Cell Size and Shape Adjustment Depending on Call Traffic Distribution. L. Du, J. Bigham, ..... International Conference on Genetic Algorithms, p. 151--157, 1991.
Cell Size and Shape Adjustment Depending on Call Traffic Distribution L. Du, J. Bigham, L. Cuthbert, C. Parini, P. Nahi Electronic Engineering, Queen Mary, University of London Mile End Road, London E1 4NS, UK Abstract- In this paper the potential of a smart antenna based dynamic cell size and shape control scheme is assessed. By intelligent control of antenna pattern, the whole cellular network performance can be improved by contracting the antenna pattern around the source of peak traffic and expanding adjacent cells coverage to fill in the coverage loss. We perform a constrained optimization of antenna patterns by using real-coded genetic algorithms (RCGA), and develop a CDMA cellular system simulator to evaluate the overall improvement of the system performance. A transformation of the problem space is used to remove the principal power constraint. A problem with the intuitive transformation is shown and a revised one is presented. This highlights a problem with transformation-based methods in genetic algorithms. While the aim of transformation is to speed convergence, a bad transformation can be counter-productive. Optimization results for two scenarios show potential capacity improvement exceeding 20%.
I. INTRODUCTION The demand for wireless mobile communications has grown at an explosive rate in the last decade and this trend is expected to continue. This is one of the main reasons for the worldwide attention that smart antennas have received recently. Smart antenna technology is considered as one of the most promising techniques for increasing the capacity of cellular systems. The basic idea of a smart antenna is to use base station antenna patterns that are not fixed, but adapt to the current radio conditions. In the context of smart antennas the term "antenna" has an extended meaning. It consists of a number of radiating elements, a combining or dividing network and a control unit. The control unit is where some of the intelligence lies. Smart antenna systems are usually categorized as either switched-beam or adaptive-array systems. Both systems attempt to increase gain in the direction of the user but only the adaptive-array system offers optimal gain. A considerable amount of work has been published in the literature on smart antenna systems, for example [1][2]. Although the benefits of using smart antenna systems are many, there are many drawbacks and cost factors. Antenna beam forming is a computationally intensive process especially if fully adaptive-array is used. The complexity and cost of the adaptive beam-former is still seen as a major disadvantage for the fully "adaptive smart antenna" system. In this work, the concept of adaptive beam forming is extended to dynamically changing cell size and shapes, to provide dynamic mobile cellular coverage. Study of dynamic cell-size control has shown that the system performance can be improved [3]. However, changing both cell size and shapes has not been widely studied so far, and hopefully, it can provide more benefits than its costs. The formation of cells is based upon call traffic needs, capacity in a heavily loaded cell can be increased by contracting the antenna pattern around the source of peak traffic and expanding adjacent antenna pattern to fill in the coverage loss as illustrated in Fig. 1.
Fig. 1 Cell size and shape control
An architecture to realize such a base station requires the capability of approximately locating and tracking mobiles in order to adapt the system parameters to meet the traffic requirements. The existing generation of cellular networks has a limited capability of mobile position location, however the next generation of cellular networks is expected to have much better mobile position location capabilities. The position location capabilities of the cellular network can be used to determine a set of gain vectors that define the desired antenna pattern level in the direction of a cluster of mobiles. These gain vectors can be thought of as sample points on the desired antenna pattern. A best-fit antenna pattern from the available antenna resource (which should be kept at minimum in order to keep the cost of the system low) is then synthesized using a combination of optimization techniques such as genetic algorithms [4] and reinforcement learning [5]. This paper concentrates on the problem of constrained optimization of antenna patterns by the use of RCGA, where the main constraint is the base station power. A system level simulation is also performed in order to evaluate performance, and results of these simulations are also presented. II. CONSTRAINED OPTIMIZATION OF ANTENNA PATTERNS The first step towards intelligent control is to know which combination of antenna patterns is suitable for a traffic distribution. Several optimization methods can be used. In this paper, we explore the use of genetic algorithms because of their robustness and efficiency. Transformations of the coordinate space that remove the central power constraint are also described. Whilst several transformations may be possible and mathematically correct, for rapid convergence care has to be taken to ensure that the
