AN ESTIMATION OF DISTRIBUTION ALGORITHM FOR THE
CHANNEL ASSIGNMENT PROBLEM
Jayrani Cheeneebash
Department of Mathematics, University of Mauritius, Reduit, Mauritius
Harry C. S. Rughooputh
Department of Electrical and Electronic Engineering, University of Mauritius, Reduit, Mauritius
Keywords: Channel Assignment problem, Estimation of Distribution Algorithm.
Abstract: The channel assignment problem in cellular radio networks is known to belong to the class of NP-complete
optimisation problems. In this paper we present a new algorithm to solve the Channel Assignment Problem
using Estimation of Distribution Algorithm. The convergence rate of this new method is shown to be very
much faster than other methods such as simulated annealing, neural networks and genetic algorithm.
1 INTRODUCTION
During the recent years, it has been observed that the
communication systems has accentuated the demand
for mobile users. However, since the number of
usable frequencies, which are necessary for the
communication between mobile users and the base
stations of cellular radio networks, is very limited,
an efficient use of the frequency spectrum or
channels is crucial to meet the increasing demands.
Therefore while assigning the frequencies to
different base stations, it is desirable to reuse the
same frequency as much as possible. On the other
hand, it is important to avoid possible interferences
between different mobile users, at the same time, the
number of frequencies assigned to each base station
must be chosen large enough to satisfy the given
demand in the corresponding cell. Regarding the
above requirements, one can formulate the
frequency assignment problem as a discrete
optimisation problem.
The channel assignment problem (CAP) is
defined as an NP-complete problem. The channel
assignment must fulfil the following three
constraints:(Sivaranjan, McElliece and Ketchum,
1986)
1. The Co-Channel Constraint (CCC): The same
channel cannot be assigned to a pair of cells
within a specified distance simultaneously.
2. The Adjacent Channel Constraint (ACC):
Adjacent Channels cannot be assigned to cells
simultaneously.
3. The Co-site Constraint (CSC): The distance by
an nn
×
symmetric matrix called the
compatibility matrix
between any pair of
channels used in the same cell must be larger
than a specified distance.
In 1982, Gamst and Rave defined the general form
of the channel assignment problem in an arbitrary
1982). In their definition, compatibility constraints
in an n-cell network are described
][
ij
cC =
. Each
non-diagonal element
ij
c in C represents the
minimum separation distance in the frequency
domain between a frequency assigned to cell i and a
frequency to cell j. The Co-Channel constraint is
represented by
1
=
ij
constraint is represented by
2=
ij
c and 0
=
ij
c
indicates that cells i and j are allowed to use the
same frequency. Each diagonal element
ii
c in C
represents the minimum separation distance between
any two frequencies assigned to cell i, which is the
co-site constraint, where
1
ii
c is always satisfied.
211
Cheeneebash J. and C. S. Rughooputh H. (2006).
AN ESTIMATION OF DISTRIBUTION ALGORITHM FOR THE CHANNEL ASSIGNMENT PROBLEM.
In Proceedings of the International Conference on Wireless Information Networks and Systems, pages 211-214
c
SciTePress
The channel requirements for each cell in an n-cell
network are described by an n element vector which
is called the demand vector D. Each element
i
d in
D represents the number of frequencies to be
assigned to cell i. When f
ik
indicates the k
th
frequency assigned to cell i, the compatibility
constraints are represented by:
ijjlik
cff for ,,,1, nji =
k = 1,…,d
i
, l = 1,…,d
j
except for i = j, k = l.
Many researchers have investigated the channel
assignment problem using non-iterative algorithms
(Gamst and Rave, 1982), (Sivaranjan, McElliece and
Ketchum1981) and iterative algorithms (Funabiki
and Takefyi, 1992), (Kunz, 1991). Neural Networks
Algorithms and Genetic Algorithms are among the
iterative algorithms in which an energy or cost
function representing frequency separation
constraints and channel demand is formulated and is
then minimised. Unfortunately, the minimisation of
the cost function is quite a difficult problem because
of the danger of getting stuck in local minima
constitutes a major problem. A much more powerful
approach to cope with the problem of local minima
has been considered in (Beckmann and Killat, 1999),
(Fu, Pan and Bourgeois, 2003).
This paper is organised as follows: section 2
gives a brief exposé on Estimation of Distribution
Algorithms and section 3 presents the new method
that we propose in this paper. In section 4 we show
the performance of our algorithms in solving some
benchmark problems and finally we conclude in
section 5.
2 SEARCH AND OPTIMISATION
ALGORITHMS
Genetic Algorithm (GA) depends to a large extent
on associated parameters like operators and
probabilities of crossing and mutation, size of
population, rate of generational reproduction, and
the number of generations. The researcher requires
experience in the resolution and use of those
algorithms in order to choose suitable values for
these parameters. All these reasons have motivated
the creation of a new class of evolutionary
algorithms classified under the name of Estimation
of Distribution Algorithms (EDAs) (Larranaga and
Lozano, 2002), which attempts to ease the prediction
the movements of the population in the search space
as well as to avoid the need of so many parameters.
EDAs are population based search algorithms based
on probabilistic modelling. The new individuals are
sampled starting from a probability distribution
estimated from the database containing only selected
individuals from the previous generation. The
interrelations between the different variables
representing the individuals are expressed explicitly
through the joint probability, associated with the
individuals selected at each iteration.
Given the population of the
l th generation , P
l,
the
N selected individuals,
Se
l
P , constitute a data set
of
N cases of ),...,,(
21 n
XXXX
=
. Denoting the
joint probability distribution of
X
by
)()( xXx
=
=
ρ
ρ
. EDAs estimate )(x
ρ
from
Se
l
P .
The pseudo-code of EDA is as follows (Larranaga
and Lozano, 2002):
1.
P
0
Generate M individuals (the initial
population) randomly.
Repeat for l =1,2,… until a stopping criteria
is met.
2.
Se
l
P
1
Select N M individuals from
1l
P according to a selection method.
3.
)|()(
1
Se
ll
Pxx
=
ρρ
Estimate the
probability distribution of an individual
being among the selected individuals.
4.
P
l
Sample M individuals (the new
population from
)(x
l
ρ
.
In the above pseudo-code of EDA there are
four main steps:
(i)
The first population P
0
of M
individuals is generated, usually by
assuming a uniform distribution (either
discrete or continuous) on each
variable, and evaluating each of the
individuals.
(ii)
A number N (N M) of individuals are
selected, usually the fittest.
(iii)
Thirdly, the n-dimensional
probabilistic model that better
expresses the interdependencies
between the n variables is induced.
(iv)
The new population of M new
individuals is obtained by simulating the
probability distribution learnt in the
previous step.
Steps (ii), (iii) and (iv) are repeated until a
stopping condition is verified. The most
important step is to find the interdependencies
between the variables (step (iii)), and this is
WINSYS 2006 - INTERNATIONAL CONFERENCE ON WIRELESS INFORMATION NETWORKS AND SYSTEMS
212
done using techniques from probabilistic
graphical models.
3 A NEW APPROACH
From literature review, it is known that call
orderings and minimisation of cost functions have
been used in solving the frequency assignment
problem (Kunz, 1991), (Sivaranjan, McElliece and
Ketchum,1986). It has been noticed that the
advantages of the method using call orderings
correspond to the disadvantages of the cost
minimising approaches and vice-versa. Using the
idea of Beckmann in (Beckmann and Killat, 1999),
we combine the two methods so as to gain their
advantages. We propose to use EDA to solve
combinatorial optimisation problems for the
determination of an optimal satisfying call list
L
and assign the frequencies using FEA (Beckmann
and Killat, 1999);this assures that in contrast to all
existing cost-minimising methods, we only get legal
frequencies without any interferences. This strategy
derives only channel allocations which do not
violate any of the interference constraints during the
search process which leads to a desirable reduction
of the search space. In our method the main
optimisation work with EDA is done to search for an
optimal frequency assignment.
Figure 1: New Method.
Our method is summarised in Figure 1. It can be
seen that new call lists are generated by the EDA.
The FEA strategy is used to evaluate the quality of
the generated call list L. This means that for each
single call list L we apply the FEA strategy in order
to determine the number of calls without an
allocated frequency; in other words we check how
many calls have been assigned a proper frequency
without violating the interference constraints; such
calls are known as blocked calls denoted by b. A
low value of b then corresponds to a high solution
quality. Then EDA is used to generate new,
hopefully better call lists during the next iteration.
This process is then repeated until the application of
the FEA strategy to a proper call list leads to a
frequency assignment with the desired minimum
frequencies
.
We now describe how EDA is applied to solve
the channel assignment problem. First, an initial
population of randomised solution vectors are
generated. Each vector is then evaluated using FEA.
