Mikael Touati
email: mikael.touati@orange.com

Jean-Marc Kelif
email: jeanmarc.kelif@orange.com

Marceau Coupechoux
email: marceau.coupechoux@telecom-paristech.fr

Impact of Mobility on QoS in Heterogeneous Wireless Networks

Keywords: heterogeneous networks, small cells, mobility, performance, quality of service, load and capacity analysis

This paper develops a model evaluating the impact of mobility in heterogeneous wireless networks by the way of analytical expressions of the spatio-temporal evolution of QoS indicators. Such formulas are obtained by introducing a multiuser averaged mobility pattern named density of users. Among the set of densities, we use a Gaussian form of this quantity, which results from a modeling method based on the maximum entropy principle. Numerical results show the temporal variations of the QoS indicators and highlight the combined effects of network heterogeneity due to the presence of macro and small cells, and traffic variations due to mobility.

I. INTRODUCTION

The recent and massive commercial release of wireless network connected advanced terminals and associated applications has changed users way of consuming network services. Customers did not only increase their data traffic and modify their usage, but have also increased their expectations in terms of Quality of Service (QoS) in mobility. This has led Mobile Network Operators (MNOs) to define new network planning strategies, such as the deployment of heterogeneous wireless networks. MNOs evaluate network performance by the way of QoS indicators (such as load, capacity, blocking probability), which can be derived using for example the processor sharing queue assumption with or without admission control [START_REF] Bonald | Wireless downlink data channels: user performance and cell dimensioning[END_REF]. These indicators are commonly derived in static configurations (fixed base stations and users) and temporal variations are usually introduced by the way of a user redistribution in the network following probabilistic or deterministic mobility patterns.

Authors of [START_REF] Kelif | Impact of transmitting power on femto cells performance and coverage in heterogeneous wireless networks[END_REF][5] analyzed the LTE wireless network heterogeneity and proposed a model evaluating the impact on coverage and throughput, showing that QoS criteria can be satisfied with reduced emitted power by the way of a spatial redeployment of stations or densification. The effect of mobility on α-fair scheduling policies is evaluated in [START_REF] Borst | Mobility-driven scheduling in wireless networks[END_REF]. In [START_REF] Ramanath | Spatial queuing analysis for mobility in pico cell networks[END_REF], authors assess the problem of mobility in a micro-cell network, each cell being divided in pico-cells. User traffic demands and mobility are defined by a triplet of the form (location of arrival, size file requirement, user velocity), each component being probabilistically distributed. They obtain expressions for some characteristic times, optimal cell sizes and maximum velocities satisfying criteria leading to a successful communication. Authors of [START_REF] Coucheney | Mobile association problems in heterogeneous wireless networks with mobility[END_REF] model mobility in an heterogeneous network (WiFi and HSDPA cells) using migration rates following Markov Modulated Poisson Process. Using the optimal policy of the association problem, authors show that mobility improves performance under certain assumptions.

One should also mention the huge work on mobility modeling in the literature around ad hoc networks. To have an overview of the field one can refer to the survey by Camp and al. [START_REF] Camp | A Survey of mobility models for ad hoc network research[END_REF] on the mobility models commonly used in simulations of ad-hoc networks. They mainly distinguish two kinds of models, the first type being entity mobility models such as random walk, random way point, Gauss-Markov or city section models and the second one being group mobility models such as exponential correlated, nomadic community or pursue mobility models. Authors of [START_REF] Silva | Magnetworks: how mobility impacts the design of Mobile Networks[END_REF] evaluate the optimal placement and the optimal number of nodes in massively dense ad-hoc mobile sensor networks. This assumption leading them to describe the nodes distribution with spatial densities.

In this paper, we jointly consider mobility, network heterogeneity and user traffic, and we introduce a multiple-user mobility pattern based on a dynamic Gaussian user density. Such a quantity allows us to use a deterministic expression of the user spatial distribution with underlying probabilistic mobility patterns. We develop an analytical model leading to simple formulas of the spatio-temporal evolution of QoS indicators. By the way of numerical evaluations, we highlight the impact in time and space of both mobility and network heterogeneity on performance.

Section II introduces the system model, particularly the heterogeneous network model (II-A) in terms of its subnetworks, the mobility model (II-B) consisting in a Gaussian density based mobility pattern and the propagation model (II-C) leading to a simple Signal to Noise plus Interference Ratio (SINR) form (II-D). Section III defines the cell-type dependent indicators (loads, maximum throughputs, capacity) formulas. Section IV presents numerical results and gives an interpretation (IV-B) of the temporal evolution of the QoS indicators. Section V concludes the paper.

II. SYSTEM MODEL A. Network Model

We consider a finite set M of omni-directional macro-cell base stations (MBSs) deployed on a hexagonal network made of several rings surrounding a central cell (see figure 1). Let R c be the half-distance between two neighboring MBSs, ρ m the MBS density and P m their common transmit power. In each macro-cell, n small-cell base stations (SBSs) are deployed with a regular pattern. Let N be the set of all SBSs of the network. Let R n be the average half-distance between two neighboring SBSs1 , ρ n the SBS density and P n their common transmit power. The superposition of the MBSs and SBSs networks results in an heterogeneous network. The coverage area of each BS is defined as the set of locations, where users are served by this station. An heterogeneous cell is made of one MBS and all the SBSs deployed in its hexagon. We define the coverage area of an heterogeneous cell as the set of locations, where users are served by the MBS or any of the SBS of the concerned heterogeneous cell. We focus our analysis on the downlink.

B. Mobility Model

In any real system, the user spatial distribution at a given instant can be seen as the realization of a random point process. The temporal evolution of this distribution follows however hardly identifiable mobility patterns. The group behavior of these users may be such that, in average, the number of users per unit surface is given by a density npr, tq (users{m2 ). Such a quantity is commonly used in physics, where systems may be described at microscopic scale (atoms, molecules) or at a larger scale while considering fluid particles whose properties result from the statistical averaging of microscopic behaviors. In this paper, we focus on this larger scale and thus do not specify any individual mobility scheme, such as random way point, random-walk or Gauss-Markov model. In other words, we consider a quantity resulting from individual mobility patterns but at a larger scale than the user point process. With this approach, we can use deterministic formulas resulting from underlying probabilistic processes. Depending on the scenarios, many forms of densities may be used.

In this paper, we will concentrate on a Gaussian density of users. With this model, the user density npr, tq has the following form:

npr, tq " A expp´p r ´µptqq 2 2σ 2 q, (1)
where r " pr, θq is a position vector in polar coordinates, A and σ are constants and µptq is a function of time.

A mathematical argument for this choice is provided in Appendix. It uses the maximum entropy principle that allows to derive the user distribution that maximizes uncertainty, when only the mean and the variance of the radial mobility pattern are known. In addition, it is interesting to focus on such a Gaussian density due to its symmetry property and the convenience it induces in terms of interpretations. Another qualitative argument in favor of this kind of density can be provided by the way of the following example. When going out of a subway entrance, users diffuse in the plane. It seems reasonable to consider in this case an average pedestrian speed of diffusion and an isotropic radial diffusion.

To give more intuition on the model, we consider here a kind of user "wave" spreading isotropically in a given area (figure 2). In such a case, the density goes through the cell at a velocity given by the temporal evolution of its mean µptq. The spatial extension of the distribution of users is given by the parameter σ. The total amount of users in a given area is obtained by spatial integration of the density. The previously mentioned symmetric diffusion in every angular direction of the plane allows us to locate those users of the distribution who generate a non-negligible proportion of the load. Most of the users concentrate around the mean (roughly in the interval rµ ´3σ; µ `3σs) and the spatial extension allows to catch the spreading trend. The slowest users locate at the back of the wave and the fastest ones locate at the front of the wave. Both cases are however less probable than the heights of the wave.

C. Propagation Model

We assume that the path-gain g j puq between a station j transmitting at power P j and a user u only depends on the distance r j puq between j and u. The power, P j,u , received by a mobile u from j can be written:

P j,u " P j K j r ´ηj j puq (2)
where K j and η j denote the path-loss propagation coefficients of station j. We assume constant coefficients for all stations of the same type 2 : K j " K m , η j " η m @j P M and K j " K n , η j " η n @j P N . Let define Cprq as the peak data rate practically obtained in r in the absence of any other user in the cell [START_REF] Bonald | Wireless downlink data channels: user performance and cell dimensioning[END_REF]. The analytical expression of this peak rate has been obtained at Orange Labs by simulations of the LTE RF chain and consists in a simple fitting of obtained curves by a modified Shannon's formula [START_REF] Shannon | A mathematical theory of communication[END_REF] of the form:

Cprq " aW log 2 p1 `bγprqq (3)
where a " 0.6, b " 1 and γprq is the Signal to Interference plus Noise Ratio (SINR) in r. The dependency of the system performance on the environment and on user mobility is taken into account by their impact on the SINR, which is detailed in the next section.

D. SINR

We assume that every user u is attached to its best server, i.e., the station providing the highest received signal power, u is thus served by j if and only if 3 :

P j g j puq ě P i g i puq @i P M Y N . (4)
Let now define k ů P M and l ů P N the closest MBS and SBS respectively to u. We can rewrite the previous association rule as follows. BS j is serving u is given by:

j " k ˚if P k ˚gk ˚puq ě P l ˚gl ˚puq, (5)
j " l ˚if P k ˚gk ˚puq ď P l ˚gl ˚puq, (6)
where we have removed the u subscript in k ů and l ů for a reason of simplicity. Finally, we can write the worst-case SINR (all stations interfering except the serving station) as:

γpuq " P m K m r ´ηm k ˚1tj"k ˚u `Pn K n r ´ηn l ˚1tj"l ˚u I N puq `IM puq `Nth (7)
where:

I N puq " ÿ lPN ztl ˚u K n P n r ´ηn l `Pn K n r ´ηn l ˚1tj"k ˚u (8)
I M puq " ÿ kPMztk ˚u K m P m r ´ηm k `Pm K m r ´ηm k ˚1tj"l ˚u, (9)
and N th is the thermal noise power. In numerical implementations presented in section IV use the fluid model [START_REF] Kelif | A fluid model for performance analysis in cellular networks[END_REF] to simplify the calculations of ( 8) and [START_REF] Coucheney | Mobile association problems in heterogeneous wireless networks with mobility[END_REF]. We now derive in the next section the QoS indicators of the heterogeneous network.

III. PERFORMANCES

Let dSprq be an elementary surface located at r " pr, θq. The total number of users in dSprq at t is given by: npdSprq, tq " npr, tq ˆdS [START_REF] Debbah | MIMO channel modeling and the principle of maximum entropy[END_REF] We assume that the traffic demand is independently and identically distributed (i.i.d) among users and arrives as a Poison process of intensity λ u . Flows sizes are i.i.d and given by the random variable σ f . We define the traffic density as the average number of bits per unit of time and surface:

ρpr, tq " npr, tqλ u Erσ f s bits{s{m 2 (11) 3 Indetermination is solved at random.

The traffic intensity generated by users in a surface dSpr, tq at t is given by: dρ dS pr, tq " npdSprq, tqλ u Erσ f s bits{s

A. Cell load

Based on [START_REF] Bonald | Wireless downlink data channels: user performance and cell dimensioning[END_REF], we define the total load generated by a domain D r " D θ ˆDR served by a unique station:

ρ Dr ptq " ż D θ ˆDR dρ dS pr, tq Cprq " λ u Erσ f s ż D θ ż D R
npr, tq Cprq dr (13) Introducing (3) in [START_REF] Silva | Magnetworks: how mobility impacts the design of Mobile Networks[END_REF], we obtain a general formula giving the instantaneous load of a bi-dimensional domain D r with users density npr, tq:

ρ Dr ptq " λ u Erσ f s aW ż D θ ż D R npr, tq log 2 p1 `bγprqq dr (14)
We now detail this load formula [START_REF] Camp | A Survey of mobility models for ad hoc network research[END_REF] for SBSs and MBSs. 1) Small-cell load: Let D l " D θ l ˆDR l be the coverage area of SBS l . In a polar coordinate system centered on SBS l , we define the angular and radial domain such that:

D θ l " r0; 2πs D R l pθq " rR N0 ; r max pθqs, (15)
where R N0 denotes the minimal distance between users and a SBS (red circles in figure 1). The extremum r max pθq of the radial domain is the solution of (6), i.e., the maximum distance such that in the angular direction θ, the power received by any user from SBS l is greater than the power received from MBS k ˚. Assuming equality of the path-loss coefficients (η m " η n " η), we obtain a simple analytical expression of r max pθq: r max,l pθq " r k ˚l rcospθ ´θk ˚l q ´acos 2 pθ ´θk ˚l q ´As A (16) with A " p1 ´p PnKn PmKm q ´η{2 q and pr k ˚l , θ k ˚l q the coordinates of MBS k ˚in the polar coordinate system centered on SBS l . We can thus write the load generated by users in the coverage area of SBS l :

ρ l ptq " λ u Erσ f s aW ż 2π θ"0 ż r max,l pθq r"R N 0
npr, tq log 2 p1 `bγprqq rdrdθ (17) 2) Macro-cell load: We now focus on the load ρ k ptq generated by the area covered by a macro-cell base station MBS k. To simplify its calculation, we subtract from the total load ρ m k generated by the hexagon cell, the load ρ m l associated to the area covered by every SBS of the heterogeneous cell. We thus obtain the following formula:

ρ k ptq " ρ m k ptq ´n´1 ÿ l"0 ρ m l ptq (18)
The load ρ m k involves a numerical integration over a convex set (the hexagon), while loads ρ m l involve numerical integrations similar to (17), i.e., over convex sets. The serving area of a MBS is a priori non-convex. On the contrary, the right hand side of (18) involves only numerical integrations over convex areas, which makes easier the load computation.

B. Cell capacity

Capacity may admit many definitions. We consider the one given in [START_REF] Bonald | Wireless downlink data channels: user performance and cell dimensioning[END_REF]. It is defined as the maximum average value of total traffic in bits/s (reached at the limit of overload, i.e., when load tends to 1). We have previously defined the demand of an area as the one resulting from the activity of users distributed according to a spatial density. Thus, we first define the maximum throughput per user of a MBS or a SBS as the maximum average value of individual traffic (bits/s/user) knowing that the service is uniformly distributed among all users. In case of heterogeneous cells, we define the capacity as the maximum total average traffic (bits/s), still knowing that the service is uniformly distributed among all users.

1) Small-cell maximum throughput: The maximum throughput per user served by SBS l is given by: Qpl, tq " sup

ρ l ă1 λ u Erσ f s " aW ˜ż 2π θ"0 ż r max,l pθq r"R N 0 npr, tq log 2 p1 `bγprqq rdrdθ ¸´1 (19)
2) Macro-cell maximum throughput: The maximum throughput per user served by MBS k is given by: Qpk, tq " sup

ρ k ă1 λ u Erσ f s " aW ˆż 2π θ"0 ż Re r"R M 0 npr, tq log 2 p1 `bγprqq rdrdθ ´n´1 ÿ l"0 ż 2π θ"0 ż r max,l pθq R N 0 npr, tq log 2 p1 `bγprqq rdrdθ ˙´1 ( 20
)
with R M0 the minimal distance between users and any MBS and R e the radius of the disk equivalent in surface to the hexagonal coverage of the macro-cell.

3) Heterogeneous-cell capacity: Finally, we define the capacity of an heterogeneous macro-cell as the total number of bits per unit of time that the system is able to transmit to users it is serving. This quantity is obtained by adding all individual maximum throughput:

C h pk, tq " n k ptq ˆQpk, tq `n´1 ÿ l"0 n l ptq ˆQpl, tq (21)
where n k ptq denotes the total number of users served by MBS k, n l ptq denotes the total number of users served by SBS l (l th small-cell station contained in the hexagon of MBS k). Equations ( 17), ( 18), ( 19), ( 20) and (21) allow us to quickly compute the loads and capacities in a heterogeneous network.

IV. SCENARIO AND RESULTS

In this section we present the numerical results obtained from the previous analytical approach. Our aim is to show the impact of mobility (due to the spatio-temporal evolution of the density of users) on QoS indicators such as loads or capacity. The first ring around each `gives the minimal distance to small-cells. Each small-cell coverage area (given by equation ( 16)) is in dotted lines.

A. Assumptions

We consider a scenario with two SBSs per heterogeneous cell. The half distance R c between two MBSs is 866 m and the distance R r between any SBS and the MBS at the center of its heterogeneous cell is 500 m. The minimum distance R M0 to a MBS is 50 m and the minimum distance R N0 to a SBS is 30 m. The MBS transmitting power P m is 43 dBm and the SBS transmitting power P n is 28 dBm. The path-loss parameter η is 3.41. The signal bandwidth W " 10 MHz. There are N tot " 30 users initially introduced in every macrocell and their individual average traffic intensity λ u Erσ f s is 100 kbits/s.

The network layout is shown on figure 3. All hexagon cells are identical in terms of SBS deployment, we thus focus on the central cell for numerical experimentations.

We propose a Gaussian density of users with mean µptq " R M0 `3σ `vt, normalized at any instant to N tot . The density goes through the cell in time at velocity v " 1 m{s (average speed of a pedestrian). Changing the average pedestrian speed v would only induce changes in the time-scales of the temporal variations of the indicators shown in this section (others such as failures if considered may behave differently). The model remains valid for any other density as long as it can be written in the form npr, tq. If the expressions in the paper are analytically tractable then the temporal evolutions are obtained, if not they can be evaluated by numerical computations.

B. Results and Interpretation

1) Load of a small-cell: Figure 4 shows the time variations of a SBS load. These variations are due to the dispersion of users in the plane via the diffusion of their density. We particularly observe a sudden jump at time t " 230 s, which corresponds to the first arrival of a user in the small cell coverage area. Then, we can observe the user "wave" traversing the small cell: the loads reaches a global maximum at t " 560 s before decreasing to zeros when the last user leaves the serving area.

Interestingly, we observe a local maximum if the load around t " 300 s. This is due to the fact that after this time instant, the peak number of users served by the SBS is reached and these users start to be in average closer to the base station. They thus tend to obtain higher physical data rates and create less load on the small cell. As the maximum of the wave starts going farther from the base station, data rates decrease and the load increases again. At t " 560 s, this effect starts to be compensated by the fact that more and more users are leaving the serving area. Figure 5 illustrates this phenomenon by showing the SBS load density at two time instants.

2) Load of a macro-cell: Recall that users diffuse from the central MBS towards the cell edge. This means that the SINR of a user going away from its initial serving MBS decreases due to the weakening of its useful signal (path-loss effect) and the growing interferences. This explains the fact that the macro-cell load is first increasing (see figure 6). At t " 230 s, the load is a bit stabilizing (at least increases with a lower slope). At this time instant, SBSs are indeed starting serving users, alleviating some load from the MBS. From 470 s to 720 s, we observe again a high load increase. This effect being due to both users going back from small-cell to macro-cell coverage area and the global mobility towards the cell edge. At t " 720 s, small cells do not serve any user and users start leaving the macro-cell coverage area.

3) Capacity of an heterogeneous-cell: The capacity variations observed on figure 7 obtained by inversion of the previously analyzed load variations. The gain in the heterogeneous-cell capacity obtained as a result of the introduction of small-cells is significant. When SBSs start serving (t " 230 s), we observe an increase of 92% in capacity, rising from 18.6 Mbits/s to 35.7 Mbits/s. Similarly, when the service of SBSs stops (t " 620 s), the capacity of the heterogeneous cell decreases of 62% from 1.61 Mbits/s to 0.61 Mbits/s.

V. CONCLUSION

In this paper, we developed an analytical model to study the impact of mobility on QoS in heterogeneous wireless networks. We introduced a multi-user averaged mobility pattern by the way of a quantity named "density of users". To illustrate this notion, we have chosen a dynamic Gaussian form. We established simple expressions of the temporal evolution of performance indicators such as load, maximum throughputs or capacity. Numerical results showed this evolution and we gave an interpretation of their variations based on the known dynamic of the Gaussian density of users. We noticed the stabilization of the load of a macro-cell and the significant increase of the capacity of an heterogeneous cell, both induced by the entrance of users in the coverage area of small-cells. Future work may include other forms of user densities and take into account more complex propagation models such as shadowing and fast-fading.

APPENDIX

We propose hereafter a proof justifying the Gaussian form of the density of users we use in the proposed model. This proof includes a modeling method using the maximum entropy principle; see [START_REF] Debbah | MIMO channel modeling and the principle of maximum entropy[END_REF], [START_REF] Guillaud | Maximum entropy MIMO wireless channel models[END_REF], [START_REF] Jaynes | Information theory and statistical mechanics[END_REF] and references therein. Basically, we are looking for the mobility model we are legitimately able to assume when only knowing the mean and variance of the radial mobility pattern of users.

We define r, the matrix containing at row i the two random polar coordinates pr i , θ i q of user i among N tot users and P prq the joint probability distribution of the random components of matrix r. Let consider the following assumptions: @i, ż dr r 2 i P prq " σ 2 ri " σ 2 and ż drP prq " 1

We are looking for the probability distribution of the coordinates of users maximizing uncertainty. Thus, we define the Lagrangian LpP q such that:

LpP q " ´ż drP prqlogpP prqq `ÿ i γ i pσ 2 ´ż drr 2 i P prqq `βp1 ´ż drP prqq

(23)
Taking the functional derivative of LpP q with respect to P : δLpP q δP " 0 ô ´1 ´logpP prqq ´ÿ i γ i r 2 i ´β " 0 (24)

We obtain the following form for the distribution: P prq " ź i P pr i , θ i q (25)

with: P pr i , θ i q " e ´pγir 2 i `β`1

N tot q (26)
Using ( 22), we evaluate the Lagrange multipliers:

e ´pβ`1q " 1 p2πσ 2 q N tot 2

π Ntot and @i, γ i "

1 2σ 2 (27)
Finally, we have: @i, P pr i , θ i q " 1 ?

2πσ 2 1 π e ´r2 i 2σ 2

(28) Now assuming the knowledge of the mean of each radial component Err i s " µ i " µ and its variance Erpr i ´µq 2 s " σ 2 , we obtain (using the same method) the following spatial distribution for the coordinates of user i: @i, P pr i , θ i q " 1 ? 2πσ

Fig. 1 .

 1 Fig. 1. Heterogeneous network obtained by the superposition of a macro-cell base stations network and a small-cell base stations network.

Fig. 2 .

 2 Fig. 2. Spatio-temporal diffusion of the Gaussian density of users in the plane. Left is at t=0s and right at t=300s.

Fig. 3 .

 3 Fig.3. Network of heterogeneous cells. Macro-stations are identified by the symbol d, small-cell stations by `. The first ring around each `gives the minimal distance to small-cells. Each small-cell coverage area (given by equation (16)) is in dotted lines.

Fig. 4 .

 4 Fig. 4. Temporal evolution of the load of a small-cell.

Fig. 5 .

 5 Fig. 5. Evolution of the load density in a small-cell coverage area (whithe circle is the SBS location). Left is at t=300s and right at t=520s. Dark shades indicate low values and bright shades indicate higher ones.

Fig. 6 .Fig. 7 .

 67 Fig. 6. Temporal evolution of the load of a macro-cell.

  Let define the average number of users in an elementary surface dSprq (section III) as:This probability of having k users (among N tot ) in dSprq being given by the following binomial distribution: Defining the density of users as the average number of users per unit area (users{m 2 ), we finally have the Gaussian form of the expected density of section II-B:

					ÿ
					N pdSprqq "	k ˆP pk users in dSprqq	(30)
					k
					P pk users in dSprqq " ˆNtot k	˙pk p1 ´pq Ntot´k (31)
					with p the probability of a user being in dSprq:
					p " P pr, θq ˆdSprq "	?	1 2πσ 2	1 π	e	´pr i ´µq 2 2σ 2	ˆdSprq (32)
					Knowing the mean of a random variable following a binomial
					distribution, we obtain:
					N pdSprqq " N tot ˆp " N tot	?	1 2πσ 2	1 π	e	´pr´µq 2 2σ 2	ˆdSprq (33)
					nprq " N tot	?	1 2πσ 2	1 π	exp	´pr´µq 2 2σ 2	(34)
	2	1 π	e	´pr i ´µq 2 2σ 2	(29)

In the sense of Voronoi cells.

For the sake of simplicity and because we focus on the mobility effect, we ignore in this paper shadowing. We will take it into account in a future work.


Mohd Shabbir Ali

Pierre Coucheney

Marceau Coupechoux

Load Balancing in Heterogeneous Networks Based on Distributed Learning in Potential Games

We present a novel approach for distributive load balancing in heterogeneous networks that use cell range expansion (CRE) for user association. First, we formulate the problem as a minimisation of an α-fairness objective function. Depending on α, different objectives in terms of network performance or fairness can be achieved. Next, we model the interactions among the base stations for load balancing as a potential game, in which the potential function is the α-fairness function. The optimal Nash equilibrium of the game is found by using distributed learning algorithms. We use log-linear and binary log-linear learning algorithms for complete and partial information settings, respectively. By running extensive simulations, we show that the proposed algorithms converge within a few tens of iterations. The convergence speed in the case of partial information setting is comparable to that of the complete information setting. We also show that the best response algorithm does not necessarily converge to the optimal Nash equilibrium.

I. INTRODUCTION

Due to the ever increasing demand for improved quality of service in terms of higher data rates and improved coverage, the conventional cellular networks are becoming heterogeneous [START_REF] Gpp | Evolved Universal Terrestrial Radio Access (E-UTRA) and Evolved Universal Terrestrial Radio Access (E-UTRAN)[END_REF]- [START_REF] Kudo | Cell range expansion using distributed Qlearning in heterogeneous networks[END_REF]. Heterogeneous networks consist of macro base stations (BSs) and small BSs that transmit with high and low power, respectively. Conventional user association rule is such that the users select a BS that provides the highest received power. This may however result in an imbalance between BSs loads because the macro BS transmits at higher power and thus associates with more users. This creates overload situation at the macro BSs and at the same time under-utilised resources at the small BSs. Therefore, a natural problem that arises is how to associate users to BSs such that the network resources are utilised efficiently and the load is shared among the BSs.

Load balancing has been extensively studied in the literature using various approaches. An overview can be found in [START_REF] Andrews | An overview of load balancing in hetnets: old myths and open problems[END_REF], [START_REF] Tonguz | The mathematical theory of dynamic load balancing in cellular networks[END_REF]. These can be broadly classified as centralised [START_REF] Stevens-Navarro | An MDP-based vertical handoff decision algorithm for heterogeneous wireless networks[END_REF], [START_REF] Elayoubi | A hybrid decision approach for the association problem in heterogeneous networks[END_REF], [START_REF] Son | Dynamic association for load balancing and interference avoidance in multi-cell networks[END_REF], [START_REF] Rengarajan | Architecture and abstractions for environment and traffic-aware system-level coordination of wireless networks[END_REF] and decentralised optimisation approaches [START_REF] Ye | User association for load balancing in heterogeneous cellular networks[END_REF], [START_REF] Niyato | Dynamics of network selection in heterogeneous wireless networks: An evolutionary game approach[END_REF], [START_REF] Aryafar | RAT selection games in HetNets[END_REF], [START_REF] Kim | Distributed alpha-optimal user association and cell load balancing in wireless networks[END_REF], [START_REF] Xu | Designing a practical access point association protocol[END_REF]. Centralised user association rules along with inter-cell interference avoidance for load balancing are proposed in [START_REF] Son | Dynamic association for load balancing and interference avoidance in multi-cell networks[END_REF], [START_REF] Rengarajan | Architecture and abstractions for environment and traffic-aware system-level coordination of wireless networks[END_REF]. Basic idea is to schedule users across the BSs in the network so that the users do not severely interfere with each other. Similar approaches where the user association decisions are modelled as Markov decision process (MDP) are presented in [START_REF] Stevens-Navarro | An MDP-based vertical handoff decision algorithm for heterogeneous wireless networks[END_REF], [START_REF] Elayoubi | A hybrid decision approach for the association problem in heterogeneous networks[END_REF]. However, centralised Mohd. Shabbir Ali and M. Coupechoux are with Telecom ParisTech and CNRS LTCI, 46 rue Barrault, 75013, Paris, France. Emails: mdshabbi-rali88@gmail.com, marceau.coupechoux@telecom-paristech.fr. P. Coucheney is with PRiSM, UVSQ / CNRS, 45 avenue des Etats-Unis, 78035 Versailles, France. Email: pierre.coucheney@uvsq.fr, This work was supported by NetLearn project ANR (ANR-13-INFR-004).

solutions are computationally intensive, require huge information exchange overhead, and are not scalable. To overcome these limitations, decentralised approaches are followed. A distributed algorithm is proposed for the load balancing, which is formulated as a convex optimisation problem in [START_REF] Kim | Distributed alpha-optimal user association and cell load balancing in wireless networks[END_REF].

A decentralised solution for convex optimisation approach for joint cell association and resource allocation problem is undertaken for the load balancing in [START_REF] Ye | User association for load balancing in heterogeneous cellular networks[END_REF]. In [START_REF] Xu | Designing a practical access point association protocol[END_REF], an online algorithm for access points association in wireless local area network (WLAN) based on the L p norm of the loads on access points in proximity is proposed. Game theoretical approaches are also proposed for decentralised solutions [START_REF] Niyato | Dynamics of network selection in heterogeneous wireless networks: An evolutionary game approach[END_REF], [START_REF] Xu | Designing a practical access point association protocol[END_REF]. Also heuristic approaches are studied in the literature. For example, in [START_REF] Bejerano | Cell breathing techniques for load balancing in wireless LANs[END_REF], an algorithm is proposed to find the optimal beacon power that minimises the load of the most congested access point (AP) in WLAN.

Another important approach for load balancing that has attracted lot of interest in the literature is user association using cell range expansion (CRE) [START_REF] Damnjanovic | A survey on 3GPP heterogeneous networks[END_REF]- [START_REF] Eguizabal | Interference management and cell range expansion analysis for LTE picocell deployments[END_REF]. According to CRE technique the users associate with a BS that provides the highest biased received power. Bias value for small BSs is greater than or equal to one and for macro BSs it is equal to one. This results into increase in the coverage regions of the small BSs and thereby associating more users to them.

However, there are several challenges for using CRE technique. One of the challenges is to avoid high co-channel intercell interference in downlink from the macro BSs experienced by the small BSs' cell edge users. To address this problem advanced interference management techniques have been studied in the literature [START_REF] Damnjanovic | A survey on 3GPP heterogeneous networks[END_REF], [START_REF] Vajapeyam | Downlink FTP performance of heterogeneous networks for LTEadvanced[END_REF], [START_REF] Oh | Cell selection for range expansion with almost blank subframe in heterogeneous networks[END_REF]. Resource partitioning is one of the techniques, in which the macro and the small BSs transmit in orthogonal time/frequency slots. Thereby, reducing the cochannel interference. Particularly, small BSs transmissions are scheduled in the almost blank subframe (ABS), in which macro BS transmit with lower power or do not transmit [START_REF] Oh | Cell selection for range expansion with almost blank subframe in heterogeneous networks[END_REF]. The advantage of using advanced interference cancellation receiver in conjunction with CRE technique is shown in [START_REF] Vajapeyam | Downlink FTP performance of heterogeneous networks for LTEadvanced[END_REF].

Another challenge, which is the focus of our work, is to determine the optimal CRE bias values for a desired optimal performance of the network. For example, the same bias values are not optimal for both rate-optimal and delay-optimal performance of the network. A set of papers determines the optimal range of bias values by simulations and experimentations for a specific performance requirement of the network [START_REF] Andrews | An overview of load balancing in hetnets: old myths and open problems[END_REF], [START_REF] Kitagawa | Performance evaluation of handover in LTE-advanced systems with pico cell range expansion[END_REF], [START_REF] Ökvist | LTE HetNet trial -range expansion including micro/pico indoor coverage survey[END_REF]. In [START_REF] Andrews | An overview of load balancing in hetnets: old myths and open problems[END_REF], the authors show that bias range of 5 -10 dB is rate-optimal when both small and macro BSs use the same frequency band. When ABS is used in conjunction with CRE the optimal bias range increases to 15-20 dB. In [START_REF] Kitagawa | Performance evaluation of handover in LTE-advanced systems with pico cell range expansion[END_REF], authors show that CRE bias values above 6 dB have a negative impact on the handover failure rate. In [START_REF] Ökvist | LTE HetNet trial -range expansion including micro/pico indoor coverage survey[END_REF], authors implemented a testbed to show that CRE increases uplink bit rates at the price of little reduction of the throughput on the downlink.

Another set of papers determines optimal bias by applying heuristic, learning algorithms, and optimisation [START_REF] Tian | An adaptive bias configuration strategy for range extension in LTE-advanced heterogeneous networks[END_REF]- [START_REF] Siomina | Load balancing in heterogeneous LTE: Range optimization via cell offset and load-coupling characterization[END_REF]. In [START_REF] Tian | An adaptive bias configuration strategy for range extension in LTE-advanced heterogeneous networks[END_REF], the bias is set according to the feedback from the network performance, which is maximisation of resource utilisation and improvement in quality of service for cell edge users. In [START_REF] Kikuchi | Proposal of adaptive control CRE in heterogeneous networks[END_REF], the authors propose an adaptive CRE scheme in which users choose among two biases depending on their signal-to-interference-plus-noise ratio (SINR). They showed through simulations that cell edge user throughput is improved while maintaining the average user throughput performance. In [START_REF] Kudo | Cell range expansion using distributed Qlearning in heterogeneous networks[END_REF], a Q-learning based algorithm is used by all users to determine the optimal bias value by minimising the cost, which is the number of users in outage, broadcasted by the small BSs. In [START_REF] Wang | Optimized fairness cell selection for 3GPP LTE-A macro-pico HetNets[END_REF], a dynamic programming and a greedy approach are proposed to associate users to small and macro BSs with the goal to achieve a global proportional fairness. In [START_REF] Siomina | Load balancing in heterogeneous LTE: Range optimization via cell offset and load-coupling characterization[END_REF], the load balancing problem is formulated as an integer optimisation problem aiming at maximising the Jain's fairness index. In [START_REF] Al-Rawi | A dynamic approach for cell range expansion in interference coordinated LTE-advanced heterogeneous networks[END_REF], a simple heuristic method is proposed to adapt the bias values. According to the relative utilities of the macro and the pico-cell the bias value is stepwise increased or decreased.

In all the above papers the optimal bias values are determined for a specific optimal performance requirement of the network. However, there is no general framework for determining the optimal biases for different optimal performance requirements of the network. This is the focus of our work. We address this problem by considering an α-fairness objective function that captures various aspects of the network performance and fairness for different α values. For α = 0 it gives the rate-optimal policy, for α = 1 gives the proportional fair policy, for α = 2 gives the delay-optimal policy, and as α → ∞ it gives the min-max load policy. We present a novel approach, according to which the BSs learn their optimal bias values given α. In contrast to the related load balancing literature our approach is unique. Although an α-fairness function is also considered for load balancing in [START_REF] Kim | Distributed alpha-optimal user association and cell load balancing in wireless networks[END_REF], our approach is entirely different in terms of the user association rule, game framework, and distributed learning algorithms. We solve an optimisation problem considering a generalised objective function that is not convex unlike in [START_REF] Ye | User association for load balancing in heterogeneous cellular networks[END_REF], [START_REF] Kim | Distributed alpha-optimal user association and cell load balancing in wireless networks[END_REF]. It is network centric, which gives full control to the network operator, as opposed to the user centric and hybrid approaches using MDP [START_REF] Stevens-Navarro | An MDP-based vertical handoff decision algorithm for heterogeneous wireless networks[END_REF], [START_REF] Elayoubi | A hybrid decision approach for the association problem in heterogeneous networks[END_REF]. It is guaranteed to converge to the optimal Nash equilibrium (NE) rather than a sub-optimal NE when compared to [START_REF] Niyato | Dynamics of network selection in heterogeneous wireless networks: An evolutionary game approach[END_REF], [START_REF] Aryafar | RAT selection games in HetNets[END_REF]. Compared to simulations/testbeds our approach is generic, i.e. not specific to a single scenario. Its distributive nature favours ease of implementation, scalability of network, and robustness to node failures.

Another distinctive feature of our approach is the use of distributed learning algorithms for solving a non-convex opti-misation problem of load balancing. Our approach is general and can be used to distributively solve other optimisation problems. The idea is to enforce the potential game structure so that the objective function is exactly the potential function.

A. Contributions

We summarise our main contributions below.

• We present a novel approach for load balancing in heterogeneous networks that uses CRE for user association.

Our approach is to distributedly minimise an α-fairness objective function that captures various performance and fairness criteria for different α. • We prove that for α = 0 the objective function captures rate-optimal policy of the network. For α → ∞, we prove that it results in the min-max load policy. We extend the classical result derived in [START_REF] Mo | Fair end-to-end window-based congestion control[END_REF] by considering a nonconvex α-fairness function.

• We solve the load balancing problem using distributed learning algorithms, where BSs learn the optimal CRE bias values. First, interactions between BSs are modelled as a potential game using wonderful life utility (WLU) structure. However, the WLU structures works only when the neighbourhood of the BSs is static. We propose a technique that allows to use the WLU structure with a time-varying neighbourhood. Next, the optimal NE of the game is determined by using log-linear learning algorithms. We consider two different settings: complete and partial information. In the former setting, we use classical log-linear learning algorithm (LLLA), whereas in the latter setting, we use binary log-linear learning algorithm (BLLLA). To the best of our knowledge, this is a novel approach in the load balancing literature.

• By running extensive simulations, we show that the proposed algorithms converge within a few tens of iterations to the optimal NE. The convergence speed of the BLLLA is comparable to that of the LLLA, meaning that partial information is sufficient in practical implementations. We also show that the classical best response (BR) algorithm does not necessarily converge to the optimal NE, although it necessitates complete information. This paper is organised as follows. In Section II, the description of system model and problem formation is given. In Section III, a potential game framework solution is presented. The various distributed algorithms that are considered in this paper are described in Section IV. In Section V, our approach is validated using extensive simulations. Finally, the conclusions and future work are given in Section VI.

II. SYSTEM MODEL A. Network Model

We consider a cellular network (typically a LTE-Advanced network) consisting of B e eNodes-B (eNB) or BSs and B s small BSs in a two dimensional region L. The set of all stations is denoted S B e ∪ B s . Every small BS maintains and can vary a parameter called CRE bias, denoted as c j for j ∈ B s . The CRE vector is c = c 1 , c 2 , . . . , c |S| , where c i is BS i's CRE bias, which for practical purposes takes discrete values from 1 to c max . The CRE biases for macro eNBs are fixed to unity, i.e., c k = 1, ∀k ∈ B e . This leads to no bias in the received power from a macro eNB.

1) Channel Model: The received power at location x from BS i is P i g i (x), where P i is the transmit power and g i (x) is the channel gain, which captures the effect of path-loss. The effect of small-scale fading is not considered because the time for user association procedure is assumed to be much larger than the channel coherence time [START_REF] Kim | Distributed alpha-optimal user association and cell load balancing in wireless networks[END_REF]. Inclusion of shadow fading increases the complexity of model and is thus left to future work. Formally, these assumptions related to channel model are summarised below.

Assumption 1: [Deterministic propagation model] The channel gain g i : R 2 → R is a deterministic function of the distance between BS i and a user.

The SINR γ i (x) at location x provided by the BS i is a random variable defined as:

γ i (x) = P i g i (x) j∈S δ j P j g j (x) + N 0 , (1)
where N 0 is thermal noise power and δ j is a random variable which is one when station j is active (i.e. transmitting) and 0 otherwise. To get more insights without much complexity due to random variable δ, we make use of the following assumption.

Assumption 2: [Worst case interference] The variables δ j are deterministic and equal to 1 for all j. This provides an upper bound on the interference, which is the worst case interference. The consequence of assumptions 1 and 2 is that γ i (x) is deterministic.

2) CRE User Association Rule: A user association rule using CRE is commonly used in the heterogeneous networks [START_REF] Damnjanovic | A survey on 3GPP heterogeneous networks[END_REF]- [START_REF] Eguizabal | Interference management and cell range expansion analysis for LTE picocell deployments[END_REF]. According to this rule, a user located at x is served by the BS that provides the highest biased received power and the SINR provided by it is greater than the minimum required SINR (γ min ). The region D i (c) served by BS i is defined as:

D i (c)={x|∀j ∈ S, P i g i (x)c i ≥ P j g j (x)c j , γ i (x) ≥ γ min } . (2)
Note that D i (c) is a bounded region because of γ min . According to assumptions 1 and 2, the association rule should be understood for a given realisation of the shadowing mask and averaged over fast fading variations.

3) Physical Data Rate: The physical data rate received by a user served by BS i with SINR γ i is denoted as ν i (γ i ), which is a non-negative and a non-decreasing function of SINR.

4) Traffic Model: Users are assumed to arrive in the system according to a spatial random process, download a file of random size and leave the system when the download is over. This is referred to as elastic traffic. At location x, the arrival rate is denoted λ(x) [arrivals/s/m 2 ] and the average file size is 1/µ(x) [bits]. If x is associated to BS i, the load generated by

x on i is i (x) λ(x) µ(x)νi(γi)
. Following [START_REF] Kim | Distributed alpha-optimal user association and cell load balancing in wireless networks[END_REF], [START_REF] Bonald | Wireless downlink data channels: User performance and cell dimensioning[END_REF], we model every BS i as a M/G/1/PS queue of load:

ρ i (c) = x∈Di(c) i (x)dx. ( 3
)
BS i is stable if and only if 0 ≤ ρ i < 1. In this work, only stable network is considered. The flow throughput of users is defined as the ratio of the mean file size to the mean file download duration [START_REF] Bonald | Wireless downlink data channels: User performance and cell dimensioning[END_REF]. Assumption 3: [Time-scale separation] The process of updating the CRE parameters is supposed to be long with respect to the traffic variations. The M/G/1/PS queues describing the BSs traffic are thus supposed to have reached their stationary regime before any new change of the CRE parameters.

B. Problem Formulation and Objective Function

Following [START_REF] Kim | Distributed alpha-optimal user association and cell load balancing in wireless networks[END_REF], we intend to minimize an α-fairness function φ α (c) over the feasible set F, which are given below.

φ α (c) = i∈S (1-ρi(c)) 1-α α-1 , α ≥ 0, α = 1, -i∈S log (1 -ρ i (c)) , α = 1, (4)
F = {ρ|0 ≤ ρ i (c) < 1, c i ∈ [1, c max ] , ∀i ∈ S}. (5)
The function φ α (c) is in general non-convex and even if it is convex the set F is non-convex because c takes discrete values. The function φ α (c) captures various aspects of fairness and so of load balancing of the network for different values of α, which are described below.

(α = 0) Rate-optimal policy: Minimising φ 0 (c) gives a rate-optimal policy. See Appendix A for the proof.

(α = 1) Proportional fair policy: In this case, φ 1 (c) captures the proportional fairness of the network [START_REF] Mo | Fair end-to-end window-based congestion control[END_REF].

(α = 2) Delay-optimal policy: It can be shown that minimising φ 2 (c) corresponds to minimising the average number of flows of the network. Little's law says that minimising the average number of flows is equivalent to minimising the average delay experienced by a typical flow. Therefore, minimising φ 2 (c) is equivalent to minimising the average delay of the network. For more detailed discussion refer to [START_REF] Kim | Distributed alpha-optimal user association and cell load balancing in wireless networks[END_REF].

(α → ∞) Minmax policy: As α → ∞ the minimiser of φ α (c) tends to the min-max load vector. It is a standard result with convex objective functions [START_REF] Kim | Distributed alpha-optimal user association and cell load balancing in wireless networks[END_REF], [START_REF] Mo | Fair end-to-end window-based congestion control[END_REF], [START_REF] Bonald | Impact of fairness on internet performance[END_REF]. We prove this result for our non-convex objective function in Theorem 1.

Definition 1: [Min-max load vector [START_REF] Bonald | Impact of fairness on internet performance[END_REF]] Let all the vectors in F be sorted in increasing order. A vector ρ ∈ F is min-max if ρ is lexicographically not greater than any vector in F. The vector ρ is lexicographically lower than y, denoted ρ ≺ y, if the first non-zero component of ρ -y is negative. We say that ρ is not greater than y, denoted by ρ y, if ρ ≺ y or ρ = y.

Let

r i (c) = 1 -ρ i (c), ∀i ∈ S. Let X = r ∈ R |S| |∃c : ρ(c) ∈ F, r(c) = r .
Load vector ρ * is a minmax if and only if r * is max-min vector.

Theorem 1: Let r α ∈argmax r∈X i∈S r 1-α i 1-α . Then, any accumulation vector of the trajectory {r α } α>1 is max-min in X .

Proof The proof is given in Appendix B.

In Fig. 1, we show an example of set F obtained with 2 BSs having different transmit powers located on a two-dimensional region. It is clear from the figure that even if the CRE set were continuous, F would not be convex. We also show the optimal loads obtained for different α values. All the optimal load points are located on the Pareto frontier. The point for α ≥ 200 in Fig. 1 is the min-max load point because a point of equal coordinates on the Pareto frontier is the min-max point.

III. POTENTIAL GAME FRAMEWORK

In this section, we present an approach using potential game framework for distributed optimisation of the objective function. We do not aim to describe and analyse selfish nature of the BSs that aim to minimise their costs. Rather, our goal is to achieve the global objective of load balancing by prescribing a cost function to the BSs. For this context, potential games provide a good framework because players of such a game distributively optimise the potential function.

We model the problem as a game, where the BSs are players and allowed CRE bias values are their strategies. Formally, a CRE game is defined by the tuple Γ = S, {X i } i∈S , {U i } i∈S , where S is the set of BSs, X = X 1 × X 2 × . . . × X |S| is a strategy set or an action set, and

U i : X → R is a cost function. X i is
: X → R such that ∀i ∈ S, ∀c i , c i ∈ X i and ∀c -i ∈ X -i , U i (c i , c -i ) -U i (c i , c -i ) = P (c i , c -i ) -P (c i , c -i ), (6)
then the game is an exact potential game. An exact potential game has at least one PNE and local optimisers of the potential function are PNEs [START_REF] Monderer | Potential Games[END_REF]. Furthermore, there exist several distributed learning algorithms that converge to a PNE when the game has a potential (see next Section). In our problem, we intend to turn our objective function (4) into a potential function by designing cost functions of the BSs.

Two ways of achieving it are by using identical interest utility (IIU) and WLU [START_REF] Li | Designing games for distributed optimization[END_REF].

Identical interest utility: In this structure, the cost function U i (c) is completely aligned with the potential function, i.e., U i (c) = φ α (c), ∀i ∈ S. However, it requires the BS i to know the loads of all the other BSs in order to compute U i (c).

Wonderful life utility: With the WLU, a BS needs only to know the loads of its neighbour BSs. The WLU structure of the individual BSs is defined as

U i (c i , c -i ) = j∈Ni (1 -ρ j (c i , c -i )) 1-α α -1 , (7)
where N i is the neighbour set of BS i,

N i = c {j ∈ S|∃x ∈ D i (c), P i g i (x)c i = P j g j (x)c j } . ( 8
) A neighbour set N i is all possible BSs that share boundary with BS i for at least one possible bias value. The motivation behind WLU is that the action of the BS i only affects its neighbour BSs.

Lemma 1: The WLU cost function ( 7) leads to an exact potential game.

Proof: Consider the strategy profiles a = (c i , c -i ) and b = (c i , c -i ). From the objective function ( 4) we have

φ α (a)-φ α (b)= j∈S (1-ρ j (a)) 1-α α -1 - j∈S (1-ρ j (b)) 1-α α -1 , (9)
= j∈Ni (1-ρ j (a)) 1-α α-1 - j∈Ni (1-ρ j (b)) 1-α α -1 .( 10
)
The second equation above is true because the CRE bias of the BS i will only affect its neighbour BSs. Hence, the utility of the BS i is aligned with the potential function and the resultant game is an exact potential game.

IV. DISTRIBUTED LEARNING ALGORITHMS

Recall that the potential function property enables finding a PNE through distributed learning algorithms. In this section, we introduce distributed learning algorithms that are used to find the PNE of the CRE game. First, we present the BR algorithm and the LLLA for the complete information setting. Next, the BLLLA for the partial information setting is described.

A. Best Response Algorithm

Best response algorithm is an asynchronous algorithm where at any given time only a single BS updates its strategy. Assume a time-varying random process with which a BS is chosen to revise its strategy 1 . At any time step t the selected player i chooses a strategy c i that minimises his cost, given the strategies c -i of other players. In other words, player i chooses a strategy from his best response set B i ,

B i (c -i ) = arg min ci U i (c i , c -i ) . ( 11
)
Note that the BR algorithm requires complete information, in which the effects of choosing all the other strategies are known, and it is not guaranteed to converge to the optimal NE [START_REF] Monderer | Potential Games[END_REF].

B. Log-linear Learning Algorithm

LLLA is a generalisation of BR. It is summarised in Algorithm 1. At each step the LLLA deviates from BR with a probability that tends to zero as the parameter τ goes to zero. However, as τ tends to infinity the LLLA selects actions randomly with uniform probability. It guarantees the convergence to the optimal NE with probability that tends to one as τ goes to zero [START_REF] Marden | Revisiting log-linear learning: Asynchrony, completeness and a payoff-based implementation[END_REF]. However, for this algorithm the BSs require again the complete information. For example, given the strategies of others, the BS has to know the cost function value for all its strategies. With this information, it selects a strategy to play according to a probability distribution. In general, acquiring this amount of information is not feasible. To overcome this difficulty in the next subsection we propose to use BLLLA.

Algorithm 1 Log-linear Learning Algorithm [START_REF] Marden | Revisiting log-linear learning: Asynchrony, completeness and a payoff-based implementation[END_REF] 1: Initialisation: Arbitrary set CRE bias c i ∀i ∈ S. 2: Set parameter τ . 3: While t ≥ 1 do 4: Randomly select a BS i. 5: Select its CRE c i (t) from X i with probability p ci i (t),

p ci i (t) = exp -1 τ U i (c i , c -i (t -1)) c i ∈Xi exp -1 τ U i (c i , c -i (t -1)) . (12)
6: All the other BSs must repeat their previous actions, i.e., c -i (t) = c -i (t -1).

C. Binary Log-linear Learning Algorithm

The BLLLA works even if only partial information about the game is available to the players. Partial information is the information that a player has about its current strategy. Unlike complete information the effect of choosing any other strategy is not known to the player. As LLLA the BLLLA is also an asynchronous algorithm. In this algorithm, whenever the BS updates its strategy it does it in two steps. In the first step, the BS tries a strategy from its strategy set to obtain its payoff. In the second step, the BS randomly chooses among the two strategies (present strategy and trial strategy) as summarised in Algorithm 2.

D. Effect of time-varying neighbours

For all the above algorithms, the BS i to know its neighbours N i to calculate its cost using the WLU structure. For a given CRE bias vector, providing to every BS the neighbours set is a standard task for network operators. This task can be performed automatically e.g., using automated neighbour relation (ANR) standardised by 3GPP [START_REF] Gpp | Evolved Universal Terrestrial Radio Access (E-UTRA) and Evolved Universal Terrestrial Radio Access (E-UTRAN)[END_REF]. The difficulty here is to deal with time-varying neighbourhood. If neighbourhood is changing then WLU doesn't lead to a Algorithm 2 Binary Log-linear Learning Algorithm [START_REF] Marden | Revisiting log-linear learning: Asynchrony, completeness and a payoff-based implementation[END_REF] 1: Initialisation: Arbitrary set CRE bias c i ∀i ∈ S. 2: Set parameter τ . 3: While t ≥ 1 do 4: Randomly select a BS i. 5: Select a trial action ĉi ∈ X i with uniform probability. Play ĉi and observe its cost. 6: Play the action c i (t) ∈ {c i (t -1), ĉi } as given below.

c i (t) =      c i (t -1), w.p. e -1 τ U i (c(t-1)) e -1 τ U i (c(t-1)) +e -1 τ U i (ĉ i ,c -i (t-1))) , ĉi , w.p. e -1 τ U i (ĉ i ,c -i (t-1))) e -1 τ U i (c(t-1)) +e -1 τ U i (ĉ i ,c -i (t-1))) .
(13) 7: All the other BSs must repeat their previous actions, i.e., c -i (t) = c -i (t -1). The algorithms above can use the WLU structure without knowing any neighbour set at the start of the algorithm. In the process of learning, whenever BS i updates its strategy if it finds new neighbours then it should remember and include them in N i . As the CRE bias set is finite, there is a time instant after which all neighbourhood sets N i , ∀i, are constant. Thus, from this time instant, the game becomes a potential game and the above algorithms will converge to the optimal PNE.

V. SIMULATION RESULTS

In this section, we show simulation results considering standard parameters as adopted in 3GPP [START_REF] Gpp | Technical specification group radio access network; evolved universal terrestrial radio access (eutra); further advancements for e-utra physical layer aspects[END_REF]. These parameters are listed in Table I. The region L considered is a square of side 1000 m. Among the 8 BSs located in L, BS 1 at the center is a macro BS that transmits with P macro and the rest are small BSs that transmit with P small . The user traffic is fixed in time but varies with location across an average traffic density of 64 bits/s/m 2 . There are two hotspots where the traffic is 5 times the average traffic, which can be seen in Fig. 2.

The following channel model is used in the simulations [START_REF] Goldsmith | Wireless communications[END_REF].

g i (x) = min 1, K |x i -x| -η , ( 14
)
where K is a constant, |x i -x| is the distance between the location x i of the BS i and the location x of the user, and η is Fig. 2: The variations of the coverage regions of BSs obtained using the optimal CRE for different α. Varying traffic, which is normalised with the average traffic of 64 Kbits/s/m 2 , is shown using varying intensity of colours.

the path-loss exponent. We use the classical Shannon formula for calculating channel capacity ν i (γ i ) at any location x.

The BSs play CRE game and learn the optimal NE using the proposed learning algorithms. The optimal coverage regions obtained using the optimal CRE bias values for different α are shown in Fig. 2. The corresponding optimal bias values and loads are shown in Table II. In Fig. 2a, the rate-optimal policy, which is obtained for α = 0, is shown. The optimal CRE bias values of all the small BSs is close to one. This is intuitive because for rate-optimal policy the bias values should be equal to one. If we take more iterations of the algorithms then the optimal bias values are guaranteed to converge to one. Therefore, this case corresponds to the classical user association without the use of CRE bias leading to a heavy load imbalance. We can observe that the load of the macro BS 1 is 92%, which is near an overload. On the other hand, the loads of all the small BSs are less than 12%, which is a heavy under-utilisation. This case serves as a benchmark for other cases of α to compare for load balancing.

In Fig. 2b, the coverage regions for delay-optimal policy, which is obtained for α = 2, is shown. As α increases to 2 the coverage regions of all small BSs increase and that of the macro BS decreases. This happens due to the increase in the optimal bias values. The load of the macro BS is decreased to 62% and the utilisation of small BSs is increased.

In Fig. 2c, the coverage regions for the min-max policy, which is obtained for α = 200, is shown. When α increases to higher value, here α = 200, the load of the macro BS 1 is further reduced to 42%. It can be observed that the loads of all the BSs are equalised except for BS 3 and 7. The utilisation of these BSs cannot be increased further because these BSs are near to the macro BS 1 that causes heavy downlink interference to their users.

The evolution of the optimal CRE biases and optimal loads for α values ranging from 0 to 200 is shown in Fig. 3. The figure shows how the optimality changes from rate-optimal to min-max optimal as α increases from 0 to 200. Therefore, initially at α = 0, the optimal bias of all BSs is close to one and correspondingly there is a heavy load imbalance. As α increases, the load of the macro BS decreases and that of TABLE II: Comparison of optimal CRE, optimal loads of BSs for different α. The evolution of the objective function for different algorithms is shown in Fig. 4. In all the cases, we know that the LLLA and the BLLLA converge to the global minimum. In Fig. 4a, we observe that BR converges much faster than the LLLA and the BLLLA, whereas it requires complete information. On the other hand, the BLLLA requires only partial information, which is an advantage for practical implementations but does not loose much in terms of convergence speed. The figure also shows that smaller values of τ result in faster convergence for the LLLA and the BLLLA.

α = 0 α = 2 α → ∞ BS i c * i ρ * i % c * i ρ * i % c * i ρ * i % 1
In Fig. 4b, we see that the BR does not converge to the global minimum because the LLLA for τ = 0.01 sometimes achieves lower values of potential. Also, the LLLA for τ = 0.001 converges to the same local minimum as BR because for smaller τ the LLLA behaviour is similar to the BR. However, the LLLA and the BLLLA for τ = 0.01 converge to the global minimum. In Fig. 4c, it is clear that the BR and the LLLA for τ = 0.001 converge to a local minimum, whereas both the LLLA and the BLLLA for τ = 0.01 converge to the global minimum.

VI. CONCLUSIONS

In this paper, a novel approach for load balancing using CRE association technique is presented. Our approach exploits the potential game structure and distributed learning algorithms. By running extensive simulations in two settings, which are complete and partial information settings, we show that the proposed algorithms converge within a few tens of iterations to the optimal NE, which is also a minimiser of a α-fairness function of the network. The convergence speed of the BLLLA that uses partial information is comparable to the LLLA that uses complete information, meaning that partial information is sufficient in practical implementations. We also show that the classical BR algorithm does not necessarily converge to the optimal NE, although it necessitates complete information. As future work we intend to extend our work for the following: 1) noisy estimation of the BSs utilities, 2) admission control policies to overcome the overload situation, 3) effect of using advanced interference management techniques in conjunction with CRE association, and 4) adaption of the algorithms to varying traffic.

Fig. 1 :

 1 Fig. 1: Feasible set F for 2 BSs.

  (α = 200) Min-max policy.

  Optimal loads.

Fig. 3 :

 3 Fig.3: Evolution of optimal CRE and optimal load with α.

  τ = 0.001 LLLA, τ = 0.01 BLLLA, τ =0.001 BLLLA, τ =0.01 (a) (α = 0) Rate-optimal policy.

  τ = 0.001 LLLA, τ = 0.01 BLLLA, τ =0.001 BLLLA, τ =0.01 (b) (α = 2) Delay-optimal policy.

  τ = 0.001 LLLA, τ = 0.01 BLLLA, τ = 0.001 BLLLA, τ =0.01 (c) (α = 200) Min-max policy.

Fig. 4 :

 4 Fig. 4: Convergence of BR, LLLA, and BLLLA.

TABLE I :

 I Simulation parameters.

	Parameter	Variable Value
	Number of BSs	Ns	8
	Transmit power of macro BS	Pmacro	46 dBm
	Transmit power of small BS	P small	24 dBm
	Average file size	1 µ	0.5 Mbytes
	Average traffic load density	λ µ	64 bits/s/m 2
	System bandwidth	W	20 MHz
	Noise power	N 0	-174+10log(W) dBm
	Minimum SINR	γ min	-10 dB
	Path-loss exponent	η	3.5
	CRE bias set	c

i {1, 1.1, 1.2, . . . , 16}

potential game. To address this problem, we propose the following technique.

Uniform probability or stationarity of the process is not required, it is only required that the probability of selecting any player is positive.

APPENDIX

A. Proof of Rate-optimal Policy

It can be shown that the minimiser of φ α (c) is same as the minimiser of the arithmetic mean of the BSs loads, i.e., i∈|S| ρ i (c). We will prove that minimising φ 0 (c) is indeed rate optimal policy by contradiction. Consider that at the minimum φ * 0 a location x 0 is associated with a BS j with rate r j (x 0 ), which is not the maximum. Then, there exist a BS k that provides the maximum rate r k (x 0 ). Let φ k 0 be the value of the objective function when the UE at location x 0 is associated with BS k. The difference of the objective function due to the loads of BSs j and k is

which is a contradiction that φ * 0 is the minimum. Therefore, we conclude that at φ * 0 all the locations will be served by the BS that provide the highest rate.

B. Proof of Theorem 1

The proof is divided into the following two lemmas. Lemma 2 gives the property that is required for proving the Lemma 3, which concludes the proof of the Theorem 1.

Lemma

Proof: Let α > 1. Without loss of generality, assume that r and y are sorted in increasing order and that r 1 > y 1 . Let δ = r 1 -y 1 . Then

Then we have f α (r) > f α (y) if and only if

Lemma 3:

Then S is non-empty and is made of max-min vectors in X . Proof:

It is a decreasing nested sequence of non-empty compact sets. By Cantor's intersection theorem, it is not empty and compact.

Let x * ∈ A>1 S A . There is an increasing sequence α(n)

there is y x * . Then, by Prop. 2, there is N such that, for every n ≥ N , f α(n) (y) > f α(n) (x * ). But, by definition of x α(n) , f α(n) (x α(n) ) ≥ f α(n) (y), which is a contradiction with the fact that f α(n) (x α(n) ) → f α(n) (x * ), which is ensured by Berge maximum theorem.


Jean-Marc Kelif

Stephane Senecal

Marceau Coupechoux

Analytical Performance Model for Poisson Wireless Networks with Pathloss and Shadowing Propagation

The SINR (signal to interference plus noise ratio) is a key factor for wireless networks analysis. Indeed, the SINR distribution allows the derivation of performance and quality of service (QoS) evaluation. Moreover, it also enables the analysis of radio resources allocation and scheduling policies, since they depend on the SINR reached by a UE (User Equipment). Therefore, it is particularly interesting to develop an analytical method which allows to evaluate the SINR, in a simple and quick way, for a realistic environment. Considering a stochastic Poisson network model, we establish the CDF (cumulative distributed function) of the SINR. We show that the shadowing can be neglected, in many cases, as long as mobiles are connected to their best serving base station (BS), i.e. the BS which offers them the most powerful useful signal. As a consequence, the analysis of performance and quality of service, directly derived from the CDF of SINR, can be established by using a propagation model which takes into account only the pathloss. Moreover, we establish that the Fluid network model we have proposed can be used to analyze stochastic Poisson distributed network. Therefore, the analysis of stochastic Poisson network can be done in an easy and quick way, by using the analytical expression of the SINR established thanks to the Fluid network model.

I. INTRODUCTION

Mobile services demand is becoming more and more important. Therefore, the estimation of performance and quality of service (QoS) has to be more and more precise in the way to answer with accuracy to the demand. Performance and quality of service evaluations of wireless networks can be analyzed by using simulations or analytical models. Several QoS parameters (like throughput, outage probability) can be derived from the SINR distribution [START_REF] Lagrange | Principes et évolutions de l'UMTS[END_REF] [2] [3] [START_REF] Viterbi | Other-Cell Interference in Cellular Power-Controlled CDMA[END_REF].

Moreover the knowledge of the signal to interference plus noise ratio (SINR) reached by a UE (User Equipment) allows to better perform the radio resource allocation and the scheduling policy. Analytical models thus try to derive simple SINR formula in order to quickly evaluate the performance of a cellular network. Therefore, their analysis need tractable and accurate models of networks. Two factors play an important role in the evaluation of the SINR : the localization of BS and the propagation phenomena. Among the network model usually considered, the Hexagonal one is the most used. However this model is based on a regular deployment of BS on an area. The Fluid model, which is another model of network, considers the interfering base stations as a continuum of infinitesimal interferers distributed in space. The main interest of this model consists in its tractability, in the possibility to establish closed form formula of the SINR, whatever the location of a UE, and to establish the SINR distribution [START_REF] Kelif | On the Dimensioning of Cellular OFDMA Networks[END_REF] [6], too.

In this paper we consider a Poisson model network: the base stations are randomly distributed according to a spatial Poisson process [7] [8]. This model allows to take into account more realistic environments than a hexagonal model of network. Indeed, the distances between base stations are not constant. The propagation is generally modeled by a term which depends on the distance from the transmitter, the pathloss. Since in a real network, the power received at any point of the system also depends on the local area, another term which characterizes local area impact can be added. The last term, the shadowing, is generally expressed as a lognormal distributed function [START_REF] Kelif | Effect of Shadowing on Outage Probability in Fluid Cellular Radio Networks[END_REF]. Papers generally analyze the shadowing impact by considering users connected to their nearest BS [START_REF] Kelif | Effect of Shadowing on Outage Probability in Fluid Cellular Radio Networks[END_REF] [START_REF] Kelif | Impact of Topology and Shadowing on the Outage Probability of Wireless Networks[END_REF]. Recent papers [START_REF] Minelli | SIR Estimation in Hexagonal Cellular Networks with Best Server Policy[END_REF] [START_REF] Blaszczyszyn | Wireless Cellular Networks with Shadowing[END_REF] analyze the impact of shadowing by considering users connected to the BS which offers the highest useful signal.

Our contribution: This paper focuses on the impact of the pathloss and the shadowing on the CDF of the SINR, considering that users are connected to their best serving station : UEs are connected to the BS which offers the highest useful signal. We establish that in a Poisson network the CDF of the SINR calculated by considering a propagation model which takes into account the pathloss and the shadowing is very close to the CDF obtained without considering the shadowing (only the pathloss is considered). This result is very interesting. Indeed, it allows to use the simple expression of the SINR given by the Fluid network model. Therefore, the determination of the quality of service and performance of a wireless network can be done quickly and in a simple way.

The organization of the paper is as follows. In Section II, we present the system model. We recall, in Section III, that performance and QoS of Poisson model can be established by using a Fluid model when the shadowing is not taken into account. Section IV expresses the best server SINR. In Section V, the impact of shadowing is analyzed, considering UEs connected to the BS which offers the highest useful signal. In Section VI, some precisions are given about the impact of the analysis we developed on the study of realistic wireless systems. A conclusion is given in Section VII.

II. SYSTEM MODEL

Let us consider a wireless network. We focus on the downlink transmission part. Our aim is to evaluate the performance and the quality of service of a single user. We consider an access technology in which the radio resources of a base station (BS) are divided in a number of parallel, orthogonal, non-interfering channels (subcarriers), i.e. OFDMA. Therefore, only inter-cell interference is considered, no intra-cell interference.

A. Network Topology : Poisson Network

In a real network, the inter site distance is variable. The Poisson model network, characterized by the density of BS, allows to take it into account by considering a Poisson distribution of base stations in a given area (Fig. 1). In this configuration, the cells of the network form a Voronoï diagram. Therefore, it becomes necessary to analyze a wide zone, with a great number of base stations, to determine the statistical characteristics of the network in terms of performance and quality of service.

B. SINR of a user

We consider a single frequency network composed of N base stations, transmitting at power P on each subcarrier. We define g i (u) the path gain between BS i and user u on a given subcarrier. The SINR u of user u served by BS i on the considered subcarrier is given by:

u = P g i (u) P j6 =i P g j (u) + N th . (1)
with N th the thermal noise on a subcarrier.

C. Performance and quality of service

The knowledge of the SINR allows to calculate the throughput that may be reached by a user. Indeed, considering any subcarrier as an AWGN (Additive White Gaussian Noise) channel, the SINR received by a mobile enables the determination of the spectral efficiency D u (in bits/s/Hz) by using the Shannon formula:

D u = log 2 (1 + u ). (2)
Let us notice that there are alternative approaches like using a modified upper bounded Shannon formula or throughput-SINR tables coming from physical layer simulations. Moreover, expressions (1) and ( 2), calculated at any location of the network, allow an evaluation of the CDF (Cumulative Distribution Function) of the SINR (or the throughput). The SINR CDF also provides the outage probability, i.e. the probability that a user cannot be accepted in the network since he cannot have a sufficient throughput. It is therefore important to develop a method which allows to determine these characteristics with a high accuracy, for a user at any distance r from his serving BS.

Moreover, since the throughput allows to know the quality of service that can be offered to a user, these methods make it possible to determine this characteristic with a high accuracy in a simple way. In particular, the minimum throughput, obtained at cell edge, can be derived. By doing an integration all over the cell range, the average throughput of the cell can be calculated, too. Dynamical analysis also need the knowledge of the SINR [START_REF] Rong | Performance Evaluation of Cellular Networks Offering TV Services[END_REF] as input.

III. REMINDER ABOUT THE FLUID NETWORK

A. Fluid Network

The fluid network model consists in replacing a given fixed finite number of transmitters by an equivalent continuous density of transmitters [START_REF] Kelif | Downlink Fluid Model of CDMA Networks[END_REF] [START_REF] Kelif | A Fluid Model for Performance Analysis in Cellular Networks[END_REF]. Given an inter site distance

B. Calculation of the SINR

We consider a path gain g j (u) = Kr ⌘ j (u), where K is a constant, r j (u) is the distance between user u and BS j and ⌘ the path loss exponent. Let us consider that a BS (eNode-B) transmits at power P on each subcarrier. Denoting r = r i , we can express (1) as (dropping u):

= r ⌘ P j6 =i r ⌘ j , (3)
We consider a urban environment, where the thermal noise can be neglected. By considering expression (3), we can see that the calculation of the analytical expression of the SINR of a user depends on its location, on the location of its serving BS, and on the location of each interfering BS.

Considering the fluid model, the SINR only depends on the distance r of the user to its serving base station [START_REF] Maqbool | Analytical Performance Evaluation of Various Frequency Reuse and Scheduling Schemes in Cellular OFDMA Networks[END_REF]:

(r) = ⌘ 2 2⇡⇢ BS r ⌘ (2R c r) 2 ⌘ (4)
If ⇢ BS is known, this analytical model allows to significantly simplify the calculation of the SINR: the only required variable is the distance of the mobile to its serving BS. This model has been proven to be reliable and close to the reality for homogeneous hexagonal networks [START_REF] Kelif | Downlink Fluid Model of CDMA Networks[END_REF] [15], as well as for heterogeneous networks [START_REF] Kelif | Impact of Transmitting Power on Femto Cells Performance and Coverage in Heterogeneous Wireless Networks[END_REF].

The cell edge throughput can particularly be calculated by setting r = R c in ( 2) and ( 4). Therefore, the minimum performance and quality of service offered to UE is evaluated in a simple way. Moreover, a simple integration over the cell range allows to calculate the average throughput of the cell.

These results on the fluid model are valid for a constant inter site distance 2R c . In the following section, we remind that we established [START_REF] Kelif | A Fluid Approach for Poisson Wireless Networks[END_REF] a correspondence between a stochastic Poisson network and the Fluid network model.

C. Comparison of the SINR CDF for the Poisson and Fluid models

Since the CDF of the SINR characterizes the performance and the quality of service of wireless systems, we established a modified expression of the SINR given by the fluid model, which allows to calculate the CDF of SINR for the Poisson network reminded hereafter [START_REF] Kelif | A Fluid Approach for Poisson Wireless Networks[END_REF]:

SIN R modif ied F luid = SIN R F luid (a⌘ + b) (5)
which yields

CDF P oisson ⇡ CDF modif ied F luid (6)
where a = 3 and b = -6, for a wide range of values of ⌘ (ranging from 2.2 to 4.2, usual range for the path-loss exponent is comprised between 2.8 and 3.6) We observed that for ⌘ = 2.8, 3, 3.6, 3.8, the CDF of the SINR established by the modified Fluid model and by the Poisson model are very close: the differences between them are less than 0.4 dB [START_REF] Kelif | A Fluid Approach for Poisson Wireless Networks[END_REF].

IV. SINR CALCULATION WITH PATHLOSS AND SHADOWING

We consider N interfering base station (BS), a mobile u and its nearest BS 0 .

A. Impact of the pathloss on the SINR By considering expression (3), we can see that the calculation of the SINR of a user depends on its location. We notice that in this case, users are connected to their best serving station which is also their nearest BS. B. Pathloss and Shadowing impact on the SINR 1) Propagation: Considering the power P j transmitted by the BS j, the power p j,u received by a mobile u can be written:

p j,u = P j Kr ⌘ j,u Y j,u , (7)
where Y j,u = 10

⇠ j,u 10
represents the shadowing effect. The term Y j,u is a lognormal random variable characterizing the random variations of the received power around a mean value. ⇠ j,u is a Normal distributed random variable (RV), with zero mean and standard deviation, , comprised between 0 and 10 dB. The term P j Kr ⌘ j,u , where K is a constant, represents the mean value of the received power at distance r j,u from the transmitter (BS j ). The probability density function (PDF) of this slowly varying received power is given by

p Y (s) = 1 a s p ⇡ exp ✓ ln(s) am p 2a ◆ 2 (8)
where a = ln 10 10 , m = 1 a ln(KP j r ⌘ j,u ) is the (logarithmic) received mean power expressed in decibels (dB), which is related to the path loss and is the (logarithmic) standard deviation of the mean received signal due to the shadowing.

2) SINR of a User connected to its nearest BS: Considering the useful power P 0 transmitted by its nearest base station BS 0 , the useful power p 0,u received by a mobile u connected to BS 0 can be written:

p 0,u = P 0 Kr ⌘ 0,u Y 0,u . (9)
For the sake of simplicity, we now drop index u and set r 0,u = r. The interferences received by u coming from all the other base stations of the network are expressed by:

p ext = N X j=1 P j Kr ⌘ j Y j . ( 10
)
The SINR at user u is given by:

= P 0 Kr ⌘ Y 0 P N j=1 P j Kr ⌘ j Y j + N th . ( 11
)
3) SINR of a User connected to its best serving BS: In this case, the user is connected to the BS which offers the highest useful signal. Therefore, the SINR at user u is given by:

= max N j=0 (P j Kr ⌘ j Y j ) P N j6 =j ⇤ P j Kr ⌘ j Y j + N th . ( 12
)
where j ⇤ is the BS which offers the highest useful signal to the user u. We notice that in this case, due to shadowing, users are connected to a BS which is not necessary the nearest one.

V. SHADOWING IMPACT ON THE SINR

In this section, we analyze the shadowing impact on the CDF on the SINR in a system modeled by a Poisson network.

A. Base stations distribution

In order to place the base stations, we set the expected half inter site distance as R c . This hypothesis fixes the density of base stations ⇢ BS . The values of ⇢ BS and of the studied surface S

A give the Poissonian characteristic of the network: the number of BS is drawn according to a Poisson distribution of parameter ⇢ BS S A . The surface is chosen to obtain in average 50 stations in the area. It allows to have a significant number of cells, representative of a realistic zone covered by BS, and a significant number of interfering BS for the computation of the SINR. Those BS are then placed in the network, with no pairwise constraint (Fig. 1): distances between neighboring base stations may be very low.

B. Poisson network SINR computation

The users are uniformly distributed on the whole area S A . Then, the SINR of a UE is computed from its definition: the best serving BS gives the power of the received signal, and all the other stations generate interference. Several Monte Carlo simulations are run. At each run, the number and locations of the BS change, whereas the set of studied points (UE) is fixed. As a result, for the set of studied points, we obtain the corresponding SINR with different configurations of BS. Therefore, it becomes easy to compute the CDF of the SINR received by UE in this zone. Considering a toroidal shape of the network allows to consider it as virtually infinite with no "edge effect" for the computation of the SINR.

Remark

We consider a model proposed in [START_REF] Viterbi | CDMA -Principles of Spread Spectrum Communication[END_REF]. The Rayleigh fading, not considered in this model, may be indistinguishable from shadowing if the fading is sufficiently slow [START_REF] Viterbi | CDMA -Principles of Spread Spectrum Communication[END_REF], as for example if the mobile travels through a region of deep fades at a very slow speed.

C. Cumulated Distributed Function of the SINR with Shadowing

We consider a Poisson model network. And we calculate the CDF of the SINR, in the zone covered by this network, by taking into account the shadowing. The UE are connected to their best serving station, i.e. the BS which offers the best signal. The SINR is calculated by using [START_REF] Blaszczyszyn | Wireless Cellular Networks with Shadowing[END_REF].

The figures 3 and 4 show the CDF of the SINR for different values of the pathloss parameter ⌘ and for standard deviation of values 0 dB (no shadowing), 3 dB, 6 dB, and 8 dB. It can be observed that for ⌘ values of 3.5 and 4 (Fig. 4), the curves between 0 and 8 dB are indistinguishable. Only for ⌘ = 2.6 (Fig. 3) it is hardly possible to distinguish between the different curves. These curves show that for typical values of the pathloss parameter ⌘, comprised between 2.6 and 4, and values of the standard deviation of the shadowing (  6 dB) the impact of the shadowing is negligible : the difference between the CDFs with shadowing and without shadowing is less than 0.6 dB. Let us notice that for ⌘ = 2.6, and for an outage of 5%, the difference , between the curve without shadowing and the curve with = 8 dB, may reach 1.3 dB.

Therefore, the impact of the shadowing is low in terms of reachable throughput, coverage (probability of outage) and QoS.

D. Explanation of these results

In the aim to understand the phenomena which induce such curves, we analyzed the impact of the shadowing on the best useful received signal received by users, and on the interferences (Fig. 5).

These curves show that the CDF of the useful signal received by users is better with shadowing than without. . .

It is due to the fact that if a user receives a low signal from its nearest BS, it has a non negligible probability to receive a better signal from another BS due to shadowing, and therefore to be connected to that BS than to be connected to its nearest BS (case without shadowing). It can be observed that the interferences increase, too. And the increase of the interferences is at the same order as the increase of the useful signal. Therefore the two increases compensate each other.

VI. IMPACT ON WIRELESS SYSTEM ANALYSIS

In the previous section, we established the CDF of the SINR in a system modeled by a Poisson network, and by taking into account the propagation in terms of pathloss and shadowing. We showed that the shadowing impact is very low for typical values of its standard deviation. Indeed, we showed that it is equivalent, in terms of performance and QoS, to consider the shadowing and a best serving BS policy, as to consider a nearest serving BS policy without taking into account the shadowing. Therefore, the analysis of performances, coverage/capacity and QoS of wireless networks can be done by taking into account the propagation in a very simple way: by only considering the pathloss.

Moreover, in Monte Carlo simulators, the implementation of the shadowing is particularly greedy in simulation time. For this reason also, it is interesting not to take it into account. Therefore a simple analytical wireless model which does not take into account the shadowing, is sufficient for most of the analysis. The CDF of the SINR can be calculated in a easy and quick way, by using the expressions (4), ( 5) and [START_REF] Kelif | Impact of Transmitting Power on Femto Cells Performance and Coverage in Heterogeneous Wireless Networks[END_REF] given by the Fluid network model.

VII. CONCLUSION

In this paper, we analyzed the joint impact of the pathloss and the shadowing in Poisson wireless networks by considering a best serving policy for users. We established that results are very close, in terms of performance and QoS, to the ones established by considering a nearest serving policy, without taking into account the shadowing. Therefore it becomes possible to analyze realistic wireless networks without considering the shadowing, in the standard range of values of propagation pathloss parameter ⌘, and standard deviation of shadowing . As a consequence the CDF of the SINR, established by Monte Carlo simulation for the Poisson model network is very close to the one calculated with the analytical expression of the SINR given by the Fluid model network when a linear function of the propagation parameter ⌘ is applied. Therefore, the analysis of performance, outage probability, throughput, the radio resource allocation, the scheduling policy can be done in an easy and quick way, by using the expressions established with the Fluid Model network.

Fig. 1 .

 1 Fig. 1. Poisson Network

  2Rc , interferers are characterized by a density ⇢ BS of BS starting at a distance of 2R c from a BS (covering a zone of radius R c ), as illustrated on Fig.2(R nw is the size of the network). The interest of this model is to establish a simple analytical expression of the SINR.

Fig. 2 .

 2 Fig. 2. Fluid model: Network and cell of interest

Fig. 3 .

 3 Fig. 3. CDF of the SINR with ⌘ = 2.6 (left) and ⌘ = 3 (right), for a Poisson model network and standard deviations of the shadowing values comprised between 0 dB (no shadowing) (curve in the left of each figure) and 8 dB (curve in the right of each figure). For readability, only the curves drawn with = 0 and 8 dB are indicated in the figure. The curves between these two curves represent the cases = 3 dB, 6 dB. They are in the line thickness

.

Fig. 4 .

 4 Fig. 4. CDF of the SINR with ⌘ = 3.5 (left) and ⌘ = 4 (right), for a Poisson model network and standard deviations of the shadowing values comprised between 0 dB (no shadowing) (curve in the left of each figure) and 8 dB (curve in the right of each figure). For readability, only the values of 0 and 8 dB are indicated in the figure. The curves between these two curves represent the cases 3 dB, 6 dB. They are in the line thickness .

Fig. 5 .

 5 Fig. 5. CDF of the useful signal and the interferences, for ⌘ = 2.6 (left) and ⌘ = 4 (right), without shadowing ( = 0 dB, blue continuous curves) and with shadowing ( = 6 dB, red dotted curves)

Jean-Marc Kelif is with Orange Labs, France Email: jeanmarc.kelif@orange.com

Stephane Senecal is with Orange Labs, France Email: stephane.senecal@orange.com

Marceau Coupechoux is with Telecom ParisTech, France Email: marceau.coupechoux@telecom-paristech.fr

Constant Bridon is with ENS Cachan and Orange Labs, France Email: constant.bridon@ens-cachan.fr

ACKNOWLEDGMENT

This work was supported by the Seventh Framework Program for Research of the European Commission under grant number HARP-318489.