IEEE WCNC’02
transformation does not distort the space in an unsatisfactory way. An example of an intuitive, but unsatisfactory transformation is given. Another transformation, which gives approximately uniform distributed output for uniform distributed, independent input data sets, is also given. This results in faster convergence. The techniques described here have application beyond the scope of this application. A. Using Genetic Algorithms Genetic Algorithms are search algorithms based on the mechanics of natural selection and natural genetics. They combine survival of the fittest among string structures. In every generation, a new set of structures is created using parts of the fittest of the previous generation with occasional random variation (mutation). Whilst randomized, genetic algorithms are not simple random walks. They have been shown to efficiently exploit historical information to find new search points with expected improved performance [4]. Since genetic algorithm has already been widely used, we only focus on how to apply it in our antenna patterns optimization problem and how to handle the central power constraint. First, we must design an efficient coding scheme to represent each possible combination of antenna patterns as chromosomes. We use a gain vector, [ g1 , g 2 , g 3 , ... g N ], in which each gain value is coded as a gene, symbolizes antenna gains along N directions (both amplitude and phase later). This determines the approximate shape of one antenna pattern. The number of gains N , and the number of bits for each gain, can be determined by the performance and precision requirements. Therefore, the chromosome for a region is formed by combining each set of genes in the region. If we have M base stations, the chromosome will be like [[ g1 , g 2 , ... g N ]1 , [ g1 , g 2 , ... g N ]2 , ... , [ g1 , g 2 , ... g N ]M ] . As many researchers [6][7][8] have reported, representing the optimization parameters as numbers, rather than bitstrings, may have some advantages. Since our problem is also real value optimization, we use a real-coded genetic algorithm, with BLX-α [9] crossover, distortion crossover (intra-crossover within a subset of genes, which represents one antenna pattern), and creep mutation operators. Since the transformation, which is explained next, handles the central constraint, it is not necessary to use other more complicated operators for our optimization.
than the methods that only map unfeasible chromosomes to feasible ones for several reasons. 1) A very simple search space can be used here, 2) it can be treated without any difference for all the feasible or unfeasible chromosomes in both spaces, and 3) this transformation can be constructed before starting optimization. In their paper [13], Koziel S. and Michalewicz Z. propose a general mapping method, which is able to map all the points in searching space into feasible space. It is a numerical method, and involves a lot of computation in searching the boundary points. To avoid this we construct a specific mapping function by analytical means, which is explained next. The feasible space here is the space includes all the legitimate values for a chromosome as that described above F = {( g1 ,...g N ) : ( g1 ,...g N ) ∈ C}, where the constrained space C ⊆ R N . A transformation is created so that the search space of the form S = { xi ∈ R : 0 ≤ xi ≤ 1} can be used. Each time that we need to calculate a fitness value for any chromosome (which is now encoded in terms of the x i ), we map the chromosome into the feasible space, calculate the fitness value for new chromosome, and then assign the value to original chromosome. In this way, we can perform genetic algorithms without any constraints. Since the RF transmitting power at a base station can be expressed as, Ptrans = δ ⋅
∑g N
1 N
i =1
'2 i
(2.1)
Where N is the number of gain, gi' is the i − th gain value along N directions, and δ is a constant. Then the main constraint of RF power available at the base station, is expressed as, Pmin ≤ δ ⋅
1 N
∑g N
i =1
'2 i
≤Pmax
(2.2)
Where Pmin is the minimum, Pmax is the maximum value of RF power for a base station. They are determined by both physical limit and call traffic density nearby. If we choose the same N for all the base stations, (2.2) can be simplified as,
It is very difficult to represent the fitness function in analytical format here, so we use a cellular network simulator to calculate it for each chromosome. Such a simulator is described in section III.
Pmin ≤ ∑ g i2 ≤ Pmax
B. Constraints Handling Method
Where gi =
N
(2.3)
i =1
N
δ
⋅ g i' .
The genetic algorithm approach is naturally an So the feasible space F is, unconstrained optimization technique. When a new chromosome is created by crossover or mutation in the N searching process, it may not be feasible. Many constraints F = gi ∈ R : Pmin ≤ ∑ gi2 ≤ Pmax , 0 ≤ g i ≤ 1 (2.4) i = 1 handling approaches have already been proposed [10][11], and recent survey papers classify them into five categories, In most cases, we will use a N -dimensional cube as the namely: use of penalty functions, special representation and search space S , operators, separation of objectives and constraints, hybrid methods and other approaches [12]. We investigate a method (2.5) S = { xi ∈ R : 0 ≤ xi ≤ 1} based on a transformation between search space and feasible space, which ensures that all the products of a crossover or We can then define a function f : S → F to map points mutation always will be feasible. This falls into the second category mentioned above. It is simpler, and usually better, in space S into space F . This mapping function must be IEEE WCNC’02
continuous, in order to satisfy the convergence requirement of genetic algorithms. Since the feasible space has the format of ∑ gi2 , inspired by polar coordinates and spherical coordinates, we can let g1 = r ⋅ cosθ1 g 2 = r ⋅ sin θ1 ⋅ cosθ 2 g3 = r ⋅ sin θ1 ⋅ sin θ 2 ⋅ cosθ 3 ... g N −1 = r ⋅ sin θ1 ⋅ sin θ 2 ⋅ ⋅ ⋅ ⋅ ⋅ cosθ N −1 g N = r ⋅ sin θ1 ⋅ sin θ 2 ⋅ ⋅ ⋅ ⋅ ⋅ sin θ N −1
∑g
2 i
The marginal PDFs of g1 , g 4 , g9 , and g12 , calculated by applying (2.8) for uniformly distributed, independent input
5
15
(2.6)
PDF
PDF
4 3
10
2
5 1 0
≡ r2 ,
r 2 = x ⋅ ( P
20
6
−P )+P
1 max min min If we let θ i = xi +1 , i = 2,3,..., N − 1
PDF
(2.7)
Then we get the mapping function f as,
0
0.5
1 1.5 g(1) values
2
0
2.5
80
80
60
60 PDF
Then,
quite coarse, it works well in our cases, since a little distortion does not affect the performance of GA optimization.
40
20
π g i = r ⋅ cos( 2 ⋅ x2 ), i = 1 i π π g i = r ⋅ cos( ⋅ xi +1 )∏ sin( ⋅ x j ), 2 ≤ i ≤ N − 1 2 2 j =2 f = N π g i = r ⋅ ∏ sin( ⋅ x j ), i = N 2 j=2 r = x1 ⋅ ( Pmax − Pmin ) + Pmin
0
0
0.5
1 1.5 g(4) values
2
2.5
0
0.2
0.4 0.6 g(12) values
0.8
1
40
20
0
0.2
0.4
0.6 0.8 g(9) values
1
1.2
0
Fig. 3. The marginal PDF of g1 , g 4 , g9 , and g12 , using (2.8), where N = 12, Pmax = 5.0, and Pmin = 0.01
(2.8)
xi , are shown in Fig. 3. It shows that the PDFs for different g i are different. g1 always has the highest probability of being the largest value, while g N has the lowest probability
We can prove that, ∀x ∈ S , there exists g = f ( x) , which obeys g ∈ F , and vice versa.
of being the largest value. This transformation causes too much distortion, and in our experiments, this results in slow convergence as seen in Fig. 6 in the next section.
However, whilst this method is mathematically correct it has a serious problem. It distorts the space in an unsatisfied way, which will be explained next.
To solve this, we chose another mapping function, inspired
To evaluate how much distortion the transformation causes, we use a PDF (Probability Density Function) based method, which check the output probability of solution points with uniformly distributed, independent input. Since the possible locations of the optimum solution are unknown in our case, the ideal transformation should generate output points uniformly distributed in feasible space. Because of the symmetry of feasible space, the marginal PDFs of g i uniformly distributed in feasible space are identical with these of g j , which i ≠ j and g i is independent of g j . One of them is shown in Fig. 2, and will be used as the criterion for evaluating transformation distortion. Whilst this method is
∑(
by the fact of
g i = r ⋅
f =
r =
xi
∑ xi2
xi
∑ x 2j N
) 2 ≡ 1 , as shown in (2.9),
, i = 1, 2,..., N
j =1
∑ x ρj
(2.9)
N
j =1
N
( Pmax − Pmin ) + Pmin
where ρ is a factor used to control the distortion of transformation. Here ρ = 8.0 is chosen to get the smaller distortion as seen in Fig. 4. In our experiments, this results in
1.4
1.2
1 1
0.8
PDF
PDF
0.8
0.6
0.4
0.4
0.2
0
0.6
0.2
0
0.2
0.4
0.6
0.8
1 1.2 g(i) values
1.4
1.6
1.8
2
0
Fig. 2. The marginal PDF of g i uniformly distributed in feasible space, where N = 12, Pmax = 5.0, and Pmin = 0.01
0
0.5
1 g(i) values
1.5
2
Fig. 4. The marginal PDF of g i using (2.9), where
ρ = 8.0, N = 12, Pmax = 5.0, and Pmin = 0.01
IEEE WCNC’02
a much faster convergence as seen in Fig. 6 in the next section.
The capacity of single cell is given by, Capacity = M max =
III. SIMULATION AND RESULTS Simulations were performed to test the efficacy of the approach and the potential of the method. To reduce the boundary effect of cellular network simulation, a 100 diamond-mesh CDMA cellular network model is used, as shown in Fig. 5.
GP +1 ( Eb / I 0 ) min
Therefore, we can simply let each base station have a fixed capacity to serve M max traffic units, and each traffic unit consume a fixed amount, namely demand, from the base station to which it is subscribed. The call assignment for i − th traffic unit is performed according to its received power from j − th base station, Pji , P(Re cv ) ji = α ij ⋅ P(Trans ) j = α ij ⋅ g ji ⋅ PBS
0,0 1,0 2,0 3,0 4,0 5,0 6,0 7,0 8,0 9,0
(3.3)
(3.4)
0,1 1,1 2,1 3,1 4,1 5,1 6,1 7,1 8,1 9,1
Where α ij is propagation factor between i − th traffic unit and j − th base station, g ji is the gain value at the direction from j − th base station to i − th traffic unit, and PBS is the transmitting power for every base station.
0,2 1,2 2,2 3,2 4,2 5,2 6,2 7,2 8,2 9,2 0,3 1,3 2,3 3,3 4,3 5,3 6,3 7,3 8,3 9,3 0,4 1,4 2,4 3,4 4,4 5,4 6,4 7,4 8,4 9,4 0,5 1,5 2,5 3,5 4,5 5,5 6,5 7,5 8,5 9,5 0,6 1,6 2,6 3,6 4,6 5,6 6,6 7,6 8,6 9,6
Since we assume no time delay, eventually, each base station only serves the best traffic units. The system uplink capacity can be calculated by counting the number of all the served traffic units from the simulation.
0,7 1,7 2,7 3,7 4,7 5,7 6,7 7,7 8,7 9,7 0,8 1,8 2,8 3,8 4,8 5,8 6,8 7,8 8,8 9,8 0,9 1,9 2,9 3,9 4,9 5,9 6,9 7,9 8,9 9,9
D. Objective function
Fig. 5. Cellular Network Model
The following is the list of simulation specifications. A. Assumptions • No shadowing or multi-path fading in radio channel. • Perfect uplink power control. • No time delay. • Each base station has the same capacity and each traffic unit has the same demand. • Uplink interference only comes from other traffic units in same cell. • Handover is not taken into account. • Voice activity factor is not taken into account. B. Propagation model Usually, the propagation for mobile radio communications is modeled by path loss, shadowing, multi-path fading. Since the simulated scenarios are static, for simplicity, we only take the path loss into account, so the propagation model can be expressed as α ij = k ⋅ rij− ρ
(3.1)
Where rij is the distance between i − th traffic unit and j − th base station, and k and ρ are propagation constants.
The objective function is shown as (3.5). The main objective for our optimization is to maximize system capacity. However, since we ignore the downlink interference from other base station, we have to minimize it by other approximate ways. One simple way is to minimize the total transmitting power of all the base stations. It is not as important as the main objective, so we give it less weight, which comes from empirical data. 1 Fitness = Capacity (dB) − RF power for all BSs( dB ) 5
(3.5)
E. Simulation configuration • • •
3000 traffic units in the whole area. 100 base stations, each has the capacity to serve 36 traffic units, i.e. (Total Capacity) = 120% ⋅ (Total Demand) . Cell radius R = 1 , path loss factor ρ = 4.0 , and propagation model factor k = −70dB .
• •
20% of traffic is uniformly distributed in the whole area. Other 80% of traffic is distributed in 40 hot spots with normal distribution (the mean value for each
C. System Uplink Capacity
hot spots, µ , is uniformly distributed over the whole
Since we assume perfect power control, the energy per chip for all traffic units is the same and equal to Ec . If we only consider the interference from traffic units in same cell, as in our assumption, the bit energy-to-effective noise ratio, Eb / I 0 at the base station is then,
area, and the standard deviation, σ = 0.2 R ).
Eb / I 0 =
Eb GP ⋅ Ec GP = = ( M − 1) Ec ( M − 1) Ec M − 1
F. Results The optimization was performed for two scenarios, and results are shown in Fig. 6. Some optimization parameters are listed in TABLE I.
(3.2) IEEE WCNC’02
the mapping can slow down, rather than speed up convergence. 2800
Scenario 1 Scenario 2
2700
The results from the GA optimizations have shown the potential of communication between adjacent base stations when dynamically computing cell size and shape. Current work is exploiting this using a cooperative agent approach. Adaptive controlling algorithms for run-time optimization, better cellular network simulation taking into consideration shadowing, multi-path fading, interference and handover, is being evaluated. Also, advanced antenna design techniques for this application need to be investigated.
System Uplink Capacity
2600 2500
Second method (2.9) 2400 2300
First method (2.8)
2200 2100
Fixed (circular) pattern
2000
REFERENCES
1900 1800 1700
0
250
500 Generation
750
1000
Fig. 6. System uplink capacity given by different optimization methods
TABLE I OPTIMIZATION PARAMETERS First method
Second method
Population Size
1500
1500
Generation
1000
1000
Elite Rate
RCGA Operator
Mapping Function
0.1
0.1
(2.8)
(2.9)
BLX-0.5 Weight
2.0
2.0
Distortion Weight
0.5
0.5
Creep Weight
0.05
0.05
BLX-0.5 Rate
0.2
0.2
Distortion Rate
0.3
0.3
Distortion Range
0.2
0.2
0.005
0.005
Creep Rate
As Fig. 6 shows, using the first mapping method is much worse than using the second one. It is sometimes even worse than not optimizing. This highlights the importance of choosing proper mapping function in transformation-based constraints handling methods. The results also show that, by the use of dynamical cell size and shape control, capacity improvements of over 20% are possible. IV. CONCLUSIONS This paper has proposed and investigated a smart antenna based dynamic cell size and shape control scheme, which can improve the system capacity for non-uniformly distributed call traffic. The results from computer simulations show that it does increase the system capacity significantly. We also describe an efficient way to handle the power constraints in antenna pattern optimization. The results show that a proper transformation is very important for constrained optimization problems. This has relevance to other mapping techniques for handling constraints in GAs, as distortion in
[1] M. Chryssomallis, “Smart Antennas”, IEEE Antennas and Propagation Magazine Vol. 42, No.3, June 2000 [2] L.C.Godara, “Application of Antenna Arrays to Mobile Communications, Part I: Performance improvement, Feasibility and System Considerations”, Proceedings of the IEEE, Vol. 85, No.7, July 1997 [3] T. Togo, I. Yoshii, and R. Kohno. “Dynamic cell-size control according to geographical mobile distribution in a DS/CDMA cellular system”, The Ninth IEEE International Symposium on Personal, Indoor and Mobile Radio Communications, Volume: 2, p. 677 -681, 1998 [4] D.E. Goldberg, Genetic Algorithms in search, optimization, and machine learning, Addison-Wesley, 1989 [5] R. Sutton and A. Barto. Reinforcement Learning: an Introduction. MIT Press, 1998 [6] D.E. Goldberg. “The theory of virtual alphabets”. In H.P. Schwefel and R. Manner, editors, Parallel Problem Solving from Nature, p. 13--22. Springe-Verlag, 1990. [7] C.Z. Janikow and Z. Michalewicz. “An experimental comparison of binary and floating point representations in genetic algorithms”. In R.K. Belew and L.B. Booker, editors, Proceedings of the Fourth International Conference on Genetic Algorithms, p. 31--36, 1991. [8] L. Davis. Handbook of Genetic Algorithms. Van Nostrand Reinhold, p. 65, 1991. [9] L.J. Eshelman, and J.D. Schaffer, “Real-Coded Algorithms and Interval Schemata”. In L.D. Whitley (Eds.), Foundations of Genetic Algorithms, p. 187-202, 1993. [10] Z. Michalewicz and C.Z. Janikow. “Handling constraints in genetic algorithms”. In R.K. Belew and L.B. Booker, editors, Proceedings of the Fourth International Conference on Genetic Algorithms, p. 151--157, 1991. [11] A. Carlos, A Survey of Constraint Handling Techniques used with Evolutionary Algorithms, Veracruz, Mexico, 2000 [12] Z. Michalewicz, A survey of Constraint Handling Techniques in Evolutionary Computation Methods, Proceedings of the Fourth Annual Conference on Evolutionary Programming, pages 135-155, The MIT Press, Cambridge, Massachusetts, 1995
IEEE WCNC’02
[13] S. Koziel and Z. Michalewicz. Evolutionary Algorithms, Homomorphous Mappings, and Constrained Parameter Optimization. Evolutionary Computation, 7(1): 19--44, 1999.
IEEE WCNC’02
