**2.2. Problem definition**

Let us consider the problem where a set of *c* channels must be assigned to *n* arbitrary cells (in this work we consider only the *fixed* CAP, where the channels are permanently assigned to each cell; the reader interested in *dynamic* and *hybrid* schemes can see (Gibson, 1996; Hale, 1980; Katzela 1996).

In our problem formulation we assume that the total number of available channels is given −it can be determined by either the available radio spectrum or the lower bound estimated by a graph-theoretic method (Mandal, 2004; Smith, 2000). Without loss of generality, channels can be assumed to be evenly spaced in the radio frequency spectrum. Thus, using an appropriate mapping, channels can be represented by consecutive positive integers. Therefore, the interference constraints are modelled by an *n×n compatibility matrix* **C**, whose diagonal elements *cii* represent the co-site constraint, i.e., the number of frequency bands by which channels assigned to cell *i* must be separated. The non-diagonal elements *cij* represent the number of frequency bands by which channels assigned to cells *i* and *j* must differ. When this compatibility matrix is binary, the constraints can be expressed more simply: if the same channel cannot be reused by cells *i* and *j*, then *cij*=1, and, otherwise, *cij*=0.

The traffic demand is modelled by means of an *n*-length demand vector **d**=[*d1,d2,...,dn*]T, whose elements represent the number of channels required in each of the cells. For instance, Fig. 1 shows the four demand vectors that will be used in the simulations' section.

The assignment to be generated is denoted by an *n*×*c* binary matrix **A**, whose element *aij* is 1 if channel *j* is assigned to cell *i*, and 0 otherwise. This implies that the total number of 1's in row *i* of matrix **A** must be *di* (see Fig. 2).

**Figure 2.** Structure of the allocation matrix **A**.

50 Simulated Annealing – Single and Multiple Objective Problems

**2. The Channel Assignment Problem (CAP)** 

specified minimum distance (Funabiki, 2000).

**2.1. Interference constraints** 

must satisfy the following constraints:

minimum spectral distance.

**2.2. Problem definition** 

1980; Katzela 1996).

and GAs: advantages and drawbacks of each of them are here explained. For the sake of comparison, a set of well-known problem instances was selected since they have been used in most of the papers related to this problem, thus allowing a direct comparison.

As mentioned in the Introduction, frequency reuse is a key issue in current mobile communication systems. It is well known that the co-channel interference caused by frequency reuse is the most restraining factor on the overall system capacity in wireless networks. Therefore, the main purpose is the simultaneous use of a given radio spectrum while maintaining a tolerable level of interferences. Specifically, each system cell is assigned a set of channels according to the expected traffic demand. This assignment of channels

• **Co-site constraint (CSC)**: channels assigned to the same cell must be separated by some

• **Co-channel constraint (CCC)**: the same channel cannot be simultaneously assigned to certain pairs of cells. The *co-channel reuse distance* is the minimum distance at which the

• **Adjacent channel constraint (ACC)**: any pair of channels in different cells must have a

The channel assignment algorithm must also take into account the specified traffic profile (number of channels) required in each cell. These non-uniform cell demand requirements imply that those cells with a higher traffic demand will need the assignment of more channels.

Let us consider the problem where a set of *c* channels must be assigned to *n* arbitrary cells (in this work we consider only the *fixed* CAP, where the channels are permanently assigned to each cell; the reader interested in *dynamic* and *hybrid* schemes can see (Gibson, 1996; Hale,

In our problem formulation we assume that the total number of available channels is given −it can be determined by either the available radio spectrum or the lower bound estimated by a graph-theoretic method (Mandal, 2004; Smith, 2000). Without loss of generality, channels can be assumed to be evenly spaced in the radio frequency spectrum. Thus, using an appropriate mapping, channels can be represented by consecutive positive integers. Therefore, the interference constraints are modelled by an *n×n compatibility matrix* **C**, whose diagonal elements *cii* represent the co-site constraint, i.e., the number of frequency bands by which channels assigned to cell *i* must be separated. The non-diagonal elements *cij* represent the number of frequency bands by which channels assigned to cells *i* and *j* must differ. When this compatibility matrix is binary, the constraints can be expressed more simply: if

the same channel cannot be reused by cells *i* and *j*, then *cij*=1, and, otherwise, *cij*=0.

same channel can be reused with acceptable interference (Katzela 1996).

The cost due to the violation of interference constraints can be written as

$$J\_1 = J\_{\rm CSC} + J\_{\rm ACC} \tag{1}$$

where *JCSC* and *JACC* represent, respectively, the costs due to the violations of the co-site and the adjacent channel constraints. The first one can be written as

$$J\_{\rm CSC} = \mathcal{A}\_{\rm CSC} \sum\_{x}^{n} \sum\_{i, i \neq j}^{n\_x} \sum\_{j}^{n\_y} \Phi(f\_i^{x}, f\_j^{x}) \tag{2}$$

where parameter *CSC* λ weighs the relative importance of CSC and (,) *x x i j* <sup>Φ</sup> *<sup>f</sup> <sup>f</sup>* is a measure of the co-site constraint satisfaction. This parameter equals 0 only if the difference between channels *i* and *j* of cell *x* is| | *x x i j xx <sup>f</sup>* − ≥ *f c* , and 1 otherwise. *<sup>f</sup>* β α represents the assigned frequency for the αth channel of cell β, and *nα* is the number of channels in the αth cell.

On the other hand, the cost due to the adjacent channel constraint violation is obtained as

$$J\_{\rm ACC} = \mathcal{A}\_{\rm ACC} \sum\_{x \bullet y}^{n} \sum\_{y}^{n} \sum\_{i}^{n\_y} \sum\_{j}^{n\_y} \Psi(f\_i^{x}, f\_j^{y}) \tag{3}$$

where

$$\Psi(f\_i^{x}, f\_j^{y}) = \begin{cases} 0 & \text{if } \quad \mathbb{I} \ f\_i^{x} - f\_j^{y} \text{ } \mathbb{I} \ x\_{xy} \\ 1 & \text{otherwise} \end{cases} \tag{4}$$

Simulated Quenching Algorithm for Frequency Planning in Cellular Systems 53

search is then performed by determining the neighbours *s*' of each solution *s*. Thus, a neighbour structure *N(s)* that defines a set of possible transitions that can be proposed by *s*

When performing local search, in each iteration of the algorithm, a neighbour *s*' of *s* is proposed randomly, and *s* will only be replaced by *s*' if cost does not increase, i.e., *J*(*s*')≤*J*(*s*). Obviously, this procedure terminates in a local minimum that may have a higher cost than the global optimal solution. To avoid this trapping in a suboptimal solution, our proposed SQ method occasionally allows "uphill moves" to solutions of higher cost using the socalled Metropolis criterion (Metropolis, 1953). This criterion states that, if *s* and *s*'∈*N(s)* are the two configurations to choose from, then the algorithm continues with configuration *s*' with a probability given by min{l,exp(-(*J*(*s*')-*J*(*s*))/*t*)}, with *t* being a positive parameter that gradually decreases to zero during the algorithm. Note that the acceptance probability decreases for increasing values of *J*(*s*')-*J*(*s*) and for decreasing values of *t*, and that cost-

**Figure 3.** SQ allows uphill moves up to a cost proportional to the instantaneous temperature *t*.

Mathematically, SA-SQ can be modelled as an inhomogeneous Markov process, consisting of a sequence of homogenous chains at each temperature level *t* (Duque, 1993). Under this framework, it has been shown (Aarts, 1989; Geman, 1984) that there exist two alternatives for the convergence of the algorithm to the globally minimal configurations. On the one hand (homogenous case), asymptotic convergence to a global minimum is guaranteed if *t* is lowered to 0, and if the homogenous chains are extended to infinite length to establish the stationary distribution on each level. On the other hand (inhomogeneous case), convergence is guaranteed, irrespective of the length of the homogenous chains, if *t* approaches 0

The problem arising here is that just the enumeration of the configuration space has an exponential time complexity and, in practice, some approximation is required. The formal

decreasing transitions are always accepted (see Fig. 3).

procedure is to choose a *cooling schedule* to decide for:

• the start condition (initial temperature, *t0*). • the rule for decreasing the temperature.

• the stop condition (final temperature, *tF*).

has to be defined.

logarithmically slow.

• the equilibrium condition.

Parameter λACC in Eq. (3) is set to weigh the relative importance of the adjacent channel constraint. Finally, the cost due to the violation of the traffic demand requirements is modelled as

$$J\_{TRAFF} = \mathcal{A}\_{TRAFF} \sum\_{i}^{n} \left( d\_i - \sum\_{j} a\_{ij} \right)^2 \tag{5}$$

Gathering all the costs, the final cost function to be minimized is

$$J = J\_{\rm CSC} + J\_{\rm ACC} + J\_{\rm TRAFF} \tag{6}$$

If the traffic demand requirements are incorporated implicitly by only considering those assignments that satisfy them, then the cost function can be expressed by *J=J1=JCSC+JACC*, subject to , *ij <sup>i</sup> <sup>j</sup> <sup>a</sup>* = ∀ *d i* . For that reason, the fitness function to be used in the algorithms is given by *ρ*=1/*J*.

Finally, the estimation of parameters λCSC and λACC has been achieved using the same inhomogeneous 25-cell network used by Kunz and Lai in (Kunz, 1991) and (Lai, 1996), respectively. After analyzing the number of iterations required for a proper convergence for different values of λCSC and λACC, the optimal values for the weights λCSC and λACC were found to be close to 1 and 1.3, respectively.

It is important to note that the most important difference between different pairs of λCSC and λACC is the required computational load for each of them, since the number of generations required to converge proportionally acts on the execution time. This way, a precise computation of both λCSC and λACC is indispensable to get an efficient allocation algorithm.
