comparative results are shown in Table 2. The performance is measured using the percentage of convergence to the solutions, defined as the ratio of the total number of

**5.2. Adjustment of parameters and convergence performance** 

parameter of the performance than the number of generations.

• Initial mutation probability, *pm(0)*: 0.04 for all the problems.

obtained.

11−16, while instances 1−4, 9 and 10 showed better results with 0.25.

search for solutions in order to avoid convergence to suboptimal local minima.

manageable, the optimal values were found to be:
