P Garambois
email: apierre.garambois@ec-lyon.fr

S Besset

L Jezequel
email: clouis.jezequel@ec-lyon.fr

Modal synthesis applied to a Reissner mixed plate finite element dynamic model

Keywords: Reissner, Mindlin, Kirchoff-Love, plate, mixed formulation, finite element, sub-structuring, vibrations

This paper deals with the application of component mode synthesis on thick and thin plate finite element model, built with a Reissner mixed formulation. The Reissner mixed formulation is a structural mechanical formulation mostly used in plate problems that uses both displacement and generalized stress variables. Thus, the access to the stress in the plate is easier than with a primal method. The Mindlin theory for thick plates, permits to take into consideration shearing phenomenon inside the plate, whereas the Kirchoff-Love for thin plates only deals with bending and twisting phenomenons. The Reissner-mixed and primal models show the exact same results, both for Mindlin and Kirchoff-Love theory. Nevertheless, the Kirchoff-Love mixed model shows a much better convergence than the Mindlin mixed model for thick plates. The inconvenience of a mixed model, whatever the chosen plate theory is, is the numerical size of the finite element model that is much bigger. That's the reason why we use modal synthesis to reduce our model. The idea is to build our model with both primal and mixed formulation at the same time. Then we reduce the mixed model using a substructuring method and bases composed of "fixed interface modes" obtained from a Craig & Bampton method applied on the primal model. That method also shows a good convergence and permits us to reduce significantly the numerical size of our model on a certain frequency band, keeping the advantages of the Reissner mixed model.

INTRODUCTION

Most of the finite element plate models used in industry are based on a dynamic primal formulation. They are fast and efficient, but they need an extra calculation and integration to get the strains and thus the constrains. Another point of view developed by Reissner [START_REF] Reissner | On a mixed variational theorem and on shear feformable plate theory[END_REF] aims at defining a new Lagrangian using both displacement fields, and generalized stress fields, called Reissner mixed function. Thus, developing this function and discretising the displacements and generalized stress fields, we obtaine a Reissner mixed finite element formulation with displacement and stress fields parameters. That method has already been used for static problems [START_REF] Eduardo | Numerical implementation of hybrid-mixed finite element model for Reissner-Mindlin plates[END_REF] [3] and more rarely for dynamic problems [START_REF] Nguyen-Xuan | Nguyen-Thoi An edge-based smoothed finite element method (ES-FEM) with stabilized discrete shear gap technique for analysis of Reissner-Mindlin plates[END_REF]. Most of the time, the Reissner mixed formulation is used for Mindlin thick plate theory. In this paper, we develop a Reissner mixed formulation both for thick Mindlin plate and thin Kirchoff-Love plate.

An inconvenient of the mixed formulation is the numerical size of the problems, due to the addition of generalized stress fields parameters to the displacements fields parameters of the primal method. Many sub-structuring method exist to reduce primal finite element such as "fixed interface mode" method (Craig & Bampton method [START_REF] Craig & Mervin | Coupling of substructures for dynamic analysis[END_REF]), free mode method (Mac Neal method [START_REF] Macneal | A hybrid methof of component mode synthesis[END_REF]) and boundary mode method (Balmes method [START_REF] Balmes | Use of generalized interface degrees of freedom in component mode synthesis[END_REF]). In this paper, the idea is to use those methods applied on the primal model to build a new reduced base for the sub-structured mixed model. First of all, the article talks about a Reissner mixed variational dynamic formulation of finite element for Mindlin thick plates and set the conditions of the following finite element model. Then a Reissner-Kirchoff-Love model is also studied. Afterwards, we study the convergence of the two mixed models. In the fifth part, we deal with the application of component mode synthesis on the finite element mixed model, based on "fixed interface modes", obtained through a Craig & Bampton method applied on the primal plate problem. In the last part, we show the convergence of our problem on a simple example and the main results.

VARIATIONAL FORMULATION OF REISSNER MIXED FINITE ELEMENT BASED ON THICK MINDLIN PLATE

The Reissner mixed formulation

The Reissner mixed function is given by:

Π RD = V -σ ij e ij + 1 2 σ ij S ijkl σ kl + b i u i + 1 2 ρ ui 2 dV
(1) considering σ ij the generalized stress, e ij the strain, u i the displacement, b i the body force, ρ the volumic mass and S ijkl the elastic compliance matrix. We consider 3 main fields in the function:

• Mixed strain energy: σ ij e ij -1 2 σ ij S ijkl σ kl • Body force work: b i u i • Kinetic energy: 1 2 ρ ui 2 2.

Definition of the Mindlin theory

The displacement is given by:

u i = w θ x θ y (2)
considering w the transverse displacement, θ x and θ y the normal rotation around the -x and -y axis. The three displacement fields are independent in the Mindlin thick plate theory whilst the rotations depend on the transverse displacement in the Kirchoff-Love theory. Thus, the shearing phenomenon doesn't exist in that theory which simplify a lot our system and gives us smaller matrix sizes and so reduce the computational time.

The strain is given by:

e ij =          ǫ xx ǫ yy γ xy γ xz γ yz          = D u i ( 3
)
where D is the following operator:

D =              0 0 ∂ ∂x 0 -∂ ∂y 0 0 ∂ ∂x ∂ ∂y ∂ ∂x 0 1 ∂ ∂y -1 0              (4)
The generalized stress field is given by:

σ ij =          M x M y M xy Q x Q y          (5)
where {M x , M y , M xy } T represents the bending and twisting moments and {Q x , Q y } T represents the transverse shear force.

The elastic compliance matrix is given by:

S ijkl =            12 Et 3 -12ν Et 3 0 0 0 -12ν Et 3 12 Et 3 0 0 0 0 0 24(1+ν) Et 3 0 0 0 0 0 2,4(1+ν) Et 0 0 0 0 0 2,4(1+ν) Et            ( 6
) where E is the Young Modulus, t is the thickness of the plate element and ν is the Poisson ratio.

The positive directions of the generalized stress field are shown in Figure 1.

Interpolation of nodal displacement in element

We assume that w, θ x and θ y , for a 4-node quadrilateral element, are interpolated in terms of nodal displacements

{w i θ xi θ yi } (i = 1,2,3,4
), as follows:

w(x, y) = 4 i=1 N i (x, y)w i (7)
θ x (x, y) = 4 i=1 N i (x, y)θ xi (8) θ y (x, y) = 4 i=1 N i (x, y)θ y i (9)
Which gives us:

u i = w θ x θ y = NU (10)
Chosing a 4-node quadrilateral element, the shape functions have 4 conditions and thus can be quadratic as follows:

N i (x, y) = a i + b i x + c i y + d i xy (11)
That condition is essential to observe shearing phenomenon.

Interpolation of stress

We assume that each field of the generalized stress is interpolated with linear function but does not depend on the nodal values, which gives a large choice of possibilities when choosing the functions. Thus, each generalized stress is interpolated with 3 parameters β, as follows:

σ ij =          M x M y M xy Q x Q y          = Pβ σ ij =         
1 x y 0 0 0 0 0 0 ... 0 0 0 1 x y 0 0 0 ... 0 0 0 0 0 0 1 x y ... 0 0 0 0 0 0 0 0 0 ... 0 0 0 0 0 0 0 0 0 ...

                  β 1 β 2 . . β 15          (12)
Those choices give us a total of 27 parameters per element matrix, as we have 12 displacement paremeters U and 15 generalized stress parameters β.

Variational function and mixed finite element formulation

Π RD = V -σ ij e ij + 1 2 σ ij S ijkl σ kl + b i u i + 1 2 ρ ui 2 dV Π RD = S 1 2 (N U ) T m(N U ) -(PU ) T (DNU ) + 1 2 (Pβ) T S(Pβ) dS + V b i U dS (13)
where

m = ρ    t 0 0 0 t 3 12 0 0 0 t 3 12    (14)
and ρ is the density. The matrix development gives us:

Π RD = 1 2 U T M U -β T GU -β T Hβ + F T U (15)
where

M = S N T mNdS (16) G = S P T DNdS (17)
H = S -P T SPdS ( 18
)
and F is the force vector applied to the mesh nodes.

Then we have:

M 0 0 0 Ü β + 0 G T G H U β = F 0 (19)

VARIATIONAL FORMULATION OF REISSNER MIXED FINITE ELEMENT BASED ON THIN KIRCHOFF-LOVE PLATE

The Reissner mixed formulation

The Reissner mixed function is still given by:

Π RD = V -σ ij e ij + 1 2 σ ij S ijkl σ kl + b i u i + 1 2 ρ ui 2 dV
(20) considering σ ij the generalized stress, e ij the strain, u i the displacement, b i the body force, ρ the volumic mass and S ijkl the elastic compliance matrix.

Definition of the Kirchoff-Love theory

The displacement is given by:

u i = w θ x θ y =    w ∂w ∂y -∂w ∂x    (21)
considering w the transverse displacement, θ x and θ y the normal rotation around the -x and -y axis. The theory of Kirchoff-Love assume that the 2 rotations θ x and θ y depend on the transverse displacement w.

The strain is given by:

e ij = ǫ xx ǫ yy γ xy = D u i ( 22
)
where D is the operator

D =    0 0 ∂ ∂x 0 -∂ ∂y 0 0 ∂ ∂x ∂ ∂y    (23)
The generalized stress field is given by:

σ ij = M x M y M xy (24)
where {M x , M y , M xy } T represents the bending and twisting moments. The transverse shearing force is not taken into consideration using the Kirchoff-Love theory.

The elastic compliance matrix is given by:

S ijkl =    12 Et 3 -12ν Et 3 0 -12ν Et 3 12 Et 3 0 0 0 24(1+ν) Et 3    (25)

Interpolation of nodal displacement in element

We assume that w, θ x and θ y , for a 3-node triangular element, are interpolated in terms of nodal displacements {w i θ xi θ yi } (i = 1,2,3), as follows:

w(x, y) = 3 i=1 N i (x, y)w i (26) θ x (x, y) = 3 i=1 ∂N i (x, y) ∂y θ xi (27) θ y (x, y) = 3 i=1 - ∂N i (x, y) ∂x θ y i (28)
Which gives us:

u i = w θ x θ y = NU =    N 1 N 2 ... N 9 ∂N1 ∂y ∂N2 ∂y ... ∂N9 ∂y -∂N1 ∂x -∂N2 ∂x ... -∂N9 ∂x    U ( 29
)
Chosing a 3-node triangular element, the shape functions have 9 conditions each and thus are 3rd order polynomials as follows:

N i (x, y) = a i +b i x+c i y+d i xy+e i x 2 +f i y 2 +g i x 3 +h i y 3 +i i xy 2 (30)

Interpolation of stress

As the Mindlin theory, we assume that each field of the generalized stress is interpolated with linear function and does not depend on the nodal values. Each generalized stress is interpolated with 3 parameters β, as follows:

σ ij = M x M y M xy = Pβ σ ij =
1 x y 0 0 0 0 0 0 0 0 0 1 x y 0 0 0 0 0 0 0 0 0 1 x y

         β 1 β 2 . . β 9          (31)
Those choices give us a total of 18 parameters per element matrix, as we have 9 displacement paremeters U and 9 generalized stress parameters β.

Variational function and mixed finite element formulation

As the Reissner-Mindlin theory, the matrix development of the Reissner-Kirchoff-Love theory gives us the same result as equation 19:

M 0 0 0 Ü β + 0 G T G H U β = F 0 (32)

CONVERGENCE STUDY OF THE MIXED MODEL

Mindlin theory

The example we study in this part is a rectangular thick plate built with Reissner-Mindlin mixed quadrangular finite elements described in the the section 2. The plate is clamped on one edge, and free on the three other edges. It is made of steel (see characteristics 1) and the thickness t of the plate is a variable of the different simulation. That example is shown in figure 2. The numerical simulations we make shows the exact same convergence for the Reissner-Mindlin mixed model and the Mindlin primal model. Nevertheless, the results obtained with the Mindlin theory for various thickness of the plate (Table 2 andTable 3) show that the convergence of the model needs a high number of elements. It appears that the characteristic size of the element needs to be at least equal or even smaller than the thickness of the plate to get acceptable results, which means that the smaller the thickness is, the smaller the elements need to be, which can cause a lot of numerical problems (mass and stiffness matrix with big numerical sizes, especially with the mixed model).

Kirchoff-Love theory

The example we study in this part is a rectangular thick plate built with Reissner-Kirchoff-Love mixed triangular finite elements described in the the section 3. The plate is clamped on one edge, and free on the three other edges.

It is made of steel (see characteristics in Table 1) and the thickness t of the plate is a variable of the different simulation. That example is shown in figure 3. The numerical simulations we make also shows the exact same convergence for the Reissner-Kirchoff-Love mixed model and the Kirchoff primal model. Nevertheless, the results obtained with the Kirchoff-Love theory for various thickness of the plate (Table 4 andTable 5) show a significantly higher convergence than the Mindlin theory. It appears that the characteristic size of the element doesn't influence the convergence of the model, as long as we stay in small thickness, as the Kirchoff-Love theory defines.

MODAL REDUCTION

As described in the first sections, the numerical sizes of the Reissner-Mindlin thin plate mixed model and the Reissner- Kirchoff-Love thick plate mixed model are significantly higher than primal models using the same theories. The idea of this part is to reduce the numerical size of the model with a sub-structuring method, using "fixed interface modes" method obtained with a primal model (considering the same characcteristics and meshes). We describe this modal reduction in the following section.

Projection of displacements: Craig & Bampton Method

The Craig & Bampton Method is a sub-structuring primal method that separates, for each sub-structure a, boundary DOF's U a j and internal DOF's U a i . It projects the initial DOF's of the sub-structure a on a new smaller base composed of the same boundary DOF's U a j (we also call them "physical DOF's") and truncated modal DOF's η a eU (which will be DOF's only for the displacements), as follows:

U a i U a j = Φ a eU Ψ a ij 0 I η a eU U a j ( 33
)
where Φ a eU is a truncated base composed of eigenmodes of the structure a assuming that the boundary nodes are held fixed (concerning the displacement fields), Ψ a ij is the attachment modes matrix

Ψ a ij = -K a ii (K a ij ) -1 .

Projection of generalized stress field: Craig & Bampton Method on the stress

For each sub-structure a, in order to access to the generalized stress field parameters, we use the relation between β a and U a :

β a = -(H a ) -1 G a U a .
We also use a Craig & Bampton method to access to the displacement fields U a corresponding to the generalized stress fields β a , as follows:

β a = P a Φ a eU 0 P a Ψ a ij I . η a eβ U a j ( 34
)
where P a = -(H a ) -1 G a , η a eβ are the modal DOF's for the generalized stress fields and Φ a eβ is a truncated base composed of eigenmodes of the structure a assuming that the boundary nodes are held fixed and concerning the generalized stress fields.

Reduction of a sub-structure

Considering the sections 5.1 and 5.2, the reduction of the whole sub-structure a is given by:

U a i U a j β a =      Φ a eβ 0 Ψ a ij 0 0 I 0 P a Φ a eβ 0 P a Ψ a ij I         η a eU η a eβ U a j    (35)

Assembly

We assemble two the sub-structures a and b considering U a j = U b j = U i , which gives us the following form:

         U a i U j U b i β a β b          = T          η a eU η a eβ U j η b eU η b eβ          (36)
where

T =                  Φ a eU 0 Ψ a ij 0 0 0 0 I 0 0 0 0 Ψ b ij Φ a eU 0 0 P a Φ a eβ 0 P a Ψ a ij I 0 0 0 P b Ψ b ij I 0 0 P b Φ b eβ 0                  (37)
6 CONVERGENCE STUDY OF THE REDUCTION

Example

This section focus on the convergence of the Reissner mixed model using the method described in section 5 with the theory of Kirchoff-Love, depending on the number of modes "fixed interface modes" kept in the truncation both for displacement fields and generalized stress fields. The example used in this part is composed of two different plates, meshed with Reissner-Kirchoff-Love triangular mixed elements described in section 3. It is made of steel (see characteristics in table 1) and we choose a thickness of 1e -3 m. That example is shown in figure 4. The first plate is clamped on one edge (102 DOF's clamped) and composed of 414 elements and 5064 DOF's (of which 1338 are displacement parameters and 3726 stress parameters), whereas the second plate is composed of 510 elements and 6294 DOF's (of which 1722 are displacement parameters and 4572 stress parameters). The jonction is composed of 102 boundary DOF's. The results are shown in figure 5. We calculate the FRF of the test structure with different modal synthesis applied on each sub-structure. In each case, the number of displacement field modes and generalized stress field modes are the same for the sub-structure 1 and the sub-structure 2. The results show a good convergence, even with a low number of modes. It appears that choosing 20 displacement field modes and 20 generalized stress field modes permits a good convergence of the model until 100Hz (27 first modes observable), whereas choosing 10 modes both for displacement field and generalized stress field give good results until 70Hz (20 first modes observable) and 5 modes until 35Hz (12 first modes observable). Those results confirm the choice of using the "Craig & Bampton method" both for displacement field and generalized stress field, and permit to greatly decrease the numerical size of the mixed model (see table 6). The figure 6 shows the Von Mises stress distribution in the plate for the 12 th mode (f = 331Hz) and the shape of the mode for our example, obtained with a modal synthesis with 20 displacement field modes and 20 generalized stress field modes.

Prospects

Even though the convergence of our mixed reduced model appears to be quite good, others modal synthesis have been studied to decrease the number of DOF's such as free modes method [START_REF] Macneal | A hybrid methof of component mode synthesis[END_REF] and interface modes method [START_REF] Balmes | Use of generalized interface degrees of freedom in component mode synthesis[END_REF]. We can also change the truncation in function of the displacements or generalized stress fields and change the types of synthesis (fixed interface modes or free modes). and a thin plate model (Kirchoff-Love theory). First of all, the mixed and the primal model show exactly the same convergence with the same amount of elements. Then, if we focus on the difference between thin plates theories (which considers only bending and twisting moments) and thick plates theory (which also considers transverse shear forces), the calculations we make with our code on simple examples show a significantly different convergence be- tween the 2 theories. The Kirchoff-Love Reissner mixed model for thin plate shows a quite good convergence as it needs about 250 elements for a thickness of t = 0.01m to get a good convergence (ǫ < 3%), and those results doesn't depend on the thickness. Nevertheless, the Mindlin Reissner mixed model for thick plate shows a much worse convergence as it needs about 1600 elements for a thickness of t = 0.05m to get good results. Furthermore, the results with the Mindlin theory look dependent on the thickness: in fact, the smaller the thickness is, the smaller the element have to be. As a results, the same example with a thickness of t = 0.01m needs more than 6400 elements to converge. As the Kirchoff-Love mixed model shows a much better convergence, even for a 0.01m thickness which is quite high for a plate model, we decide to choose it for the rest of the study.

The inconvenience of a mixed model, be it for Kirchoff-Love or Mindlin theory, is the numerical size of the finite element model that is much bigger as it uses parameters for the generalized stress fields and the elementary mass and stiffness matrix are bigger. That's the reason why we try use modal synthesis to reduce our model. The reduction we make aims at using modes obtained with a primal model of the same meshes and same theory, which means we build both primal and Reissner mixed assemblage at the same time for each examples. We reduce the mixed model using a sub-structuring method and bases composed of "fixed interface modes" obtained from a Craig & Bampton method. That modal synthesis, both applied on displacements and generalized stress fields parameters show a good convergence with our case (double plate with a thickness t = 1e -3 m) as we can reduce the size from about 11 000 DOF's to about 200 DOF's and still be able to observe the 30 first modes. It improves considerably the computation time and, keeps the advantages of a Reissner mixed model with access to both the displacements and generalized stresses.

Figure 1 .

 1 Figure 1. Bending and twisting moment and transverse shear force

Figure 2 .

 2 Figure 2. thick plate built with Reissner-Mindlin quad elements

Figure 3 .

 3 Figure 3. Rectangular thin plate built with Reissner-Kirchoff-Love triangular elements

Figure 4 .

 4 Figure 4. Rectangular thick plate built with Reissner-Mindlin quad elements

Figure 5 .

 5 Figure 5. FRF of the test structure calculated with modal synthesis on the mixed model in function of the truncation

Figure 6 .

 6 Figure 6. Von Mises stress distribution in the 2 plates for the 12 th mode (f = 331Hz) obtained with the β parameters of the model with reduction figure 5a

Table 1 .

 1 Steel characteristicsYoung Modulus (P a) 2.1 × 10 11

	Poisson ratio	0.33
	Density (kg.m -3 )	7.5 × 10 3

Table 2 .

 2 Relative error for the first eigenfrequencies for the example 1 with t = 0.05m and Reissner Mindlin mixed model

			Element size (cm)
		9.5	4.8	3.2	2.4	1.9
	P P P P P P P Mode Elt	100	400 900 1600 2500
	1	44.3 12.7 5.7	3.2	0.01
	2	15.5	1.9	0.6 0.16	0.00
	3	46.2 12.9 5.9	3.2	0.02
	4	12.0	3.2	1.4	0.7	0.00
	5	19.0 13.5 6.0	3.3	0.002
	6	15.21 5.9	2.5	1.4	0.01
	7	26.3 14.9 9.0 0.13	0.00
	8	29.5	5.3	1.7	0.9	0.00
	9	34.0	8.6	3.7	2.0	0.01
	10	16.50 2.4	1.0 0.48	0.00
	Table 3. Relative error for the first eigenfrequencies for the example 1 with t = 0.01m and Reissner Mindlin mixed model
			Element size (cm)
		2.4	1.9	1.6	1.35 1.19
	P P P P P P P Mode Elt 1600 2500 3600 4900 6400
	1	64.2	43.7	32.4 24.2 19.1
	2	8.1	5.6	4.0	2.9	2.3
	3	100	44.5	32.4 32.4 19.3
	4	24.9	10.3	7.3	7.3	4.2
	5	100	44.2	32.2	32	19.0
	6	44.5 18.06 12.7 12.8	7.4
	7	35.8	20.4	17.2 17.2 13.2
	8	45.6	33.9	19.3 19.3	8.7
	9	46.9	26.0	8.5	5.5	3.0
	10	59.6	25.3	18.0 18.0 10.4

Table 4 .

 4 Relative error for the first eigenfrequencies for the example 1 with t = 0.01m and Reissner Kirchoff-Love mixed model

			Element size (cm)
		47.5 23.8 32.9	5.9
	P P P P P P P Mode Elt	16	64	256 1024
	1	2.94 1.47 0.53 0.21
	2	8.13 1.90 0.42 0.10
	3	6.80 1.19 0.40 0.13
	4	-	7.58 1.63 0.33
	5	-	2.87 0.24 0.06
	6	-	11.59 2.28 0.41
	7	-	1.72 0.01 0.01
	8	-	4.90 0.27 0.00
	9	-	3.35 0.06 0.00
	10	-	20.35 3.29 0.43

Table 5 .

 5 Relative error for the first eigenfrequencies for the example 1 with t = 0.001m and ReissnerKirchoff-Love mixed model

			Element size (cm)
		47.5	23.8 32.9	5.9
	P P P P P P P Mode Elt	16	64	256 1024
	1	3.16	2.11 1.05 0.00
	2	8.14	1.67 0.42 0.00
	3	6.68	1.17 0.33 0.00
	4	43.16 7.55 1.63 0.32
	5	24.84 2.86 0.24 0.18
	6	-	11.56 2.26 0.38
	7	-	1.77 0.01 0.00
	8	-	4.85 0.26 0.00
	9	-	3.60 0.29 0.06
	10	-	20.36 3.72 0.41

Table 6

 6

	. Maximum frequency of convergence and number of observable modes in function of the number of modes chosen for the truncation
	DOF's	Displacement & stress modes	Maximum frequencies (Hz)	Observed modes
	11256	Non-reduced model
	502	200 & 200	1700	>200
	182	20 & 20	100	27
	142	10 & 10	70	20
	122	5 & 5	35	12

AKNOWLEDGEMENT

The first author gratefully acknowledges the French Education Ministry which supports this research.


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Non-Cooperative Inter-Cell Interference Coordination Technique for Increasing Throughput Fairness in LTE Networks

Keywords: Long Term Evolution, inter-cell interference, ICIC, FFR, SINR, fairness

One major concern for operators of Long Term Evolution (LTE) networks is mitigating inter-cell interference problems. Inter-Cell Interference Coordination (ICIC) techniques are proposed to reduce performance degradation and to maximize system capacity. It is a joint resource allocation and power allocation problem that aims at controlling the trade-off between resource efficiency and user fairness. Traditional interference mitigation techniques are Fractional Frequency Reuse (FFR) and Soft Frequency Reuse (SFR). FFR statically divides the available spectrum into reuse-1 and reuse-3 portions in order to protect cell-edge users, while SFR reduces downlink transmission power allocated for cell-center resources to protect vulnerable users in the neighboring cells. However, these static techniques are not adapted to non-uniform user distribution scenarios, and they do not provide guarantees on throughput fairness between user equipments. In this paper, we introduce a non-cooperative dynamic ICIC technique that dynamically adjusts resource block allocation according to user demands in each zone. We investigate the impact of this technique on throughput distribution and user fairness under non-uniform user distributions, using an LTE downlink system level simulator. Simulation results show that the proposed technique improves system capacity, and increases throughput fairness in comparison with reuse-1 model, FFR and SFR. It does not require any cooperation between base stations of the LTE network.

I. INTRODUCTION

The exponential increase in the number of mobile users, the proliferation of mobile applications and the huge demand for data in mobile networks have led to the introduction of new radio access technologies such as Long Term Evolution (LTE) [1] of Universal Mobile Terrestrial Radio Access System (UMTS). However, wireless communications are characterized by the scarcity of frequency resources required to increase network capacity and to satisfy throughput demands.

3 rd Generation Partnership Project (3GPP) LTE chooses frequency reuse-1 model to make maximum use of the available spectrum. Although Orthogonal Frequency Division Multiple Access (OFDMA) technique [START_REF] Schulze | OFDM[END_REF] eliminates intra-cell interference, due to the orthogonality between subcarriers, Inter-Cell Interference (ICI) problems always exist. They have a negative impact on system performance, since they reduce network capacity and User Equipment (UE) throughput, especially for cell-edge UEs located at the border of the cell and receiving high power interfering signals.

Inter-Cell Interference Coordination (ICIC) techniques are required to improve system performance. Some techniques perform ICI mitigation by adjusting resource allocation between cell zones, or between LTE base stations called evolved-NodeBs (eNodeBs). Other ICIC techniques use power allocation strategies to reduce interference in a non-cooperative manner [START_REF] Yassin | A Downlink Power Control Heuristic Algorithm for LTE Networks[END_REF], or in a coordinated manner, when eNodeBs cooperate together in order to perform power allocation. ICIC is also seen as a joint resource and power allocation problem, where the objective is to find the optimal resource and power distribution. Fractional Frequency Reuse (FFR) [START_REF] Kwan | A Survey of Scheduling and Interference Mitigation in LTE[END_REF] is a static ICIC technique, where restrictions on Resource Blocks (RBs) usage are imposed to protect cell-edge UEs. Another static ICIC technique is Soft Frequency Reuse (SFR); it controls the downlink transmission power allocated for each set of RBs. FFR and SFR are considered as reference state-ofthe-art techniques for research on ICI mitigation.

In this paper, we introduce a non-cooperative dynamic ICIC technique that does not require any cooperation between LTE base stations. Our aim is to improve system performance, and to increase cell-edge UEs throughput. Resource and power allocation decisions are made locally by the scheduler of each eNodeB, in order to respond dynamically to throughput demands in each zone. An LTE downlink system level simulator developed by Vienna University of Technology is chosen to simulate our technique, and to compare its performance to that of reuse-1 model, FFR and SFR schemes. Simulation results show that our proposed technique reduces ICI, improves cell-edge UEs performance and increases throughput fairness among all the existing UEs in the network.

The rest of the paper is organized as follows: in section II, we describe the existing ICIC techniques. Section III presents system model. In section IV, we introduce and explain our proposed ICIC technique. Section V gives information about simulation environment, and simulation results are reported in section VI. Concluding remarks are given in section VII.

II. RELATED WORK

The scarcity of radio resources is becoming more severe with the huge increase in data demands, and with the proliferation of mobile applications. LTE network operators are committed to reduce ICI, and to improve cell-edge users performance, while keeping spectral efficiency at an acceptable level. Another concern that we particularly address in this paper is increasing throughput fairness among all the existing users. Interference mitigation in Global System for Mobile communications (GSM) networks is achieved through the cellular concept [START_REF] Macdonald | The Cellular Concept[END_REF] as well as frequency planning [START_REF] Pasquale | Optimizing Frequency Planning in the GSM System[END_REF]. Within a cluster of N cells, reuse-N model specifies the number of frequency resources to be used in each cell. However, network capacity and spectral efficiency are reduced.

Classic ICIC techniques for LTE networks divide each cell into two zones: a cell-center zone containing UEs close to the serving eNodeB, and a cell-edge zone containing UEs located at the edge of the cell. FFR and SFR are frequency reuse-based static ICIC techniques. FFR [START_REF] Hassan | Optimal Fractional Frequency Reuse (FFR) and Resource Allocation in Multiuser OFDMA System[END_REF] reduces ICI by creating restrictions on RB usage within each cell. The cell-center zone is allowed to use half of the available spectrum according to reuse-1 model. However, reuse-N model is used to allocate the remaining bandwidth among adjacent cell-edge zones. UEs having low Signal-to-Interference and Noise Ratio (SINR) are protected, since they do not receive interfering signals from their neighboring cells. For SFR [START_REF] Umeda | Efficient Adaptive Frequency Partitioning in OFDMA Downlink with Fractional Frequency Reuse[END_REF], ICIC is seen as a joint RB and power allocation problem. Reuse-1 model is maintained; however, cell-edge zones in adjacent cells are allocated disjoint portions of the available spectrum to avoid ICI. In addition, the remaining bandwidth allocated for cell-center zones is used at a lower transmission power. The basic principles for FFR and SFR techniques are illustrated in Fig. 1a and Fig. 1b respectively. LTE architecture does not include a centralized entity that controls the entire network. However, neighboring base stations exchange information about RB usage over X2 interface. Coordinated ICIC techniques make use of the cooperation between adjacent eNodeBs to adjust RB distribution, power allocation, or both according to radio conditions, user demands, or network load. For instance, power allocation is dynamically adjusted according to SINR for each RB in [START_REF] Aboulhassan | Novel Resource Allocation Algorithm for Improving Reuse One Scheme Performance in LTE Networks[END_REF]. Note that this cooperation creates an additional signaling traffic that is exchanged between adjacent eNodeBs, and it increases computational complexity in comparison with non-cooperative techniques.

The introduction of LTE-Advanced (LTE-A) architecture and Coordinated Multi-Point (CoMP) transmission and reception techniques [START_REF] Sawahashi | Coordinated Multipoint Transmission/Reception Techniques for LTE-Advanced[END_REF] gave birth to a new category of ICIC techniques. These techniques are qualified as centralized, since there is a central entity that sets restrictions on RB usage within multiple network cells. A centralized downlink ICIC technique is introduced in [START_REF] Rahman | Enhancing Cell-Edge Performance: A Downlink Dynamic Interference Avoidance Scheme with Inter-Cell Coordination[END_REF] where each cell defines RB restrictions for each of its dominant interferer neighbors. A restriction list is sent to the central entity, that makes RB restriction decisions and sends them to network schedulers. The different classes of ICIC techniques are summarized in Table I.

III. SYSTEM MODEL

OFDMA is the multiple access technique chosen for the downlink of the radio interface in LTE. The spectrum is divided into several channels, where each channel consists of a number of consecutive orthogonal subcarriers. One RB consists of 12 consecutive subcarriers in the frequency The geographical partitioning of each cell into cellcenter and cell-edge zones assumes that cell-center UEs are characterized by high SINR values due to their proximity to the transmitting antennas, and that cell-edge UEs have lower SINR levels since they are far from the serving base station. However, we might have cell-center UEs suffering from interference problems, as well as cell-edge UEs with good radio conditions due to fast fading and other propagation issues. This approach also requires the knowledge of the exact position of each active UE. For these reasons, we perform UE classification according to mean wideband SINR values. An SINR threshold (SIN R th ) is defined to classify UEs: when mean SINR of a UE is higher than the predefined SIN R th , it is considered as a Good Radio (GR) conditions UE; otherwise, it is considered as a Bad Radio (BR) conditions UE. GR UEs are commonly taken for as cell-center UEs, and BR UEs as cell-edge UEs. Our classification is more accurate than the traditional approach, and it does not require any localization information. SINR of a UE k attached to cell i and allocated RB n is given by:

SIN R i k, n = P i n • G i k, n j =i P j n • G j k, n + P T N (1)
Where P i n is the downlink transmission power allocated by cell i for the RB n, G i k, n is channel gain for UE k served by eNodeB i on RB n, and P T N is the thermal noise power on the considered RB. Indexes i and j refer to useful and interfering signals respectively.

IV. NON-COOPERATIVE ICIC TECHNIQUE

The main disadvantage of FFR, SFR and other stateof-the-art techniques is that the available spectrum is statically allocated for cell-center and cell-edge zones. These resource allocation policies are not adapted for non-uniform UE distributions between cell zones. We might improve BR UEs throughput, while causing system performance degradation. We introduce a non-cooperative dynamic ICIC technique for multiuser OFDMA networks such as LTE. Originally, RB and power allocation for GR and BR zones are performed according to the SFR scheme. Periodic interventions are made by the scheduler of each eNodeB in a distributed manner to find out whether Keep the same RB distribution 14: end if GR or BR UEs are unsatisfied. Intervention period is 25 ms, so that the scheduler has enough time to receive UE feedbacks, and to calculate mean throughput for each zone. The proposed technique is illustrated in Fig. 2.

: CQI n (t) = K k=1 CQI k n (t) K 5: CQI n (t) = γ × CQI n (t-1) + (1 -γ) × CQI n (
In LTE, Channel Quality Indication (CQI) feedbacks [14] sent from UE to eNodeB reflect the level of SINR for a certain frequency band. We use narrowband CQI reports to classify RBs in each zone: the higher the mean received CQI value is, the better the RB is. CQI k n denotes CQI feedback sent by UE k for RB n, and CQI n is the mean CQI value for RB n over all UEs. In addition, we define a throughput threshold value ∆ th to decide when UEs of a cell zone are unsatisfied. Algorithm 1 shows how our ICIC technique operates at each scheduler.

Our objective is to increase throughput fairness among all the existing UEs. When mean throughput per GR UE (R GR ) exceeds by ∆ th the mean BR throughput (R BR ), the BR zone selects the best RB from GR zone and borrows it to improve BR UEs performance. Reciprocally, when R BR exceeds by ∆ th the mean GR throughput, the worst BR RB (having the lowest mean CQI value) is borrowed by the GR zone. Power allocation mask is kept the same, so that we do not create additional ICI problems when modifying RB allocation. ∆ th is used to specify the tolerated throughput difference between cell zones. It is a tuning parameter that could be adjusted by network operator according to quality of service requirements. Our technique operates locally independently of the scheduler type, and it does not require any additional signaling messages to be exchanged between eNodeBs.

V. SIMULATION PARAMETERS

A MATLAB-based LTE downlink system level simulator [START_REF] Ikuno | System Level Simulation of LTE Networks[END_REF] developed by Vienna University of Technology is chosen as the simulation platform. It allows to investigate several system-level issues such as resource allocation and ICI mitigation. We consider an LTE network of several adjacent hexagonal cells. Each cell is equipped with 120 • directional transmit antennas with an azimuth offset of 30 • . The original version of the simulator includes frequency reuse-1 model as well as FFR technique. However, homogeneous power allocation is considered.

We integrated SFR scheme within the simulator, and we adjusted the power allocation scheme so that the power mask can be modified. Finally, we integrated our proposed distributed ICIC technique. The simulated network includes several adjacent hexagonal LTE cells, with a 5 MHz operating bandwidth. Since the total bandwidth per RB equals 180 kHz, we have 25 RBs available in each cell. SINR threshold is used to classify active UEs into GR and BR UEs. Its value is chosen in order to have indirect control of the percentage of GR UEs when UE positions are generated. Traffic model is full buffer; thus, Fig. 2: Distributed ICIC technique the available spectrum is permanently used to serve UEs once they exist in the network. Simulation parameters are given in Table II. We use Jain's fairness index [START_REF] Jain | A Quantitative Measure of Fairness and Discrimination for Resource Allocation and Shared Computer System[END_REF] to measure throughput fairness among all UEs existing in the network. It is given by:

J(x 1 , x 2 , ..., x n ) = ( n i=1 x i ) 2 n • n i=1 x 2 i (2)
Where J rates the fairness of a set of throughput values; n is the number of UEs, and x i is the throughput of UE i. Jain's fairness index ranges from 1 n (worst case) to 1 (best case). It reaches its maximum value when all UEs receive the same throughput.

VI. SIMULATION RESULTS

A. Impact on system throughput

First, we study the impact of our ICIC technique on GR and BR throughputs. We simulate a network of seven LTE cells with 10 UEs randomly placed in each cell. The throughput difference used to decide whether to borrow RBs or not equals 512 kbit/s. Simulation time equals 1000 TTIs, and interventions occur every 25 TTIs to adjust RB allocation between cell zones. Mean throughput for GR and BR zones as well as throughput difference are reported in Fig. 3.

During the first 100 TTIs of simulation time, throughput difference between GR and BR zones exceeds the predefined threshold (512 kbit/s). ICIC algorithm implemented at the scheduler decides to borrow RBs characterized by the highest CQI feedback values from GR to BR zone. Throughput difference is therefore reduced, and we improve quality of service for BR UEs. Our algorithm succeeds in increasing throughput fairness among all UEs existing in the network. ∆ th value allows to define the tolerated throughput difference between GR and BR zones.

B. Throughput cumulative distribution function

The simulated network consists of seven adjacent LTE cells with 10 UEs randomly placed in each cell. Simulation For a given throughput value, CDF represents the probability to find a UE served with a low throughput. The lower the CDF is, the better the quality of service is. We notice that our proposed ICIC technique shows a lower CDF for throughput values less than 2 Mbit/s, when compared to FFR and SFR schemes. It also shows the lowest CDF for throughput values less than 1 Mbit/s in comparison with reuse-1 model. Consequently, the number of UEs suffering of bad quality of service is reduced. For throughputs higher than 2 Mbit/s, FFR and SFR are slightly better than the distributed ICIC technique. In fact, our objective is to increase throughput fairness among all UEs by reducing the number of RBs allocated for UEs having relatively high throughputs, in order to improve quality of service for vulnerable UEs.

C. Jain's fairness index

We simulate the same scenario as in the previous subsection. The scheduler used for GR and BR zones is round robin, and ICIC algorithm interventions occur periodically (every 25 TTIs). We generate UEs with uniform and nonuniform distributions between GR and BR zones. For each user distribution, simulations are repeated 30 times, and results are shown in Fig. 5. Simulation results reported in Fig. 5 show that FFR and SFR techniques have a lower fairness index in comparison with frequency reuse-1 model, especially for nonuniform user distributions between cell zones. In fact, static RB classification is not adapted for situations where the majority of UEs served by an eNodeB are located in only one of the two zones. FFR improves throughput fairness in comparison with reuse-1 model only when RB classification between GR and BR zones matches UE demands in these zones. SFR's fairness index is lower than that of FFR, since RBs used in the BR zone are permanently used by GR UEs of the neighboring cells. However, it becomes better when the majority of UEs are GR UEs, since FFR provides less RBs for GR UEs. Our proposed non-cooperative dynamic ICIC technique adjusts RB allocation according to UE demands, without any cooperation between eNodeBs. Therefore, it shows the highest Jain's fairness index compared to SFR, FFR and frequency reuse-1 model.

VII. CONCLUSION

Due to the scarcity of frequency resources required to increase network capacity and to satisfy throughput demands, frequency reuse-1 model is proposed for LTE networks. However, inter-cell interference problems have a negative impact on UE throughput and system performance. Several ICIC techniques are introduced to reduce interference, and to improve quality of service for BR UEs. For instance, FFR adjusts RB distribution between cell zones, while SFR considers ICIC as a joint RB and power allocation problem.

In this paper, we introduced a non-cooperative ICIC technique that dynamically adjusts RB allocation according to UE demands in each zone. It operates locally independently of the scheduler type, and it does not require any cooperation between network base stations. Simulation results show that the proposed ICIC technique reduces throughput difference between cell zones, and improves quality of service for vulnerable UEs. It increases throughput fairness among all UEs existing in the network under uniform and non-uniform user distributions.

Fig. 1 :

 1 Fig. 1: FFR and SFR schemes

4

 4

Fig. 3 :

 3 Fig. 3: GR and BR throughputs with time

Fig. 4 :

 4 Fig. 4: Throughput cumulative distribution function

Fig. 5 :

 5 Fig. 5: Fairness index with percentage of GR users

TABLE I :

 I ICIC Techniques Classification

	ICIC Class	Description	Example
	Frequency reuse	Static RB and power allocation	FFR, SFR
	Non-cooperative	RB and power allocation without cooperation	Techniques in [12, 13]
	Coordinated	Cooperation between neighboring eNodeBs	Using X2 interface
	Centralized	Central control entity	LTE-A architecture
	domain, and 7 OFDM symbols (respectively 6) in the
	case of normal cyclic prefix (respectively extended cyclic
	prefix) in the time domain. The scheduling period in LTE
	networks is called Transmit Time Interval (TTI = 1 ms),
	and the smallest scheduling unit is the RB.

  Borrow best RB from GR to BR zone10: else if (R BR -R GR > ∆ th ) then

		t)
	6: end for
	7: Every 25 TTIs:
	9:
	11:	Borrow worst RB from BR to GR zone
	12: else
	13:

8: if (R GR -R BR > ∆ th ) then

TABLE II :

 II Simulation Parameters

	Parameter	Value	Description
	Inter-eNodeB distance	500 m	Urban area
	Operating bandwidth	5 MHz	--
	Number of RBs (N )	25	In the 5 MHz bandwidth
	TTI	1 ms	Transmit Time Interval
	Pathloss model	TS 25.814	Same as in HSDPA
	Feedback delay	3 ms	3 TTIs
	Scheduler	Round Robin	--
	Traffic model	Full buffer	--
	eNodeB max. power (P )	20 W	43 dBm
	Max. RB power (P RB )	0.8 W	P N
	SINR threshold	3	UE classification
	SFR power ratio (α)	0.25	P GR = P RB 4
	∆ th	512 kbit/s	UE satisfaction


Mohamad Yassin

Samer Lahoud

Marc Ibrahim

Kinda Khawam

A Downlink Power Control Heuristic Algorithm for LTE Networks

Keywords: Long Term Evolution, OFDMA, power control, resource block, Inter-Cell Interference, CQI

The recent development of mobile terminals, the proliferation of mobile applications and the increasing need for mobile data have led to a dense deployment of mobile networks. In this context, the Long Term Evolution (LTE) standard is adopted by a large number of mobile network operators. LTE uses Orthogonal Frequency Division Multiple Access (OFDMA) technique on the downlink of the radio interface along with frequency reuse-1 model. However, Inter-Cell Interference (ICI) and system power consumption will cause limitations in terms of mean user throughput and system performance. Indeed, several recent works focus on the minimization of ICI and power consumption in multi-user OFDMA networks. In this paper, we propose a distributed heuristic power control algorithm that aims at minimizing the total downlink power of an LTE system. We also study the impact of the power control algorithm on ICI and system performance. Simulation results show that the proposed algorithm largely reduces the downlink power consumption without degrading system performance. In addition, it increases the mean throughput for cell-edge users that are mainly affected by ICI problems.

I. INTRODUCTION

Power control mechanism in Long Term Evolution (LTE) [START_REF]Physical Layer Aspects for Evolved Universal Terrestrial Radio Access (UTRA) (Release 7)[END_REF] adjusts the transmission power on the different frequency resources to improve system performance. Power allocation varies from cell to cell. It can also be device specific, and it has an impact on data throughput, which affects user experience. In fact, downlink transmission power is an important issue in distributed networks with dense frequency reuse schemes. The excessive use of downlink power causes interference problems to the neighboring cells. Thus, it degrades system performance, and lowers spectrum profitability. Several works have proposed power control algorithms in multiuser Orthogonal Frequency Division Multiple Access (OFDMA) [START_REF] Prasad | Performance Analysis of OFDMA in LTE[END_REF] systems such as LTE. For instance, authors in [START_REF] Bisaglia | Resource Allocation and Power Control in a TDD OFDM-Based System for 4G Cellular Networks[END_REF] investigate a power control mechanism that tries to minimize the requested power under user rate constraints in a noncollaborative system. In addition, [START_REF] Al-Kebsi | The Impact of Modulation Adaptation and Power Control on PAPR Clipping Technique in OFDM of 4G Systems[END_REF] introduces a hybrid algorithm that combines adaptive modulation along with power control (i.e., power is increased when the order of the modulation scheme is reduced and vice versa).

The transmission power allocated for frequency resources on the downlink affects mobile users in the neighboring cells. In fact, the 3GPP LTE [START_REF]Physical Layer Aspects for Evolved Universal Terrestrial Radio Access (UTRA) (Release 7)[END_REF] is introduced to provide higher bitrates for the mobile users and to improve the quality of service for both Downlink and Uplink traffic. It is characterized by the full frequency bandwidth reuse in all the system cells (frequency reuse factor = 1). Therefore, Inter-Cell Interference (ICI) problem arises due to the usage of the same frequency resources during the same time interval in adjacent cells. ICI decreases the SINR of the users and consequently the total system bit rate. Each cell is divided into two zones: cell-center zone (containing users that are close to the base station) and cell-edge zone (containing users that are close to the edge of the cell). Cell-edge users are more affected by ICI than cell-center users (i.e., they have lower SINR) due to propagation loss and due to their proximity to the neighboring interfering cells. For example, authors in [START_REF] Schoenen | Adaptive Power Control for 4G OFDMA Systems on Frequency Selective Fading Channels[END_REF] noticed that high transmission power for cell-center users in 4G systems degrades the performance of cell-edge users in the neighboring cells. Therefore, an Adaptive Power Control (APC) is proposed to reduce ICI.

A distributed power control is used in [START_REF] Kasparick | Autonomous Distributed Power Control Algorithms for Interference Mitigation in Multi-Antenna Cellular Networks[END_REF] to reduce ICI especially when there is a lack of cooperation between base stations. Power control mechanisms were already proposed in [START_REF] Stolyar | Self-Organizing Dynamic Fractional Frequency Reuse for Best-Effort Traffic Through Distributed Inter-Cell Coordination[END_REF] and [START_REF] Wunder | Self-Organizing Distributed Inter-Cell Beam Coordination in Cellular Networks with Best Effort Traffic[END_REF] where they lead to dynamic Soft Frequency Reuse schemes. In addition, authors in [START_REF] Wong | Multiuser OFDM with Adaptive Subcarrier, Bit and Power Allocation[END_REF][START_REF] Choi | Adaptive Subcarrier, Bit and Power Allocation Algorithm for MIMO-OFDMA System[END_REF] have proposed dynamic subcarrier, bit and power allocation for multiuser OFDM systems. In fact, frequency resources, modulation order and transmission power are dynamically assigned depending on channel conditions. This flexibility allows to reduce the transmission power while guaranteeing the required bitrates and bit error rate for all the users. Therefore, it is more advantageous than static access schemes such as time division multiple access and frequency division multiple access.

ICI mitigation is also the objective in [START_REF] Fodor | On the Impact of Uplink Scheduling on InterCell Interference Variation in MIMO OFDM Systems[END_REF], where the authors use a proportional fair scheduler along with an open loop power control to reduce ICI on the uplink of multi-user OFDMA system. Their objective is to reduce SINR variation in order to increase average user throughput and cell-edge user throughput on the uplink. Power is allocated to each user depending on the number of used Resource Blocks (RB), on cell-specific characteristics and on path loss parameters. Authors in [START_REF] Pischella | Power Control in Distributed Cooperative OFDMA Cellular Networks[END_REF] introduce a cooperation between base stations of the OFDMA cellular systems in order to couple the resources allocated by the source and relay base stations for a mobile user. They also propose a distributed power control mechanism that operates independently of the information received from neighboring cells. The influence of cooperation and power control is tested over four different schedulers.

In LTE, interference mainly affects cell-edge users that receive high interfering signals due to their proximity to the neighboring cells. In this context, we propose a distributed heuristic power control algorithm that operates along with the scheduler deployed at the base station of an LTE network. The objective of this algorithm is to avoid power wastage especially for cell-center users (that are close to the base stations) or users having good radio conditions. In fact, resources are allocated to mobile users on a timescale of one Transmit Time Interval (TTI = 1ms). Numerous scheduling techniques can be used together with our power control algorithm such as Proportional Fair, Round Robin, Maximum SNR and many others. Particularly, we introduce a heuristic power control algorithm that computes power allocation to the different resource blocks once they are allocated to the users.

We tackle the power allocation problem as a method to reduce ICI in multi-user OFDMA systems such as LTE. In fact, LTE system is based on the reuse-1 model where all the available frequency resources are used in the adjacent cells during the same time interval. Power control does not only reduce the power levels of interfering signals (signals usually belonging to cell-center users), but it can also increase the power levels on resource blocks that suffer of bad radio conditions (usually RB allocated for celledge users). Therefore, it is considered as a method for Inter-Cell Interference Coordination (ICIC) [START_REF] Daeinabi | Survey of Inter-Cell Interference Mitigation Techniques in LTE Downlink Networks[END_REF].

The rest of the paper is organized as follows: the downlink power control in LTE is described in section (II). We explain the proposed power control heuristic algorithm in section (III). Simulation environment is described in section (IV), and simulation results are reported in section (V). Conclusion is given in section (VI).

II. DOWNLINK POWER CONTROL IN LTE

In LTE, downlink Reference Signal (RS) exists only in the physical layer. There are two types of RS: cell specific and user specific. The former is transmitted every subframe, and it spans the operating bandwidth. However, the latter is only transmitted within the resource blocks allocated to a specific UE. RSs are inserted every 6 subcarriers in the frequency domain [START_REF] Roessler | Understanding Downlink Power Allocation in LTE[END_REF]. Each RB contains 2 RSs in the first and the fifth OFDM symbols.

The power level for the RS is signaled within system information to the device. It is cell-specific [15], and it ranges between -60dBm and +50dBm per 15kHz. It is a requirement that the LTE base station transmits all reference signals with constant power over the entire bandwidth. RS is an important element for downlink power allocation, which can be done on a 1ms basis. In fact, it delivers the reference point for the downlink power.

Traditionally, downlink power control for LTE femtocells is performed according to distance-based algorithms. The femtocell pilot power is configured such that it is received on average with equal strength as the pilot power received from the strongest macrocell at a defined target cell radius, subject to its maximum downlink power [START_REF] Claussen | Self-Optimization of Coverage for Femtocell Deployments[END_REF]. LTE femtocell transmission power is estimated using path-loss models. In measurement-based techniques, the received power is measured using the built-in measurement capability of the femtocell. Then, the downlink pilot power is calculated.

Authors in [START_REF] Xu | Autonomous Downlink Power Control for LTE Femtocells Based on Channel Quality Indicator[END_REF] propose a downlink power control algorithm based on Channel Quality Indication (CQI). Users with different types of service are studied. The Algorithm 1 Downlink power control end if 8: end for downlink power allocated for all the RBs is initialized with the minimum transmission power. For voice-over-IP and data users, transmission power is increased until their data requirements are met. However, an additional offset throughput is set upon the minimum data rate requirement for web users. The base station stops increasing the transmission power when the additional throughput is achieved.

III. HEURISTIC POWER CONTROL ALGORITHM

We propose a distributed heuristic power control algorithm that aims at reducing power wastage especially for cell-center users that usually have good radio conditions. Moreover, this algorithm increases downlink transmission power on the RB allocated for cell-edge users that suffer the most from interference and path loss problems. The downlink channel dependant scheduling in LTE requires specific information to be sent by the terminals to the network. Such information is transmitted through Channel State Reports that contain CQI feedback. CQI represents the highest modulation and coding scheme that guarantee a block error rate less than 10% for physical downlink shared channel transmissions [START_REF] Dahlman | Power Control, Scheduling, and Interference Handling," in 4G LTE/LTE-Advanced for Mobile Broadband[END_REF]. There are two types of channel state reports in LTE: either periodic configured by the network to be delivered with a certain periodicity or aperiodic delivered when explicitly requested by the network. Moreover, several reporting modes are supported. For example, wideband CQI feedbacks reflect the average channel quality across the entire cell bandwidth while configured reports require the transmission of one CQI per configured sub-band.

We use the CQI feedbacks as an entry to solve the downlink power allocation problem. In this context, we propose a distributed heuristic downlink power control algorithm that computes downlink power allocation on the different resources according to CQI values received from the users. The heuristic power control algorithm is compared with the scenario where no power control is applied i.e., the scheduler allocates permanently the maximum downlink transmission power for each RB.

In the proposed algorithm, the scheduler of each base station (e.g. Proportional Fair, Round Robin or Best CQI) performs both RB and power allocation each TTI, independantly of the other base stations in the network. The idea behind this algorithm is to minimize downlink transmission power without degrading the performance of cell-center users. In addition, transmission power allocated for RBs having low SINR values is increased in order to guarantee a better usage of the available spectrum. If the received CQI feedback is higher than a predefined CQI threshold , downlink transmission power is decremented as long as it is higher than a predefined minimum power P min . However, if CQI feedback is lower than CQI threshold , downlink transmission power is incremented as long as it is lower than the maximum allowed transmission power P max , where:

P max =
Sector M aximum Downlink P ower N umber of Available RB

The minimum transmission power P min is a predefined parameter that guarantees an acceptable bitrate over the considered RB. CQI threshold can take one of fifteen possible integer values since 1 ≤ CQI ≤ 15. When CQI threshold = 15 the power control algorithm allocates the maximum downlink transmission power for each RB. Moreover, P max (given in ( 1)) guarantees that the maximum transmission power allocated for all the RB is always less than or equals the maximum transmission power of the cell.

Algorithm 1 describes how downlink transmission power attributed for each RB is adjusted by the scheduler according to the last received CQI feedback. The objective of this algorithm is to avoid power wastage especially for cell-center users (that are close to the base stations) or users having good radio conditions (high CQI values). In addition, this power decrease will reduce the ICI that mainly affects the cell-edge users of the neighboring cells. We also improve the throughput of cell-edge users by increasing their downlink transmission power.

IV. SIMULATION ENVIRONMENT

In order to study the performance of our power control algorithm, several LTE system level simulations are performed. Comparison parameters are: downlink transmission power and mean user throughput. We simulate our proposed algorithm using a MATLAB LTE system level simulator [START_REF] Ikuno | System Level Simulation of LTE Networks[END_REF], and we compare its performance with other power control strategies.

We adapted the chosen LTE system level simulator before performing the required simulations. In fact, SINR calculations were made assuming the maximum downlink transmission power is attributed permanently for all the RB. In addition, no downlink power control mechanism was implemented. Therefore, we implemented the MAT-LAB open-source code of the simulator to ensure that the effective power levels on each RB are taken into account when calculating SINR levels. We choose the proportional fair scheduler to implement the proposed power control algorithm. Note that this algorithm can also be integrated within other schedulers such as round robin or best CQI. Simulation parameters for the simulated LTE system and the power control algorithm are summarized in Table I.

V. SIMULATION RESULTS

A. Downlink transmission power

First, we simulated a basic scenario where an LTE system contains one cell-center user and one cell-edge user in two adjacent cells. The objective of the simulation is to show the impact of our proposed heuristic algorithm on the downlink transmission power for cell-center and cell-edge users. The simulated LTE system bandwidth is 5M Hz; therefore, we have 25 RBs available in each cell. Simulation time equals 500 TTI which is equivalent to 500ms, and CQI feedback reception at the base station is delayed by 3 TTI. The proportional fair scheduler is used, and it allocates the RBs based on the full buffer model. We report the variation of the total downlink transmission power allocated for each user along with time. These variations are illustrated in Fig. 1.

Fig. 1. Downlink transmission power variation with time

According to Fig. 1, we notice that the downlink transmission power allocated for the cell-center user is reduced after receiving the CQI feedback at the serving base station. In fact, the power control algorithm implemented at the scheduler will decrease the downlink power attributed for the RBs allocated to this user since the received CQI is higher than the predefined CQI threshold . Downlink power is decreased as long as the CQI received is higher than the predefined threshold until it reaches the minimum downlink transmission power. However, when the received CQI value is lower than the CQI threshold , downlink transmission power allocated for the corresponding RB is increased. The oscillations observed for the downlink transmission power of the cell-edge user are due to successive increase and decrease in the downlink transmission power. Indeed, when the received CQI becomes higher than the threshold, the downlink transmission power is decreased. Consequently, this power reduction will affect the current CQI value that will also decrease. Thus, downlink power value along with time is subject to oscillations that are ruled by the received CQI feedback values. To reduce these oscillations, we may define a CQI interval around CQI threshold where we tolerate CQI variations (i.e., we do not modify the allocated downlink power).

B. System performance

We simulate an LTE system having 7 hexagonal adjacent cells and 10 mobile users in each cell. Therefore, the simulated LTE system contains 70 mobile stations. The scheduler used at each base station is Proportional Fair, and it allocates resources for the users with a period of one TTI. The heuristic power control algorithm adjusts downlink transmission power according to the CQI feedback received from the mobiles. This simulation is repeated 100 times. Downlink parameters of our simulated system, such as mean throughput per user and power consumption, are averaged over the 100 simulations. Path loss and radio conditions are randomly generated. Users in the central cell of the system are cell-edge users; however, all other users are cell-center users. Simulation results are reported in Fig. 2 to 5. In Fig. 2, we report the mean throughput per user with and without the power control heuristic algorithm. It shows that the system performance remains approximately the same when using our proposed algorithm. In fact, downlink transmission power for resource blocks having good Fig. 3. Cell-center users mean throughput with time radio conditions (CQI relatively high) is decreased; however, the transmission power allocated to RB characterized by low CQI feedback values is increased to compensate throughput loss that might occur due to propagation loss and ICI problems. Globally, the mean throughput per user remains approximately the same for the entire system. Fig. 3 shows the impact of downlink transmission power adjustments on cell-center users. Such mobile users are geographically close to the base station. Thus, they are characterized by a lower path loss. In addition, cell-center users are less affected by inter-cell interference since the interfering signals will experience important degradation before reaching the mobile. For these reasons, resources allocated for cell-center users show relatively high CQI feedback values. Hence, downlink power allocated for these resources is decreased, and the mean throughput per cell-center user is slightly reduced. Simulation results reported in Fig. 4 show the average throughput per cell-edge user with and without the power control heuristic. We notice that the mean throughput is increased for cell-edge users when applying the proposed algorithm. In fact, these users suffer the most from intercell interference problems for two reasons: first, their signal path loss is important; second, they are highly affected by interfering signals transmitted by the neighboring cells.

Since resource blocks allocated for cell-edge users are characterized by low CQI values (due to propagation loss and inter-cell interference), the power allocated for these RB is incremented. Thus, the signal loss caused by free space propagation is reduced. Furthermore, resource blocks used by cell-center users in the neighboring cells have higher CQI values. According to our power control algorithm, the downlink power allocated to these RB is decreased. Hence, we reduce the amount of inter-cell interference that mainly affects cell-edge users. These power adjustments will improve the channel quality for edge users allowing them to get higher throughputs and better performance. Curves in Fig. 5 show that the power control heuristic largely reduces the total downlink transmission power. Indeed, the LTE system consumes only 25W instead of 70W (which is the maximum power consumption for an LTE system having 7 cells transmitting at 10W each). Thus, the total downlink transmission power is reduced by 64%. Joining these results along with the results obtained in Fig. 2 (system performance is approximately the same) allows us to exhibit the main advantage of our algorithm: we avoid power wastage without degrading system performance. Moreover, cell-edge users mean throughput is improved when using the power control algorithm.

Note that in the special case where all the users are located at the edge of the cells, downlink transmission power is increased (as long as the received CQI feedback is lower than the predefined CQI threshold) until we reach the maximum transmission power. Thus, user position has an important impact on power consumption.

VI. CONCLUSION

Multiuser OFDMA systems, such as LTE, are becoming widely deployed to fulfill the increasing demand for mobile data. However, inter-cell interference problem arises due to the dense reuse of frequency resources. Cell-edge users are mainly affected by ICI due to their proximity to the neighboring cells. Thus, we proposed a distributed power control heuristic algorithm that can be implemented within several schedulers such as proportional fair, round robin or best CQI.

The proposed algorithm adjusts the downlink power allocated for each RB according to the received CQI feed-back. Simulation results show that the main advantage of the algorithm is reducing power wastage without degrading system performance. It also improves the performance of cell-edge users by decreasing the transmission power on resource blocks allocated to cell-center users since they are the main source of interference between neighboring cells.

Fig. 2 .

 2 Fig. 2. Mean throughput per user with time

Fig. 4 .

 4 Fig. 4. Cell-edge users mean throughput with time

Fig. 5 .

 5 Fig. 5. System power consumption with time

1 :

 1 Each user sends CQI feedback about all available RBs to the serving base station 2: for each RB ∈ RB pool do 3: if ((CQI < CQI threshold ) and (P t-T T I < P max perRB )) then P t ← P t-T T I + P ower Control Step

	4:
	5:	else if ((CQI > CQI threshold ) and (P t-T T I >
		P min perRB )) then
	6:	P t ← P t-T T I -P ower Control Step
	7:


Mohamad Yassin

Marc Ibrahim

Samer Lahoud

A Hybrid Approach for RAT Selection in Wireless Heterogeneous Networks

Keywords: RAT selection, heterogeneous wireless network, hybrid approach, radio resources I

Given the diversity and heterogeneity of the existing wireless radio access technologies, a key problem that arises in this context is the radio access network selection. The purpose of radio access network selection is to improve the quality of service experienced by clients and optimize the use of available radio resources. In this context, two approaches can be adopted: either a centralized or a distributed approach. In this paper, based on the two existing radio network selection approaches, we propose a new hybrid approach that combines the distributed one with periodic centralized interventions. We introduce the concept of the proposed hybrid approach, and study its performance. In the hybrid approach, client inbound sessions perform a distributed network selection while the system regularly intervenes to change client associations and/or to modulate the distributed algorithm in order to guarantee an acceptable level of performance. The results show that the hybrid approach performance is close to that of the optimal solution, and is better than that of the distributed approach in specific scenarios. Moreover, the hybrid approach reduces the overall signaling load for the system by decreasing the number of required handovers.

INTRODUCTION

Nowadays, wireless networks have become very common. A wireless network is based on a radio access technology that provides mobile users the ability to access the network while they are moving, and to use the various available services via a wireless connection.

Wireless networks are experiencing a wide diversification and a large-scale deployment. Indeed, in addition to mobile networks (2/3/4G) [START_REF] Lescuyer | Evolved Packet Systems[END_REF] that are deployed on a large scale there are also many broadband technologies of the IEEE family (IEEE 802.11, IEEE 802.16, etc.) [START_REF] Matthew | 11 Wireless Networks: The Definitive Guide[END_REF].

We can benefit from the coexistence of different technologies [START_REF] Labiod | Wi-Fi, Bluetooth, Zigbee and WiMAX[END_REF] to improve the overall system performance, to ensure users' growing demands for resources, to benefit from the available radio resources and to improve the quality of service experienced by clients (in terms of offered bitrates and transmission delays). In this context, a client has the ability to connect to different wireless access networks.

The operator can benefit from this heterogeneity by providing a diversified access via different radio interfaces coexisting in the same geographical area [START_REF] Hossain | Heterogeneous Wireless Access Networks Architectures and Protocols[END_REF].

In this context, two approaches can be adopted: either the centralized optimal approach or the distributed approach. The centralized solution controlled by the operator has the advantage of a better view of the network and the possibility of reaching an optimum set by the operator [START_REF] Coupechoux | Network Controlled Joint Radio Resource Management for Heterogeneous Networks[END_REF]. The second approach is to let the mobile choose its network based on certain quality measures [START_REF] Antoniou | A Game Theoretic Approach for Network Selection[END_REF] or based on game theory techniques [START_REF] Antoniou | 4G Converged Environment: Modeling Network Selection as a Game[END_REF]. This approach is simpler than the previous one, but induces an individual optimality at the expense of an overall network sub-optimality [START_REF] Kumar | User-Network Association in an 802.11 WLAN & 3G UMTS Hybrid Cell: Individual Optimality[END_REF].

Several previous researches have studied the problem of radio access network selection. In [START_REF] Khawam | Individual vs. Global Radio Resource Management in a Hybrid Broadband Network[END_REF] authors developed and searched to minimize a function that gives the overall system cost. Power optimization in heterogeneous networks is the objective in [START_REF] Sabbagh | A Power Efficient RAT Selection Algorithm for Heterogeneous Wireless Networks[END_REF] where authors propose a hybrid power efficient Radio Access Technology (RAT) selection algorithm. In [START_REF] Moety | Distributed Heuristic Algorithms for RAT Selection in Wireless Heterogeneous Networks[END_REF] authors have developed and evaluated four distributed heuristic algorithms approaching the optimal solution.

Based on the previous work, we develop in this paper a dynamic model of the hybrid system where we implement the hybrid approach algorithms. In the centralized approach, the system uses a common radio resource management strategy in order to minimize the overall cost function given for all the users ; however, in the distributed approach each client aims at optimizing its own cost function individually, regardless of the other clients and the system state. The distributed approach is an implementation of the 'Always Best Connected ABC' concept [START_REF] Gazis | Toward a Generic 'Always Best Connected' Capability in Integrated WLAN/UMTS Cellular Mobile Networks (and Beyond)[END_REF]. Always Best Connected (ABC) is to select and always connect to the most appropriate network when multiple networks are available.

In this paper, we define a new hybrid approach that lies between the centralized [START_REF] Coupechoux | Network Controlled Joint Radio Resource Management for Heterogeneous Networks[END_REF] and the distributed [START_REF] Antoniou | A Game Theoretic Approach for Network Selection[END_REF][START_REF] Antoniou | 4G Converged Environment: Modeling Network Selection as a Game[END_REF] approaches. The proposed approach uses a distributed algorithm to associate the client sessions that arrive to the heterogeneous system. Moreover, client associations are periodically adjusted according to a centralized approach. This new approach is studied in different scenarios. The main simulation parameter is the client peak rates perceived from the existing wireless access networks. This paper gives significant arguments to answer the question: in which cases and for what scenarios could the hybrid approach be more advantageous than the two existing approaches used separately?

The rest of the paper is organized as follows: the system model and the cost calculation are described in (II). The different approaches for radio access network selection and the new proposed hybrid approach are presented in (III). The simulated scenarios are described in (IV) and the results are reported in (V). Conclusion is given in section (VI).

II. SYSTEM MODEL AND COST CALCULATION

The studied geographic area could be simultaneously covered by different types of wireless access networks. These heterogeneous technologies have different characteristics such as coverage, offered bitrates, capacity, etc. The main parameter is the peak rate perceived by the users.

We consider a geographic area entirely covered by a primary radio access technology (i.e., it contains a base station belonging to the primary access technology) and partially covered by a secondary radio access technology (i.e., several access points of the secondary radio access technology). Only clients within the coverage areas of the secondary access points can balance their traffic between the two technologies. Fig. 1 shows a geographic area covered by two radio access technologies with clients that can balance their traffic between the two technologies.

Figure 1. Heterogeneous system with hybrid clients [START_REF] Khawam | Individual vs. Global Radio Resource Management in a Hybrid Broadband Network[END_REF] The model used can be applicable to different types of wireless technologies. In each cell, the peak rate of each user depends on its radio conditions. We adopt a simple model that takes into account client peak rates as the main parameter. We consider only the downlink traffic and we neglect the waiting times (i.e., DIFS and SIFS for Wi-Fi). Therefore, the access scheme (i.e., CSMA/CA for Wi-Fi) leads to a fair rate sharing of the channel among the existing users [START_REF] Moety | Distributed Heuristic Algorithms for RAT Selection in Wireless Heterogeneous Networks[END_REF]. All users have the same throughput that is less than the minimum peak rate that a single user can perceive when alone in the cell [START_REF] Heusse | Performance Anomaly of 802.11b[END_REF].

We denote by n the number of clients that will be concerned in radio access technology selection. Those users will be indexed by k = 1, 2, ... n. We denote by h(k) the secondary technology cell (hotspot) in which the client k exists.

The total cost for a client k, defined as the time required for sending a data unit in a heterogeneous environment, will be given by [START_REF] Moety | Distributed Heuristic Algorithms for RAT Selection in Wireless Heterogeneous Networks[END_REF]:

k C = k k h k k k T T θ θ . ) 1 .( ) ( , 0 , + - (1)

Given that

x k T , is the service time required for sending a data unit in a cell x, and θ k is the fraction of time during which the client k is associated to the access point of the secondary technology h(k).

III. RAT SELECTION APPROACHES

A. Existing approaches: description and weakpoints

In the following, we analyze the centralized and the distributed approaches:

In the centralized approach, the system load balances the downlink traffic of every user between the two radio access technologies in a way to privilege the overall system performance. It gives the optimal solution -the best association of the clients to the existing wireless networks -by minimizing the sum of all the individual cost functions given by equation ( 1). Client association occurs each time we have a change in the system state: either a client arrival or a departure. The centralized approach requires knowledge of the peak rates of all the users in the system. The main drawback of this approach is the number of handovers to be carried out for each change in the system state (when an arrival or a departure occurs). In fact, the centralized approach will search for the optimal client association profile after each client arrival/departure. It will then modify client association according to the new profile. This will induce an additional number of handovers to achieve the profile modification. These handovers will generate additional signaling traffic that increases network congestion and degrades its performance.

In the distributed approach, each user selfishly strives to improve its own performance regardless of the other users and the system state. It is a simple approach where each client associates individually to the network that offers him the best peak rate. Client association occurs only once, when the client arrives to the system. The distributed approach requires only the knowledge of the peak rates perceived via the two existing radio access technologies. This approach is characterized by a very low processing load (since the network selection is done in a distributed way). In addition, no handover is required because the client decides its association upon his arrival to the system and remains associated to the selected network throughout his session. However, the distributed approach will degrade the performance since it induces a sub-optimality for the overall system: the given solution is not the one with the lowest overall cost.

B. The new hybrid radio access technology selection approach

In the centralized approach, the system allocates the downlink traffic between the existing radio access technologies. It balances users' traffic between wireless technologies in order to optimize the total cost. For each change in the network state (client arrival/departure) the system searches for the optimal association profile and associates the clients according to it.

However, in the distributed approach each client aims at minimizing its individual cost function regardless of the other clients and the system state. Hence, each client decides of his association individually when he arrives to the system in order to minimize his own cost.

The hybrid approach consists in adopting the distributed approach, but with periodic system intervention to adjust the client association profile to the existing wireless access networks.

Compared to the centralized approach, the hybrid approach has a lower processing load. In fact, the centralized approach requires a global knowledge of the entire system state and induces a huge processing cost when a change in the system state occurs (arrival or departure). However, hybrid approach interventions occur at regular time intervals (periodic intervention) to adjust the client association to the existing radio access technologies in accordance with the centralized approach.

In addition, intervening periodically to apply the centralized approach may improve the performance of the distributed solution. When the clients choose their association individually the distributed association profile tends to differ from the optimal one since each client aims at choosing the network that suits him the most regardless of other clients and system's state. However, periodic intervention forces the clients to adopt the optimal association profile when it occurs. Therefore, the client association profile will remain close to the optimal association profile.

The hybrid approach works as follows (Fig. 2): The hybrid approach performs periodic interventions to adjust the association of clients to the existing networks. When a new arrival occurs (a new client arrives) if the intervention period has not expired, it is the distributed approach that will decide of his association. In addition, if the intervention period has not elapsed and a client leaves our system, the distributed approach does not induce any change in the client association profile.

Consequently, whenever the intervention period elapses (at T, 2T, 3T, ..., nT ...) we adopt a centralized approach to update the client association profile to the wireless access networks (even if no arrival has occurred since the last intervention). This intervention corrects association errors that may be committed by the distributed decision.

The hybrid approach algorithm must be implemented with a dynamic system model. This algorithm is a compromise between the centralized and the distributed approaches algorithms.

In the hybrid approach, several distributed algorithms can be used when adopting the distributed decision: deterministic or probabilistic [START_REF] Moety | Distributed Heuristic Algorithms for RAT Selection in Wireless Heterogeneous Networks[END_REF]. The system intervention is performed on a periodic basis to adjust client associations when necessary.

IV. SIMULATED SCENARIOS

Various scenarios are proposed in order to compare the performance of the different implemented algorithms. For each scenario, we perform a random draw to determine the number of clients that will be located in each of the secondary cells present in the geographic area (one primary and two secondary cells). Then, we perform another random draw weighted by the percentage of clients with higher primary peak rates to determine whether the client has better primary or secondary peak rate. Finally, we perform a random draw to associate each client with a couple of peak rates (primary peak rate, secondary peak rate). These random draws will guarantee a wide variety of distributions during the generation of each scenario. In addition, each scenario is iterated several times, and the average cost is adopted. For each scenario, several trust points are plotted to assess the validity of results.

V. SIMULATION RESULTS

In order to compare the different approaches that can be adopted in radio access technology selection, simulations on several scenarios were performed. All the approaches that have already been mentioned in this paper will be processed and analyzed.

The main parameter for the system model used is client peak rates. IEEE 802.11b and 802.11g are the two radio access technologies used for the simulation since each one have its own peak rates. The Table 1 shows the main characteristics of Wi-Fi technology (IEEE 802.11b and IEEE 802.11g): To better analyze the simulation results, we define the following ratios (Table II): Where:

• R denotes the cost for the distributed deterministic peak rate based algorithm.

• PR denotes the cost for the distributed probabilistic peak rate based algorithm.

• O refers to the cost of the centralized optimal solution.

• HR denotes the cost for the hybrid deterministic peak rate based algorithm.

• HPR denotes the cost for the hybrid probabilistic peak rate based algorithm.

These ratios are the cost ratios of the different algorithms implemented. The cost is the time required by a client to send a data unit via its wireless access network.

If the ratio is greater than 1, the algorithm in the numerator is more expensive. The solution is more efficient when it has a lower cost. However, when the ratio is less than 1, the algorithm of the numerator will be the most efficient. It will improve the QoS and the client experience, and it will reduce the total cost (of the system) allowing the operator to make better use of its network, to introduce new services and to take better advantage of the available bandwidth.

A. Cost ratios

Simulations are performed for scenarios where 500 clients arrive to the system according to a Poisson process of parameter λ (1 client/s). Session durations follow a Pareto distribution, and the system performs periodic interventions with a period T = 1 second. The results are reported below (Fig. 3): The curves obtained in Fig. 3 show that the distributed deterministic approach's algorithm (without any intervention by the system) leads to a performance that is very close to the centralized approach for equitable client peak rates distribution between the two radio access technologies existing in the coverage area. In fact, the distributed decision is very close to the optimal one, and it will associate the clients to the adequate wireless access network so that we will have the lowest cost.

The hybrid approach's algorithm that performs periodic interventions to adjust client association profile has its performance degraded for equitable distributions of peak rates between the two wireless access networks. In fact, the distributed approach's decision is very close to the centralized decision. Thus, the periodic intervention (to apply the centralized approach's association profile) will adversely affect the performance of the system over time. Therefore, the hybrid solution cost will be greater than that of the centralized approach (and even greater than that of the distributed approach) when the client peak rates distribution is equitable between the two wireless access networks. The hybrid approach's algorithm shows better performance than the distributed one for extreme peak rates distributions since the latter's decision is not close to the optimal one. In this case, the intervention will be interesting since it will adjust the client association profile to the optimal one, which will reduce the cost and improve the performance.

B. Number of handovers

We also study the number of handovers (HO) that must be performed for each approach. The more we have handovers the more we need signaling and the more we have valuable resources that are consumed. Therefore, we will have fewer resources for the signaling traffic. This will adversely affect the performance of the system. Simulations (Fig. 4) are made to highlight the cost of each approach in terms of number of handovers: The curves show that the centralized approach has the largest number of handovers. Whereas the number of handovers for the hybrid deterministic or probabilistic peak rate based approach is smaller.

It is important to note that for the distributed approach no handover is required, since the client association to the system is done only once (when the client arrives at the system). The client will always be associated to the chosen network until its session expires. Thus, no handover is done.

The centralized approach requires a calculation of the best association profile at each change of the system state in order to find the profile with the lowest cost. Thus, the number of handovers made will increase since the association of the clients is modified at each event (arrival/departure).

When the hybrid approach is used, a change in the client association profile will occur for each centralized intervention. Therefore, we will need to perform a number of handovers to modify the client associations. However, the number of required handovers is less than that required for the centralized approach since the association changes does not occur very frequently. The hybrid approach will reduce the number of handovers and the signaling traffic needed to carry them out.

C. Effect of the intervention period

Simulations results (Fig. 5) show the influence of the intervention period variation on the hybrid approach performance. These curves (Fig. 5) show that the centralized approach has always the best performance since it provides the lowest cost. The distributed approach is the furthest from the centralized approach. The hybrid approach is a compromise between the two extreme approaches in terms of cost, quality, service and processing load.

The intervention period is the main parameter for the hybrid approach. The more the intervention period is small, the more the interventions are frequent, and the more the hybrid approach is close to the centralized one (optimal). However, this will increase the number of handovers and the signaling load. Furthermore, the cost function increases when the intervention period increases, but the signaling load decreases.

The centralized approach is indeed a special case of the hybrid approach but with an intervention period that tends to zero (or an intervention frequency that tends to infinity). Therefore, when the intervention period decreases (the intervention frequency increases), the hybrid approach's performance will be closer to that of the centralized approach.

VI. CONCLUSION

In this paper, a hybrid approach was introduced and simulated. In fact, instead of adopting a unique radio access technology selection approach, the hybrid approach allows the use of a distributed solution with periodic system intervention to adjust the client association profile. This hybrid approach is a compromise between the two generic approaches (centralized and distributed approaches). It can improve the overall system performance (and the quality of service perceived by the clients. This solution will also induce additional processing costs and calculation charges due to the use of an exhaustive search at each intervention. It also reduces the signaling load by reducing the number of required handovers.

Operator's choice remains the key factor to monitor the system performance, to reduce costs experienced by mobile users, to reduce resource consumption and to make the most of those available. This choice is governed by several parameters which are mainly the characteristics of the system and the clients such as: the client arrival rate to the system, their session durations, the distribution of their peak rates, etc.

Figure 2 .

 2 Figure 2. Heterogeneous system adopting the hybrid approach

Figure 3 .

 3 Figure 3. Performance variation depending on clients peak rates distribution

Figure 4 .

 4 Figure 4. Performance variation depending on clients peak rates distribution

Figure 5 .

 5 Figure 5. Performance variation in time

TABLE I .

 I

		WIFI TECHNOLOGY'S CHARACTERISTICS
				Frequency
	Technology	Bitrates (Mb/s)	Range (m)	Band
				(GHz)
	802.11b	36, 48, 54 6, 9, 12, 18, 24,	100 From 35 to	2.4
	802.11g	1, 2, 5.5, 11	75 From 25 to	2.4

TABLE II

 II

	Ratio 1	Ratio 2	Ratio 3	Ratio 4	Ratio 5	Ratio 6
	R/O	PR/O	PR/R	HR/O	HPR/O	HR/R
						.	CONSIDERED RATIOS