Alexander V Bulgakov

Anton B Evtushenko

Yuri G Shukhov

Wladimir Igor Ozerov

Wladimir Marine

Cluster Generation Under Pulsed Laser Ablation Of Compound Semiconductors

Keywords: Laser Ablation, Clusters, Compound Semiconductors, Zinc Oxide, Indium Phosphide, Congruent Vaporization, Laser Plume Expansion, Mass Spectrometry PACS: 79.20.Ds, 36.40.-c, 52.38.Mf, 81.07.-b

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INTRODUCTION

Efforts to develop methods for controllable synthesis of nanoclusters and nanostructured thin films of multicomponent materials are motivated by both fundamental science and the prospects for applications. In particular, nanostructured materials of II-IV and III-V compound semiconductors like ZnO, CdTe, InP have great potential in various technological applications such as solar cells, chemical sensors, photocatalysis, and nanoscale optoelectronic devices [1][START_REF]Zinc Oxide Bulk, Thin Films and Nanostructures[END_REF][START_REF] Kumar | [END_REF][4]. All these applications require a very high quality of the materials with respect to purity, defect concentration, nanoscale uniformity. However, producing multicomponent nanoclusters with desirable properties is still challenging. As compared to singlecomponent materials, there is an additional key parameter in this case, cluster composition, which strongly affects cluster properties and makes the fabrication process more difficult to control and optimize.

Pulsed laser ablation (PLA) has proven to be an efficient and flexible method for synthesis of nanostructures of various materials including simple and compound semiconductors. Although the physical processes involved in laser ablation are very complex, important progress has been achieved in recent years in PLA synthesis of nanoclusters of simple semiconductors with controlled size, shape and optical properties [5,6] and in characterization of the cluster formation process [7][8][9]. However, mechanisms of compound cluster generation under PLA conditions are still poorly understood. Despite of numerous reports on PLA-produced nanostructures and nanostructured films of binary semiconductors (see, e.g., [START_REF]Zinc Oxide Bulk, Thin Films and Nanostructures[END_REF][10][11][12]), little is known of the origin of the nanoclusters which can be either directly ejected from the target or formed in the ablation plume via a condensation process during the plume expansion [START_REF] Kumar | [END_REF]9]. The existing data on the composition of compound clusters in the PLA plume are rather contradictory. Thus, highly off-stoichiometric (zinc-rich) clusters were observed in Ref. [13] under PLA of ZnO in vacuum while other works report on production of stoichiometric [14] and even oxygen-rich [15] clusters under similar ablation conditions. Indium phosphide clusters produced by PLA in a carrier gas were found to be highly indium-rich [16] and essentially stoichiometric [17]. The cluster size and stoichiometry depend on various PLA parameters such as laser intensity and wavelength as well as ambient gas environment [10,[18][19][20]. Moreover, the composition the irradiated sample can vary from pulse to pulse due to non-congruent vaporization resulting in enrichment of a surface layer with a less volatile component [21,22]. The existing data on thermal evaporation of compound semiconductors demonstrate that there are two different ways of their decomposition. Some twocomponent materials, e.g., ZnO, vaporize congruently over a wide temperature range [23] whereas other semiconductors, for instance, InP and GaAs, exhibit a disproportionate loss of high-volatile components when temperature exceeds a critical point [24]. Detailed mechanisms of the non-congruent vaporization are still not clear even for the relatively simple situation of thermal decomposition [25].

In this work, we have performed a comparative mass-spectrometric study on generation of clusters of two binary semiconductors, zinc oxide and indium phosphide, under PLA in vacuum with IR (1064 and 800 nm) and visible (532 nm) laser pulses. We focus on investigation of small clusters, both charged and neutral, which are formed in early ablation stages and play a role of nucleation centres for further nanocluster growth [7]. Particular attention is paid to the expansion dynamics of different plume constituents and to the plume composition change during multi-pulse irradiation.

EXPERIMENTAL

The apparatus used for laser ablation and cluster production and detection was described earlier [9,[26][27][28]. The target (a sintered polycrystalline ZnO pellet or a (100) InP wafer) was placed in a rotating holder in a vacuum chamber (base pressure 10 -5 Pa) and irradiated by a laser pulse. Three different laser systems operating at 1064 nm (9 ns pulse, Nd:YAG laser), 532 nm (7 ns, 2nd harmonic of Nd:YAG laser), and 800 nm (100 fs, Ti:sapphire laser) were used for ablation. The laser fluence at the target was varied in the range 0.3 -8 J/cm 2 for ZnO ablation and 0.05-1 J/cm 2 for InP ablation. The relative abundances of neutral and cationic particles in the PLA plume were analyzed by a reflectron time-of-flight mass spectrometer (RETOF MS). The apparatus allows studying neutral and charged species independently. The neutrals were pulse-ionized in the ion source of the MS either by electron impact (1 μs, 90 eV) or by 193-nm photons (5-ns laser pulse of an ArF laser). When investigating neutrals, two ion-repelling plates [28] were used to prevent the proper plume ions from reaching the ion source of the RETOF MS.

The plume expanded under field-free conditions over a distance of 6 cm (at nsablation) or 11 cm (at fs-ablation) towards a repeller grid where the plume ions (or pulse-ionized neutrals) were sampled by pulsing the grid at a time delay t after the laser pulse. Variation of the time delay allowed the characterization of the plume expansion dynamics and temporal evolution of the plume composition. When studying clusters, a mass filter in front of the MS detector was used in order to suppress the huge atomic (Zn + or In + ) peaks and thus to avoid detector saturation. A voltage pulse applied to the filter blocked the detector for a time interval when the main plume ions arrived. High resolution of the RETOF MS (~1000) allowed us to distinguish between Zn n O m and Zn n+1 O m-4 according to their isotope distributions. Every mass spectrum was averaged over 200 laser shots unless otherwise specified.

RESULTS AND DISCUSSION

We carried out three kinds of experiments: (a) studies of relative abundances of neutral and charged plume particles as a function of laser fluence with a focus on cluster distributions; (b) mass spectrometric investigations of the expansion dynamics of the major plume constituents, and (c) measurements of the yields of various plume particles as a function of the number of consecutive laser pulses applied at a fixed fluence.

Clusters Produced by Laser Ablation of Zinc Oxide

For all studied conditions, neutral Zn atom and O 2 molecule were the dominant plume particles and positively charged Zn + was the dominant ion though the plume ionization degree was not exceeded ca. 1% throughout the laser fluence ranges studied. Other species, less abundant but always presented in the plume, were neutral O atom, and O + , O 2 + , and Zn 2 + ions. At relatively narrow ranges of laser fluences, rich spectra of neutral and charged zinc oxide clusters were observed for both IR-and visible-laser ablation. The optimum laser fluence for cluster production under vacuum conditions is found to increase with wavelength being in the range approx. 1-3 J/cm 2 for 532 nm and 2-5 J/cm 2 for 1064 nm.

Figure 1 shows a typical mass spectrum of cationic clusters observed at 1064-nm laser ablation when cluster ions are especially abundant. Under optimal ablation conditions, fairly large Zn n O m + clusters (up to n = 35) are registered and the cluster fraction can rich up to 10% of the total number of charged particles in the plume. The clusters are produced in abundance not only with a virgin target surface, as was in the case of UV-ablation [20], but also after etching the sample with several thousand pulses (though we avoided considerable cratering at the surface). The measurements with varying time delay t show that the observed clusters have near equal expansion velocities ~ 1.5 km/s in spite of rather large difference in their mass. This implies a gas-phase aggregation mechanism for their formation rather that direct ejection [9,28]. The distribution has no obvious preferential (magic) numbers but the cluster composition strongly depends on the cluster size. The smallest species (n = 2, 3) are essentially substoichiometric (Zn-rich). The intermediate-size particles (n = 4-9) occur mainly as a twin peak, the first peak corresponding to Zn n O n-1 + clusters with one missing oxygen atom while another peak is for stoichiometric species. The twin clusters seem to follow a rule that the first peak is higher at n = odd, while the second peak dominates at n = even. Large clusters (n > 9) are preferentially stoichiometric. Clusters with a distribution similar to that shown in Fig. 1 (though less abundant) were produced in this work by 532-nm ablation with laser fluence around 2 J/cm 2 and were observed for 337-nm ablation of ZnO [14]. On the other hand, a completely different (much poorer) cluster distribution with preferential sizes was reported for 1064-nm ablation of ZnO in vacuum [15]. This is probably due to the high laser fluence (7.7 J/cm 2 ) used in [15] which is beyond the optimum fluence range found in this work. Figure 2 shows the mass spectrum of neutral particles obtained with electron impact ionization and at the same "optimum" ablation conditions of ZnO as for cationic cluster generation (Fig. 1). The neutral Zn, O 2 and O species are totally dominant in the plume but, using the mass filter, we were able to detect neutral Zn n O m clusters up to n = 7. The clusters are essentially stoichiometric at n > 3 while substoichiometric Zn 4 O 3 and Zn 5 O 4 species are registered. The most intriguing feature of the spectrum shown in Fig. 2 is the strong dominance of the Zn 4 O 4 tetramer over the neighbors. It is rather surprising since recent DFT calculations of zinc oxide cluster structures [29] do not predict special stability for the tetramer. We suggest that the appearance of the Zn 4 O 4 specie as a magic cluster in the mass spectrum is due to fragmentation of larger plume clusters upon electron impact. An argument for the fragmentation origin is that the Zn 4 O 4 unit is one of the main building blocks of Zn n O n clusters [29]. We performed a mass spectrometric study of neutral clusters under the same ablation conditions using the photo ionization technique with 6.4-eV photons (ArF laser) instead of electron impact. However this resulted in even stronger decomposition and we were not able to detect particles larger than dimers. Such a strong photofragmentation of zinc oxide clusters indicates that the ionization potential (IP) of large clusters (which presumably exist in the plume) is still higher than 6.4 eV and thus the probability of cluster decay due to absorption of two photons on the ns timescale is very high. Indeed, according to [30], IP of Zn n O n clusters is about 7.7 eV in the n = 10-16 range). It would be interesting to investigate neutral particles in the ZnO ablation plume under single-photon ionization conditions, e.g. with F 2 laser (photon energy 7.9 eV). Preparation of such experiments is under progress.

Clusters Produced by Laser Ablation of Indium Phosphide

Similar to ZnO ablation, three particles are dominant in the PLA-produced plume above the InP target throughout the irradiation conditions studied, namely, neutral In atom and P 2 molecule, and In + ion. Other species presenting in the plume at lower abundances are neutral P 3 and P 4 molecules, P atoms (the latter is observed only at fairly high fluences), and In 2 + ions. Due to a low ionization potential of indium (5.8 eV), the fraction of charged particles in the plume is considerably higher than that under ZnO ablation and the plume ionization degree can reach up to 10% for the highest laser fluences investigated. Correspondingly, the thresholds for ablation and plasma formation are relatively low being ca. 150 mJ/cm 2 for 532-nm pulses, ca. 300 mJ/cm 2 for 1064-nm pulses, and ca. 100 mJ/cm 2 for 800-nm fs-laser pulses.

Again similar to ZnO ablation, at narrow fluence ranges of around 200-400 mJ/cm 2 for 532-nm and 400-700 mJ/cm 2 for 1064 nm, compound In n P m clusters are observed in the plume. At fs-laser ablation, indium phosphide clusters are particularly abundant in the plume and registered in a wide fluence range starting almost from the ablation threshold (~100 mJ/cm 2 ). However, in contrast to the ZnO case, the clusters are now highly off-stoichiometric and their generation depends strongly on the laser wavelength. Under visible-laser ablation, charged clusters have been produced only with a relatively fresh target exposed to less than ~ 20 laser shots. By measuring the plume composition after every consecutive laser shot applied to the same spot we have found that the most pronounced cluster peaks are observed after 3rd and 4th laser pulses and drop drastically in intensity upon further laser exposure. The first two pulses serve to prepare the surface for cluster emission and to remove trace contaminations. Figure 3 shows a mass spectrum of cationic particle observed at 532nm laser ablation of InP after 4th pulse. In contrast to the rather smooth distribution of zinc oxide cluster ions (Fig. 1), only four magic In n P m + clusters with n = 4-6 and m = 1-3 are present in the plume. Interesting that namely these clusters were observed in a mass spectrometric study of neutral products of 532-nm laser ablation of InP in a carrier gas with an ArF ionizing laser [16]. Therewith no neutral In n P m clusters were observed in the present work under 532-nm PLA of InP in vacuum. The situation is completely different under IR-laser ablation of InP (with both ns and fs pulses) resulting in rich spectra of neutral and charged clusters. Indium oxide clusters are now observed not only with a fresh target surface but also after etching the surface with several thousand pulses. Figure 4 shows the distribution of cationic In n P m + clusters produced under 800-nm fs-laser ablation when they are especially abundant. All observed clusters are essentially indium-rich and contain no more than 3 phosphorus atoms even for largest clusters. Depending on the In/P ratio the clusters may be gathered into two major series, In n P and In n P 2 with (n+m) = odd (except for In 6 P 2 ). The most abundant (magic) cationic clusters are the same as observed at 532nm ablation (see Fig. 3), In 4 P + , In 4 P 3 + , In 5 P 2 + , and In 6 P 2 + . Similar to zinc oxide clusters, all observed indium In n P m + cations have near equal expansion velocity ~ 2 km/s that again suggests a gas-phase aggregation mechanism for their formation. Ablation of InP by 1064-nm laser pulses results in generation of ionized clusters with a similar distribution but considerably less abundant. This is not surprising since fs ablating pulses are generally more favorable for gas-phase cluster production as compared to ns-laser pulses due to absence of plasma absorption effects and a more forward-directed plume resulting in more particle collisions during plume expansion [9,31]. Figure 5 shows a mass spectrum of neutral particles obtained with photoionization (ArF laser) at IR-laser ablation of InP. The ionization potentials of In n P m clusters appear to be below 6.4 eV (at least for n > 3) [16] and thus ionization of the clusters is a single-photon process without their considerable fragmentation. So the observed cluster distribution is expected to correspond to the true distribution of neutrals. Again the neutral clusters are strongly off-stoichiometric. By comparing the neutral cluster compositions with those of cationic clusters (Fig. 4) one can see that, for the same n value, most of ions have an additional P atom. However, the two magic clusters, In 4 P and In 5 P 2 are dominant among both neutrals and ions. The enrichment of the observed In n P m clusters with indium is explained by peculiarities of laser-induced vaporization and plume expulsion under PLA of InP as discussed below.