The solution vector is then sorted out with respect to
the number of blocked calls and half of the best
solutions is retained. We then estimate the
probability distribution of an individual being
among the selected individuals and then sample a
new population with the same number of individuals
so as to keep the number of individuals in the
population same. The described procedure is
repeated until we get a solution of the desired quality
or the maximum number of iterations is reached.
4 SIMULATION RESULTS
The algorithm described in section 3 has been
applied to the well known benchmark problem, that
is, the 21-cell system shown in Figure 2. This
special example which has been dealt by many
researchers working in the field of frequency
assignment, allows us to compare the results
obtained from the new algorithm with published
results. Different set of problems have been
considered by choosing different demand vectors
D1 and D2 as shown in Table 1 and different
interference conditions summarised in Table 2.
Figure 2: Cells Layout.
Table 1: Frequency Demand Vectors D1 and D2.
d
1
d
2
d
3
d
4
d
5
d
6
d
7
d
8
d
9
d
10
d
11
d
12
d
13
d
14
d
15
d
16
d
17
d
18
d
19
d
20
d
21
D1 8 25 8 8 8 15 18 52 77 28 13 15 31 15 36 57 28 8 10 13 8
D2 5 5 5 8 12 25 30 25 30 40 40 45 20 30 25 15 15 30 20 20 25
AN ESTIMATION OF DISTRIBUTION ALGORITHM FOR THE CHANNEL ASSIGNMENT PROBLEM
213
Table 2: Interference Conditions.
Problem Case 1 2 3 4 5 6 7 8 9 10 11 12
N
c
7 7 7 12 12 7 7 7 7 12 12 7
ACC 1 1 2 1 1 2 1 1 2 1 1 2
c
ii
5 7 7 5 7 5 5 7 7 5 7 5
To evaluate the performance of the new
algorithm, we solve the problem with 50 different
seed values. We choose a maximum of 25 iterations
to stop the algorithm if no solution is obtained. At
each iteration only half of the best solutions are
retained and a new population is sampled. Each
problem case was run 20 times from the different
initial seed values from random generators and the
average generation number is shown in Table 3. To
evaluate the performance of the new algorithm, we
solve the above mentioned problem with 50 different
seed values. Convergence rate is an important factor
to compare the efficiency of a
method which is
shown in Table 3. Problems 3 and 9, converge in
only one generation, this result is comparable to that
obtained in (Beckmann and Killat, 1999). The other
problem cases converge in average of two iterations.
Table 3: Average Number of Generations.
Problem Case Ave Gen Num
1, 5 1.7
2, 4, 6, 7 1.6
3, 9 1
8, 10 1.55
11 1.75
12 1.6
5 CONCLUSION
In this paper we have presented a new method to
solve the frequency assignment problem, which is a
blend of the frequency exhaustive strategy and EDA.
Our results show that EDA can be applied for
solving the channel assignment problem in mobile
cellular environment. EDA has an advantage over
other methods like Neural Networks and Genetic
Algorithms in terms of rapid convergence to the
optimal solution. Hence the new algorithm
presented in this paper seems promising in solving
the CAP problem
.
REFERENCES
Beckmann, D. and Killat, U., 1999. A new strategy for
the application of genetic algorithms to the channel
assignment problem, IEEE Trans Veh Tech. Vol. 48,
No. 4, , pp. 1262-1269.
Funabiki, N. and Takefyi, Y., 1992. A neural network
parallel algorithm for channel assignment in cellular
radio networks, IEEE Trans Veh Tech. Vol. 41, , pp.
430-437.
Gamst, A. and Rave, W., 1982 On frequency
assignment in mobile automatic telephone systems,
Proceedings of GLOBECOM'82. IEEE., pp. 309-315.
Gamst, A., 1986. Some Lower bounds for a class of
frequency assignment problems, IEEE Trans Veh
Tech. Vol. 35, , pp. 8 – 14.
Kim, J. S., Park, S., Dowd, P. and Nasrabadi, N., 1996.
Channel Assignment in Cellular Radio using Genetic
Algorithm, Wireless Personnal Communications. Vol.
3, , pp. 273-286.
Kunz, D., 1991, Channel Assignment for cellular radio
using neural networks, IEEE Trans Veh Tech. Vol. 40,
, pp. 188-193.
Larranaga, P. and J. A. Lozano., 2002. Estimation of
Distribution Algorithms; A new tool for Evolutionary