Shot Number Dependence of the PLA Plume Composition

To follow the pulse-to-pulse evolution of the vaporization process, we measured the yields of the main ablation products starting from virgin surfaces. The experiments were performed at fixed laser fluences and at time delays t corresponding to the highest signals of neutral particles (at maxima of the TOF distributions). Figure 6a shows the results for Zn atoms and O 2 molecules produced by 1064-nm laser ablation of ZnO at a fluence of 3 J/cm 2 corresponding to the optimum conditions of cluster generation. The particle yields decrease first upon laser exposure, reach a minimum at ~30 shots, then increase again reaching a quasistationary level after ~1000 shots. Similar dependences were obtained at lower laser fluences (when, however, more laser shots are needed to reach a stationary value) and with 532-nm ablating pulses. Such a nonmonotonic dependence can be attributed to a complicated modification of the ZnO surface during irradiation resulted in variations of its optical properties (e.g. due to laser-induced generation of oxygen vacancies [10]). More important in the context of cluster generation is that the ratio of the zinc and oxygen yields remains constant (~ 1 within the experimental error). This means that vaporization of the ZnO target under these conditions occurs congruently, much like as thermal vaporization [23]. In its turn this implies a thermal nature of ZnO ablation with IR-and visible laser pulses. Note that this does not appear to be the case for UV-laser pulses when the surface is enriched with zinc upon irradiation [32] and Zn-rich clusters are observed [20]. A different evolution of the plume composition is observed at InP ablation. Figure 6b illustrates the yields of the main plume particles in this case, neutral In and P 2 , as a function of the laser shot number for 1064-nm pulses. At a virgin InP surface, the P 2 signal is considerably stronger than that of In atoms (the P 2 /In ratio is about 2 for this conditions and even higher for lower fluences). Upon irradiation, the phosphorus yield drops quickly while that of indium increases and after several hundred of shots both signals reach quasistationary levels with the P 2 /In ratio being about 0.5. Similar shotnumber dependences at InP ablation were obtained in this work with 532-nm pulses and in [33] at 337-nm ablation of InP where such a behavior was explained by the preferential emission of more volatile phosphorus that makes the irradiated surface Inrich. Indeed, the disproportionate loss of phosphorus and alteration of the surface stoichiometry were previously observed for thermal heating [24] and visible-laser irradiation [34] of InP samples. We note, however, that the observed quasistationary value of the P 2 /In ratio after a fairly large number of laser shots (Fig. 6b) suggests equal overall yields of phosphorus and indium under these conditions even with the In-reach surface. This can be possible if phosphorus and indium are released from the surface at different timescales that is the case as we will see below.

Plume Expansion Dynamics

To further elucidate the laser-induced vaporization and cluster formation process, the temporal evolution of the dominant plume particles was examined by varying the time delay t. Figure 7a shows the time-of-flight (TOF) distributions of neutral Zn and O atoms and O 2 molecules obtained under "optimum" conditions for ZnO ablation (1064 nm, 3 J/cm 2 ) resulting in the spectra shown in Figs. 1 and2. As seen, the main plume components have essentially the same expansion velocity (around 2 km/s) and expand jointly. This implies a hydrodynamic regime of expansion during a prolonged time [26] that is favorable for stoichiometric cluster formation. Gas-phase collisions of different plume particles under these conditions are equiprobable and thus clusters have a high probability to be self-organized into most stable structures which are apparently stoichiometric in this case. In contrast to the zinc oxide PLA plume, the main products of InP ablation products exhibit very different expansion behaviors. Figure 7b shows the TOF distributions of In atoms and P 2 molecules obtained with 1064-nm laser pulses at a fluence corresponding to maximum cluster yields (see Fig. 5). The In distribution is quite similar to those for particles produced under ZnO ablation (see Fig. 7a): it is singlepeaked and rather narrow, is maximized at ca. the same arrival time of ~ 30 µs (corresponding velocity ~2 km/s) and can be fitted with the Maxwell-Boltzmann distribution with a reasonable effective plume temperature of about 15000 K (solid line in Fig. 7b). The lighter P 2 molecules arrive therewith to the detector at considerably later times and their TOF distribution has a clear delayed tail corresponding to a fraction of very slow molecules. This delayed fraction of P 2 particles is found to increase with laser fluence and become dominant at above ~700 mJ/cm 2 . Similar TOF distributions were obtained at 532-nm ablation of InP. Such peculiarities of the plume expansions under multi-pulse ablation of InP can be explained using above considerations of the overall stoichiometric vaporization from the irradiated In-rich surface. Indeed, indium is released via the normal vaporization mechanism [35], mainly during the laser pulse action when the surface temperature is maximal. The heating of the sample stimulates diffusion processes within an offstoichiometric surface layer [22] resulting in a partial compensation of the phosphorus lack at the surface. Then the low-volatile phosphorus evaporates for a quite long time during cooling down the sample, leaving thus the surface again essentially indiumreach. Such a diffusion-limited vaporization produces rather slow particles and manifests itself by the delayed tail in the TOF distributions. The higher is the laser fluence, the stronger is the surface enrichment with indium and thus the larger fraction of phosphorus is evaporated in the delayed manner. The essentially different processes responsible for indium and phosphorus vaporization during PLA of InP result in the fact that the main ablation products fly separately in the plume as seen in the experiments (Fig. 7b). Thus indium atoms and clusters have little chances to undergo collisions with P n molecules during the plume expansion while indium-indium collisions are much more probable. This explains why the clusters produced under laser ablation of InP are highly indium-rich.

The performed experiments have demonstrated that the character of laser-induced vaporization compound semiconductors (congruent or non-congruent) is much the same as that under simple thermal heating. This offers a possibility to predict the composition of the laser-generated clusters, whether they are stoichiometric or not, based on thermodynamic properties of the ablated materials. A fundamental question arises now why some of compound materials, like ZnO, vaporize congruently whereas others, e.g., InP, lose a more volatile component during vaporization. The experiments of the present work do not give an irrefragable answer to this issue which needs obviously more detailed experimental and theoretical studies of the vaporization process. We believe, however, that the difference in vaporization behavior is associated with the composition range in the phase diagram where the irradiated compound remains thermodynamically stable. The broader is the range, the higher is the loss of stoichiometry during vaporization. Physically, this probably means that such compounds with a variable composition can be structurally transformed upon heating in order to release a high-volatile component. In particular, in binary semiconductors heated up to the melting points, like-atoms can form branched chains and clusters [36]. As for the materials considered here, the ZnO crystalline structure is stable in a very narrow composition range (less than 0.1 at. % [37]) and zinc oxide evaporates congruently. In contrast, indium phosphide remains stable over a wide range of the In/P ratio (at least from 0 to 55 mole % phosphorus [38]) that results in strongly non-congruent vaporization.

FIGURE 1 .

 1 FIGURE 1. Mass spectrum of cationic clusters produced by ZnO ablation with 1064-nm laser pulses at 3 J/cm 2 . The pairs of numbers (n-m) above the peaks correspond to the numbers of Zn and O atoms in Zn n O m + clusters. The single numbers above the peaks stand for n in stoichiometric (ZnO) n + clusters.

FIGURE 2 .

 2 FIGURE 2. Mass spectrum of neutral particles produced by 1064-nm laser ablation of ZnO and pulse-ionized by electron impact at 90 eV. Ablation conditions are same as for Fig. 1.

FIGURE 3 .

 3 FIGURE 3. Mass spectrum of cationic clusters produced by InP ablation with 532-nm laser pulses at 280 mJ/cm 2 . The spectrum was collected after 4th laser pulse applied to a virgin InP surface.

FIGURE 4 .

 4 FIGURE 4. Mass spectrum of cationic In n P m + clusters produced by InP ablation with 800-nm fs-laser pulses at 260 mJ/cm 2 . The numbers (n-m) above the peaks stand for the n and m values.

FIGURE 5 .

 5 FIGURE 5. Mass spectrum of neutral clusters produced by InP ablation with 1064-nm laser pulses at 500 mJ/cm 2 . The clusters were ionized by unfocused 193-nm laser pulses at ~20 mJ/cm 2 . The In + ions were blocked from reaching the detector using a filter (the blocked interval is shown by the arrows).

FIGURE 6 .

 6 FIGURE 6. Yields of the main plume particles under 1064-nm laser ablation as a function of the number of laser shots. (a) is for neutral Zn atoms and O 2 molecules produced at ZnO ablation at a laser fluence of 3 J/cm 2 ; (b) is for neutral In atoms and P 2 molecules produced at InP ablation at a laser fluence of 500 mJ/cm 2 . The yields for all particles are obtained at a time delay t = 30 µs.

  λ = 1064 nm, 500 mJ/cm2

FIGURE 6 .

 6 FIGURE 6. Time-of-flight distributions of the main plume particles under 1064-nm laser ablation. (a) is for neutral Zn and O 2 produced at ZnO ablation at a laser fluence of 3 J/cm 2 ; (b) is for neutral In and P 2 produced at InP ablation at a laser fluence of 500 mJ/cm 2 . The solid line in (b) corresponds to a Maxwell-Boltzmann fit of the indium distribution with an effective plume temperature of 15000 K.

ACKNOWLEDGMENTS

It is a pleasure to thank N.M. Bulgakova and J. Bonse for useful discussions. The InP samples were kindly provided by J. Bonse. This work was supported by the Russian Foundation for Basic Research (Project 09-02-91291-CSIC).


B Baynat

R-M Indre

N Nya

P Olivier
email: phil.olivier@orange.com

A Simonian

Impact of mobility in dense LTE-A networks with small cells

In this paper, we investigate the impact of mobility on user performance in the context of dense LTE-A networks. To this end, we propose simple analytical models that capture mobility through the distribution of the mobile users sojourn time, i.e., the time a mobile user is physically present in a given cell. We use these models to derive the throughput of users who remain spatially static during their whole data transmission and the amount of handovers generated by moving users. We analyze the impact on performance of some key parameters such as the size of the small cell, the speed and proportion of mobile users and the distribution of their sojourn time. Numerical evaluation and simulation results are provided to assess the accuracy of the latter model and gain insight into the global system performance.

I. INTRODUCTION

To cope with the continuous increase in mobile data traffic, network operators envisage massive deployment of small cells so as to enhance the capacity and/or coverage of LTE-A networks [1]. In such dense LTE-A networks, the proportion of mobile users and their impact on system performance is expected to increase. Indeed, moving users will experience various transmission conditions in the overlapping zones between macro-cells and small cells. As a consequence, increased overhead has to be handled due to a higher handover frequency between neighboring cells. In the meantime, static users may benefit from the fact that mobile users traverse the cell. Indeed, the effective load decreases when moving users leave the cell.

Accounting for such complex mobility effects in a simple manner represents an important challenge. As motivated above, this can be formulated as a trade-off between an enhanced performance for static users versus an increase of handovers generated by mobile users. In this context, the present paper addresses the impact of user mobility on the performance of small cells in LTE-A networks. Specifically, we study the influence of mobile users on the performance of static users and the amount of generated handovers.

Although revisited in the present context of LTE-A networks, various user mobility patterns for wireless networks have been previously proposed in order to define the relevant modeling characteristics [START_REF] Schindelhauer | Mobility in wireless networks[END_REF]. Two main categories of models can be identified. First, geometric models explicitly represent the random paths of moving users in the plane, e.g., by means of the so-called Random Waypoint (RW) model. In [START_REF] Lin | Towards understanding the fundamentals of mobility in cellular networks[END_REF], the RW model enables the authors to show that the handover rate is proportional to the square root of the cell density. However, these geometric models do not typically account for the dynamicity of communication traffic (i.e., flow arrivals and departures). On the other hand, non-spatial models address this dynamicity at the expense of a simplified mobility description. Indeed, mobility is simply modeled by assuming that moving users jump between distinct capacity zones according to a Markov process. The capacity gain due to mobility is then estimated through upper and lower bounds associated with fluid limits and quasi-stationary regimes, respectively [START_REF] Bonald | How mobility impacts the flow-level performance of wireless data systems[END_REF], [START_REF] Bonald | Flow-level performance and capacity of wireless networks with user mobility[END_REF], [START_REF] Borst | Capacity of wireless data networks with intra-and inter-cell mobility[END_REF]. Instead of deriving bounds, approximate expressions for the mean flow throughput are derived in [START_REF] Karray | User's mobility effect on the performance of wireless cellular networks serving elastic traffic[END_REF] by means of a Markovian model for the moving users' displacement.

In this paper, we propose to decompose the mobility problem as follows. We first capture mobility through the distribution of the mobile users sojourn time in a given cell, i.e., the time a mobile user is physically present in the cell. Such a distribution is the output of a chosen mobility model, be it RW, discrete or continuous Markovian, etc. Given this distribution, we then construct a queuing model that allows us to derive performance metrics at the flow level for each user class, namely the mean throughput of static users and the handover probability for mobile users.

II. NETWORK MODEL

Consider a macro-cell containing several small cells. Each small cell being of limited range,we consider here that its transmission capacity C (Mbit/s) is spatially constant, which represents a reasonable assumption for small cells. The case of macrocells with varying capacity will be considered in future work. The macro-cell is visited by both static and mobile users; the latter move inside the macro-cell and may cross one or several small cells before completing a data transmission. We address the impact of these mobile users on the performance of users who remain static inside the small cell, and the amount of generated handover. In the following, we interchangeably use the terms "small cell" and "cell".

A. Assumptions and notation

The transmission capacity C of each small cell is equally shared among all users present in its service area. This can be implemented by means of a Round-Robin discipline and the occupancy of this service system can be modeled by a Processor-Sharing (PS) queue [START_REF] Bonald | Wireless downlink data channels: user performance and cell dimensioning[END_REF]. In other words, at any time t, each user is given the instantaneous service rate C/N (t) if the total number of active users in the cell is N (t).

Considering downlink traffic only, we assume that static and mobile users generate requests for transmission in the cell according to independent Poisson processes with rates λ s and λ m , respectively. Note that λ m accounts for users that become active inside the considered cell as well as already active users coming from neighbouring cells. Both kinds of requests have an identically distributed volume Σ of data to be transferred. As soon as a new request arrives (either from a static or a mobile user), it triggers the start of a new data transmission. For a static user, this transmission lasts until the completion of the whole data transfer; the volume eventually transferred thus equals Σ. For a mobile user, this transmission can end either because the user has completed its transfer before leaving the cell, or because he has left the cell before completing its transfer. As a result, the volume transferred to a mobile user is always less than (or equal to) Σ. We denote by Θ the remaining sojourn time of an active mobile user, that is, the time duration he physically stays in the cell once the transmission has started; note that Θ can be expressed as Θ = D/v where D is the random distance the mobile user travels in the cell (from its activation point) and v is the speed of the user, here assumed to be constant. Finally, we denote by µ = C/E(Σ) (s -1 ) the service rate of the cell, i.e., the departure rate of full transmissions that the cell can handle. A "full transmission" refers either to the transmission of a static user or to that of a mobile user that has not left the cell before completing its transfer. Due to mobility, µ does not correspond to the actual number of transfers in the cell.

B. Markovian model

Given the above description, the occupation state of the cell can be described by the bi-dimensional process (N s (t), N m (t)) t≥0 , where N s (t) (resp. N m (t)) is the number of static (resp. mobile) ongoing data transfers in the cell at time t. The evolution of this process is exactly represented by a so-called Processor-Sharing queue with impatience [START_REF] Coffman | Processor-shared buffers with reneging[END_REF], the "impatient" customers here corresponding to mobile users that may leave the system before their service completion.

To simply characterize the evolution of this PS queue with impatience, we now assume that the volume Σ is exponentially distributed with parameter 1/E(Σ) and that the duration Θ is exponentially distributed with parameter θ = 1/E(Θ), the impatience rate. The process (N s (t), N m (t)) t≥0 is then clearly Markovian with transition diagram illustrated in Figure 1; from any state (n s , n m ), n s > 0, n m > 0, we can reach state (n s + 1, n m ) with transition rate λ s , or state (n s -1, n m ) with transition rate n s µ/(n s + n m ), corresponding to the completion of the transmission of one of the n s static users among a total of n s + n m users. Similarly, we can reach state (n s , n m + 1) with transition rate λ m , or state (n s , n m -1) with transition rate n m µ/(n s + n m ) + n m θ; the second term corresponds to one of the n m mobile users leaving the cell before completing its transfer. By using an appropriate Lyapunov function, it can be simply shown that the occupancy process (N s (t), N m (t)) t≥0 has a stationary regime if ρ s = λ s /µ < 1. Note that this stability condition does not depend on the traffic intensity of mobile (i.e., impatient) users. Intuitively, this can be understood by the fact that the latter always leave the cell after a finite time and therefore cannot cause a system overload. Now fixing ρ s < 1, it is easily verified from the transition diagram in Figure 1 that the process (N s (t), N m (t)) t≥0 is not reversible and thus its stationary distribution p(n s , n m ), n s , n m ≥ 0, is not amenable to a simple product form. We can nevertheless determine this stationary distribution by solving numerically its associated system of global balance equations, truncating both dimensions of the state space and using any appropriate numerical technique (e.g. Gauss-Seidel or Least mean square). From this distribution, we can derive the performance indicators of interest. First, we calculate the average throughput γ obtained by any static user; by Little's law, E(N s )/λ s is the mean time to transfer the average volume E(Σ) hence

n s , n m -1 n s , n m n s , n m +1 n s n s -1, n m λ s λ s λ m λ m n s n s + n m µ n s +1 n s +1+ n m µ n m n s + n m µ + n m θ n m +1 n s + n m +1 µ + (n m +1)θ
γ = λ s E(Σ) E(N s )
.

(1)

We further derive the proportion H of mobile users that exit the cell before the completion of their transmission (H stands for "Handover"), that is,

H = ns≥0,nm≥1 q(n s , n m ) (n s + n m )θ µ + (n s + n m )θ , (2)
where q(n s , n m ) = p(n s , n m )/P(N m ≥ 1).

C. A Fixed-Point approximation

Although the Markovian model can be solved numerically, it does not lend itself to explicit expressions for either the mean throughput γ or the handover probability H. Moreover, it is limited to exponentially distributed variables Σ and Θ. We now propose an alternative approximate model that allows deriving such expressions in a wider assumption framework.

Instead of viewing mobile users as impatient customers for the Processor-Sharing queue, we can consider them as belonging to another class similar to that of static users but with a distinct service rate µ m = µ which accounts for the early departures due to their mobility. Specifically, we represent the cell occupancy by a multi-class Processor-Sharing queue with 2 classes of customers, a static class and a mobile class, as illustrated in Figure 2. As in Section II-A, static (resp. mobile) users still generate requests according to a Poisson process with a rate λ s (resp. λ m ) and the service rate of static users is µ s = µ = C/E(Σ). We are now left to estimate the equivalent service rate of mobile users µ m = µ. Let us first assume that the average number of bits x transferred by a mobile user before he physically leaves the cell is known. As argued in Section II-A, we have x ≤ E(Σ); this quantity is obviously related to the total load of the cell and to the physical sojourn time Θ of mobile users in the cell. From that value of x, we can write the service rate of mobile users in the PS queue as

µ m = C x . (3)
Given parameters λ s , λ m , µ s and µ m , standard results for the stationary multi-class Processor Sharing queues can be readily applied to calculate the average throughput γ obtained by both static and mobile users during their transfer, that is,

γ = C(1 -ρ) (4)
where ρ = ρ s + ρ m , with

ρ s = λ s µ = λ s E(Σ) C , ρ m = λ m µ m = λ m x C . (5)
Recall that the stability condition for that multi-class PS queue is ρ < 1, and it is thus necessary that ρ s < 1 to ensure the system stability. The sufficiency of condition ρ s < 1 is less straightforward. However, as argued below, x and therefore ρ m will be defined as a function of ρ s , say ρ m = f (ρ s ). Based on the numerical evaluations of γ in Section III-A below, we can conjecture that ρ s < 1 is sufficient to ensure that γ > 0.

We now provide an estimation for the missing parameter x, required in the expression of µ m . Consider a mobile user with a given volume Σ = y to transfer. If he stays in the cell a time t ≥ y/γ, he can complete a full transfer. On the other hand, if he stays in the cell a time t < y/γ, this time only enables him to download γt bits in average. Given Σ = y, the mean transferred volume x(y) can therefore be expressed as

x(y) = y γ 0 γt f Θ (t) dt + +∞ y γ y f Θ (t) dt (6)
where f Θ is the probability density of the time Θ spent by a mobile user in the cell after the transmission has started. Now, deconditioning expression ( 6) with respect to the distribution of Σ (with probability density f Σ ), we obtain

x = +∞ 0 x(y)f Σ (y) dy (7)
with x(y) expressed in [START_REF] Coffman | Processor-shared buffers with reneging[END_REF]. We thus end up with a system of four dependent equations ( 3)-( 4)-( 5)-( 7) that provide a Fixed-Point equation for unknown µ m , or equivalently γ. Once γ is determined by the Fixed-Point equation, the handover probability H can then be estimated in turn by

H = +∞ 0 P Θ ≤ y γ f Σ (y) dy. ( 8
)
Despite its simplicity, the above Fixed-Point approach is, however, an approximation in that we implicitly assume in both relations [START_REF] Karray | User's mobility effect on the performance of wireless cellular networks serving elastic traffic[END_REF] and ( 8) that a mobile user always receives an average throughput γ along its entire sojourn in the cell.

If we assume, as previously, that Σ and Θ are exponentially distributed; integral [START_REF] Karray | User's mobility effect on the performance of wireless cellular networks serving elastic traffic[END_REF] then readily gives

x = σγ σθ + γ , (9)
where σ = E(Σ) and θ = 1/E(Θ). This relation enables us to easily reduce the latter set of equations ( 3)-( 4)-( 5)-( 7) to a simple quadratic equation in γ with a unique positive solution, thus providing the average throughput

γ = C -σ(λ s + λ m + θ) + √ ∆ 2 , ( 10
)
where ∆ = (C -σ(λ s + λ m + θ)) 2 + 4θσ(C -λ s σ). Finally, applying [START_REF] Schindelhauer | Mobility in wireless networks[END_REF] to exponentially distributed Σ and Θ gives

H = σθ σθ + γ . ( 11
)
Note however that, unlike the first Markovian model, this second PS queue model can account for wider range of assumptions for the distributions of both the sojourn time Θ of a mobile user in the cell and the volume Σ of data a user has to transfer, while staying very easy to implement.

III. PERFORMANCE RESULTS

We have proposed two models that account for the mutual impact of static and mobile users in a radio cell, namely a Markovian model and a Fixed-Point approximation. As long as the assumptions of Section II-B are satisfied, the Markovian model is exact. However, if one of the above-mentioned assumptions is not satisfied anymore (e.g., one of the random variables is no longer exponentially distributed), exact results can no longer be derived using the Markovian model and we need to resort to simulation to validate both models.

A. Model validation

We first compare the results provided by the two models in the case where Θ and Σ are both exponentially distributed. We set C = 40 Mbit/s for the cell capacity and σ = 10 MB for the mean flow volume. The mobile users speed is set to v = 50 km/h and the mean distance crossed by mobile users is E(D) = 100 m, yielding a mean exit rate θ = v/E(D) = 0.14 s -1 . Figure 3 compares the mean throughput of static users γ obtained by the two models, in terms of the load ρ s induced by static users. The continuous (resp. dotted) lines correspond to the results of the exact Markovian model (resp. the Fixed-Point approximation). Each of the three sets of curves corresponds to a proportion of mobile users p ∈ {0.2, 0.5, 0.8}. We observe that the results provided by the Fixed-Point approximation match the exact results with a good accuracy (with a relative error less than 10% in most cases), and provide a lower bound (i.e., conservative) for the static user throughput in all cases. From these results we note, as expected, that the throughput of static users degrades considerably as the proportion of mobile users increases, thus increasing the total load ρ = ρ s + ρ m .

Figure 4 represents the handover probability H of mobile users, still as a function of the load ρ s of static users. We use the same numerical values for all parameters, but for sake of clarity we just represent two sets of curves corresponding to a proportion of mobile users p ∈ {0.2, 0.5}. When the load of static users is very low, i.e., ρ s → 0, then γ → C and in view of (11), we have H → σθ/(σθ +C); with the considered parameters this yields H → 0.22 when ρ s → 0. The handover probability then increases from this initial value as the load of the static users increases and H tends to 1 when ρ s approaches 1. Note that the Fixed-Point approximation is fairly accurate at relatively low loads and becomes slightly less accurate as the load increases. We now evaluate the robustness of models with regard to the distribution of the sojourn time Θ of mobile users in the cell. The exponential distribution previously considered for Θ is probably not a realistic assumption. Indeed, if we consider a mobile user moving in the cell along a diameter at constant speed and whose initial position is uniformly distributed on that diameter, it can be easily shown in this case that Θ is uniformly distributed. Figure 5 thus depicts the throughput of the static users as a function of the static user load for different distributions of the mobile user sojourn time. Values for the uniform and deterministic distribution are obtained through simple event-driven flow-level simulations, while those for the exponential distribution are obtained by numerically solving the Markovian model. We note that the throughput performance is only marginally impacted by the sojourn time distribution indicating that derived results remain valid for more realistic sojourn time distributions.

B. Impact of key parameters

We now use the Fixed-Point approximation to discuss the impact of some key system parameters such as the speed of mobile users or the cell size. Figure 6 of the static load ρ s ∈ {0.2, 0.5, 0.8} (from top to bottom) and a fixed proportion of mobile users p = 0.5. Note from (4) that the bounds γ(0) ≤ γ ≤ γ(+∞) hold, where γ(0) = C(1 -ρ) with ρ = (λ s + λ m )σ/C corresponds to the case when all users are static (θ = 0), and γ(+∞) = C(1 -ρ s ) corresponds to mobile users with infinite speed (θ = +∞) and a vanishing impact on static users. The variations of the throughput in terms of θ are therefore significant for relatively small values of ρ s . Note also that, as expected, the throughput γ of static users increases when the speed v increases, validating the fact that static users benefit from increased mobility. In Figure 7, we assess the impact of the cell size by considering a typical Micro, Pico or Femto cell with a diameter of 1 km, 100 m or 10 m, respectively; we also consider that the mean sojourn durations are proportional to these values (v is fixed to 5 km/h, e.g. typical of a pedestrian). As expected, users in Femto cells experience the highest throughput since they benefit from the least interaction with the mobile users.

IV. CONCLUSION

We have analyzed the impact of mobility on user performance in LTE-A networks with small cells. Two simple The first one is exact as long as all distributions involved in the modeling process are assumed to be exponential, but relies on a bi-dimensional Markov process that has to be solved numerically. The second one can manage a wider range of assumptions for the distributions of variables and is very easy to implement, but remains an approximation even when all distributions are exponential. In cases where the first model is exact, we have shown that the second one is able to predict user performance with a good accuracy. Based on the proposed models, we were able to quantify the impact of key mobility parameters such as speed, proportion of mobile users and cell size. By means of simulations, we have shown that the derived results are robust with respect to different distributions of the mobile users' sojourn time.

Fig. 1 .

 1 Fig. 1. Transition diagram for the Markov process (Ns(t), Nm(t)) t≥0

Fig. 2 .

 2 Fig. 2. Multi-class PS queue with Fixed-Point approximation

Fig. 3 .Fig. 4 .

 34 Fig. 3. Static user throughput as a function of load ρs for different proportions of mobile users (v = 50 km/h and mean crossed distance E(D) = 100 m)

Fig. 5 .

 5 Fig. 5. Static user throughput as a function of load ρs for different distributions of mobile users' sojourn duration

Fig. 6 .

 6 Fig. 6. Static user throughput as a function of mobile users' speed for different loads ρs ∈ {0.2, 0.5, 0.8} (from top to bottom)

Fig. 7 .

 7 Fig. 7. Static user througput as a function of load ρs for different cell sizes

* This work has been carried out in the framework of IDEFIX project, funded by the ANR under the contract number ANR-13-INFR-0006.


Bruno Baynat
email: bruno.baynat@lip6.fr

Nya Narcisse
email: narcisse.nya@lip6.fr

Performance Model for 4G/5G Networks Taking Into Account Intra-and Inter-Cell Mobility of Users

In this paper, we investigate the influence of intraand inter-cell mobility of users on performance of 4G/5G cellular networks, such as LTE and LTE-A. To this end, we develop a multi-class PS queue model that captures mobility of users between zones of a cell and between cells, through a simple mobility model, that is decoupled from the cell model itself, enabling to directly apply the approach to more realistic mobility patterns. We first show that this model is consistent with known analytical bounds corresponding to a system with either static users or users having an infinite speed. We then compare our model to simulations for more realistic speeds, and show that it provides user and cell performance with a very good accuracy. The outcomes of our model confirm that mobility may improve both users and cells performance, and enable to quantify the gain as a function of users speed.

I. INTRODUCTION

In LTE, like in other wireless communication systems, Adaptive Modulation and Coding (AMC) has become a standard approach. The idea behind AMC is to dynamically adapt the modulation and coding scheme to the channel conditions so as to achieve the highest spectral efficiency at all times [START_REF] Goldsmith | Adaptive coded modulation for fading channels[END_REF]. As such, modulation and/or coding can change with time for a given user depending on its location and its channel quality. This is especially true when users are mobile, highlighting the influence of mobility on user and cell performance. The resulting variations in the transmission rates is exploited by opportunistic schedulers to increase overall throughput of data transmissions [START_REF] Ayesta | A modeling framework for optimizing the flow-level scheduling with time-varying channels[END_REF], [START_REF] Bonald | A score-based opportunistic scheduler for fading radio channels[END_REF], [START_REF] Borst | Capacity of wireless data networks with intra-and inter-cell mobility[END_REF]. But even schedulers with fair resource sharing strategy can take advantage of users mobility.

The fact that mobility may improve performance has already been observed in the literature (e.g., in [START_REF] Bonald | How mobility impacts the flow-level performance of wireless data systems[END_REF], [START_REF] Bonald | Flowlevel performance and capacity of wireless networks with user mobility[END_REF], [START_REF] Borst | Mobility-driven scheduling in wireless networks[END_REF], [START_REF] Karray | User's mobility effect on the performance of wireless cellular networks serving elastic traffic[END_REF], [START_REF] Kim | Stability of flow-level scheduling with markovian time-varying channels[END_REF]). These papers mainly present theoretical properties and performance bounds. For example, in [START_REF] Bonald | How mobility impacts the flow-level performance of wireless data systems[END_REF], authors identify two limit regimes of infinitely fast and infinitely low channel variations, and show that these limit regimes provide simple bounds on performance at a flow-level. [START_REF] Bonald | Flowlevel performance and capacity of wireless networks with user mobility[END_REF] also develops lower and upper bounds for the flow-level performance measures, and show that mobility tends to increase the overall capacity of the network. In [START_REF] Borst | Mobility-driven scheduling in wireless networks[END_REF], authors analyze networks with several interacting base stations, and show that mobility increases the stability region of the system. Authors of [START_REF] Jdidi | Flow-level performance of proportional fairness with hierarchical modulation in ofdma-based networks[END_REF] model an OFDMA system jointly using Proportional Fairness (PF) and Hierarchical Modulation (HM), by a multi-dimensional Markov chain, and show that in the presence of HM, a simple cyclic service, such as Round Robin, yields better performance than PF. The most related work is [START_REF] Abbas | Opportunistic gains of mobility in cellular data networks[END_REF], in which authors assess the impact of users mobility on cell performance, under a fair and an opportunistic scheduling scheme. They show in particular that under both scheduling policies, mobility improves throughput performance at cell edge. But as the Markovian process associated with their model is no longer reversible in the case where mobile users fairly share resources, they can only develop closed-form expressions in two limiting cases, namely when users are static and when users have a theoritecal infinite speed. For more realistic speeds, their analysis relies on the numerical solution of multi-dimensional Markov chains.

In this paper we develop a multi-class Processor Sharing (PS) queue model, that captures mobility of users through the distribution of the time a given user physically stays in the different coding zones of cells. Contrarily to previous studies, our model does not rely on the numerical analysis of complex Markov chains, or on limiting assumptions such as infinite speed of users, and as such is one of the first tractable and accurate approximations for 4G/5G cellular networks with mobile users. The originality of the approach is to decouple the mobility model from the cell model itself, by relating input parameters of the PS queue to physical mobility parameters. We show that our model is consistent with aforementioned analytical bounds for realistic speeds of users. Thanks to our model, we quantify the gain of speed on both the performance of the cell and the end-to-end performance of users, and investigate the influence of intra-and inter-cell mobility.

The paper is organized as follows. Section II presents system and trafic assumptions used in the model. Section III develops the PS queue model and all performance parameters of interest. The model is validated through simulation in Section IV that also investigates the impact of users speed on performance. Finally, Section V concludes the paper.

II. SYSTEM AND TRAFFIC ASSUMPTIONS

A. The system: a LTE macrocell divided into J zones

We consider a LTE macrocell with a round robin scheduling discipline. For a given number of active users, resources (Resource Blocks in LTE) are equally divided among users. A user that is alone in the cell will have different bit rates if he is close to the base station, compared to the case where he is far from it. The cell can thus be divided into J zones of equal radio conditions, or classes, each characterized by an achievable throughput, i.e., a throughput that can be achieved by a user when scheduled by the base station [START_REF] Bonald | Wireless downlink data channels: user performance and cell dimensioning[END_REF]. A user of class j, j = 1, ..., J, i.e., currently present in zone j, will obtain a throughput C j , if he is alone in the whole cell. Even if it is not necessary for the developments in our work, we can assume as, e.g., in [START_REF] Tarhini | Qos-oriented resource allocation for streaming flows in ieee802. 16e mobile wimax[END_REF], that the J zones form concentric circles of radius R j , where zone 1 is the central zone and zone J is the peripheral zone.

B. Traffic assumption and mobility model

We consider that connection demands arrive to the cell according to a Poisson process of rate Λ. We assume that a user that carries a new connection demand has a probability p j to be initially of class j and thus to start its download in zone j. As a result, new connection demands arrive in zone j according to a Poisson process with a rate Λ j = p j Λ.

Contrarily to previous classical works on PS queue models (e.g., [START_REF] Bonald | Wireless downlink data channels: user performance and cell dimensioning[END_REF]), we assume that users are mobile, and can thus change zone during their download or leave the cell before completing their transfer. The mobility model we consider is the following. It is important to emphasize that it is a physical mobility model of users that is decoupled from the utilization of the resources of the cell by users. We denote by Θ j the physical sojourn time of a user in zone j at each visit of the zone, that is, the time duration he physically stays in zone j starting form the moment he appears in zone j (either because he begins its download in zone j, or because he enters zone j from another neighboring zone or cell), and ending when he leaves zone j (still being active or not) for another neighboring zone or leave definitively the cell. We then define P jk , the probability that a user that physically (i.e., active or not) exists zone j, move to a neighboring zone k. If the outside of the cell is labelled 0, P j0 is the probability that a user exits the cell from zone j, and P 0j is the probability that a user enters the cell by zone j. Note that when zones form concentric circles, users can only leave and enter the cell from zone J.

We assume that each new connection demand (regardless of its initial class) brings an identically distributed volume Σ of data to be downloaded. As soon as a new request arrives, it triggers the start of a new data transmission (in the zone where the request appears). This transmission ends either because the user has completed its transfer before leaving the cell (whatever the number of zones he has visited meanwhile), or because he has left the cell before completing its transfer. As a result, the volume actually transferred by a given user in the cell is, in average, less than Σ.

We consider in this paper that random variables Σ and Θ j are exponentially distributed. The first assumption is necessary for the derivations presented below. As a matter of fact, and because of the memoryless property of the exponential distribution, assuming an exponential volume enables us to forget the amount of data already transferred by a user in previous zones, as long as he is still active and moves to another zone or to another cell. This drastically simplify the analysis. On the other hand, assuming exponential sojourn time in each zone is only made for simplification purposes, and other more realistic distributions can be considered.

C. Estimation of parameters when zones are concentric circles

We now consider the special case where the J zones form concentric circles of radius R j , and see how we can estimate the traffic and mobility parameters. First, the surface of zone

j is S j = π(R j 2 -R j-1 2
), for j > 1, and S 1 = πR 1 2 . If arrivals of new connection demands are uniformly distributed over the whole surface of the cell, the probability p j that a new connection demand appears in zone j, is proportional to the surface of zone j:

p j = S j πR J 2 . (1)
Concerning the mean sojourn time in zone j, E(Θ j ), we can reasonably assume that it is proportional to the square root of the surface of the zone and inversely proportional to the speed V of users:

E(Θ j ) = K S j V , (2)
When zones form concentric circles, a user that physically exists zone j, j = 2, ..., J -1, has a probability P j j-1 to move to zone j -1, and a probability P j j+1 to move to zone j + 1 (with, of course, P j j-1 + P j j+1 = 1). For zone 1, obviously, P 12 = 1. And from zone J, a user can either move back to zone J -1 with a probability P J J-1 , or exit the cell with a probability P J0 . All these probabilities clearly depend both on the radius R j of zones and on the real mobility of users. However, without additional assumptions on physical mobility of users, we can use the following approximation:

P j j-1 = R j-1 2R j , j > 1. (3)
This is a linear approximation that respects obvious limits: when R j-1 → 0, P j j-1 → 0, and when R j-1 → R j , P j j-1 → 1 2 . Note that probabilities p j and P ij , as well as mean sojourn times E(Θ j ), are input parameters for our PS queue model. Any alternative expressions resulting from a realistic physical mobility model of users, can be alternately used without changing the development presented below.

III. MODEL

A. Model of an isolated cell

To simplify the presentation, we first consider in this section the cell as isolated from the outside, and do not explicitly take into account active users that make a handover from neighboring cells. This assumption will be relaxed in Section III-B.

The model represents the cell occupancy by a multi-class Processor Sharing queue, as illustrated in Figure 1. Each class corresponds to a zone of the cell. Customers of class j arrive to the queue according to Poisson process of rate λ j , j = 1, ..., J.

It is important to note that, contrarily to [START_REF] Baynat | Impact of mobility in dense lte-a networks with small cells[END_REF], λ j is different from the rate Λ j of new connection requests that appear in zone j, as it must include arrival of users that move from the other zone while still being active. If we denote by Λ ij the average number of active users moving from zone i to zone j by unit of time, we have:

   λ 1 = Λ 1 + Λ 21 λ j = Λ j + Λ j-1 j + Λ j+1 j , j = 2, ..., J -1 λ J = Λ J + Λ J-1 J (4)
Note that, relaxing the assumption that the cell is isolated from the outside, will consist in adding to λ J a term that will represent the rate of active users making a handover from the outside. For the time being, we can just consider that this rate is part of the supposedly known rate Λ J of new connection demands appearing in zone J.

PS μ 1 μ 2 μ J : λ 2 λ 1 λ J

Fig. 1: PS queue model

Class-j rate can in turn be expressed as in [START_REF] Baynat | Impact of mobility in dense lte-a networks with small cells[END_REF]:

µ j = C j x j , (5)
where x j is defined as the average number of bits transferred by an active user in zone j, for each visit of the zone, i.e., between the moment he enters the zone (or appears in the zone) and the moment he leaves it (or finishes his transfer in the zone).

Before going further, it is important to emphasize that this PS queue is always stable, i.e., there is no stability condition for this system. This can be intuitively explained by the fact that all users are mobile and will eventually leave the cell after a finite time. When the rate of news connection demands increases and tends to infinity, users are very likely to leave the cell before downloading a single bit.

As a result, we are left to estimate all the input parameters of the PS queue, namely λ j and µ j , for j = 1, ..., J, or more precisely all Λ ij and x j . Note that in [START_REF] Baynat | Impact of mobility in dense lte-a networks with small cells[END_REF], only one parameter was left to estimate.

Given parameters λ j and µ j , standard results for the stationary multi-class Processor Sharing queues can be readily applied to calculate the average throughputs γ j obtained by users in zone j during their transfer, that is,

γ j = C j (1 -ρ), (6)
where ρ = J j=1 ρ j and ρ j = λj µj . In order to estimate the first missing parameters x j , required in the expression of µ j , we use the methodology developed in [START_REF] Baynat | Impact of mobility in dense lte-a networks with small cells[END_REF] for a cell with a single zone, and apply it to each zone individually. As shown in [START_REF] Baynat | Impact of mobility in dense lte-a networks with small cells[END_REF] in the special case where Σ and Θ j are exponentially distributed, x j is related to the average throughput γ j obtained by users in zone j and to the average time E(Θ j ) a user physically spends in zone j, as:

x j = E(Σ)E(Θ j )γ j E(Σ) + E(Θ j )γ j . ( 7
)
We now need to estimate the missing parameters Λ ij appearing in the expressions of λ j . If we denote by h j the handover probability from zone j, i.e., the probability that an active user in zone j leaves the zone without having finished its transfer, we can express Λ j j+1 and Λ j j-1 as:

Λ j j+1 = λ j h j P j j+1 , j = 1, ..., J -1 Λ j j-1 = λ j h j P j j-1 , j = 2, ..., J (8)
The handover probabilities h j can in turn be expressed as:

h j = +∞ 0 P Θ j ≤ y γ j f Σ (y) dy. ( 9
)
Assuming that Σ and Θ j are exponentially distributed, the previous integral readily gives [START_REF] Baynat | Impact of mobility in dense lte-a networks with small cells[END_REF]:

h j = E(Σ) E(Σ) + E(Θ j )γ j . ( 10
)
We finally end up with a system of 6 dependent equations (4-8, 10) that will be solved using a fixed-point iterative technique.

B. Extension to take into account handovers from the outside of the cell

Consider the cell as isolated is not realistic. Indeed, if some users can exit the cell without having completed their transfer (which is what we consider from the beggining), it is normal to consider that some users can enter the cell from the outside, with some data volume remaining to be transferred. In order to account for outside handovers, we can simply add to the rate λ J of the external zone of the cell, the rate of users making a handover from the outside, denoted as Λ 0J :

λ J = Λ J + Λ J-1 J + Λ 0J (11)
Now, the question becomes: how can we estimate Λ 0J ? The solution mentioned earlier, that consists in considering that Λ 0J is a known input parameter is not satisfying. We now propose an alternative solution that consists in assuming that the considered cell is involved in a network of statistically equivalent cells. If this is true, the average number of active users by unit of time that make a handover from the cell to the outside, must be equal to the average number of active users by unit of time that make a handover from the outside to the cell. In other words, we must have Λ 0J = Λ J0 , and this last quantity can simply be related to the handover probability of zone J of the considered cell as: Λ J0 = λ J h J P J0 . As a result, to take into account a cell involved in a network of equivalent cells, the last equation of system 4 must be replaced by:

λ J = 1 1 -h J P J0 (Λ J + Λ J-1 J ) (12)
However, it is very important to note that this system taking into account handovers from the outside, is no longer always stable. As shown in [START_REF] Abbas | Opportunistic gains of mobility in cellular data networks[END_REF], the stability condition of this system is independent of the speed V of users and is equivalent to the stability condition of a system where users have an infinite speed. It can be expressed as:

Λ < C∞ E(Σ)
, where C ∞ is the equivalent capacity of the system where users have an infinite speed and is defined in the Section III-D.

C. Performance of users in the cell

We now see how we can derive from the model the performance of an active user in the considered cell. First, we can redraw the PS queue model as in Figure 2 for the special case where J = 2, by clearly showing the feedback loops of active users making a Handover from one zone to the other one, and active users coming from the outside.

Let us denote by λ the total arrival rate of connection demands in a cell. λ is different from the rate Λ of new connection requests that appear in cell, as it must include users that move from the outside of the cell, while still being active:

λ = Λ + Λ 0J = Λ + Λ J0 . (13)
From classical results of PS queues, we can calculate the average number of customers of each class, corresponding to the average number of active users in each zone:

Qj = ρ j 1 -ρ , (14)
and derive from Little law, the average time R spent by an active user in the cell, starting from the moment where a new active user appears in the cell (either because of a new connection demand or by handover of an active user from the outside), and ending either when the user finishes its transfer in the cell before leaving it, or leave the cell before the completion of its transfer:

R = J j=1 Qj λ . (15)
We define q j as the probability that an active user is in zone j:

q j = Qj J i=1 Qi . ( 16
)
We can then calculate the average throughput γ obtained by an active user during it whole sojourn in the cell:

γ = J j=1 q j γ j . (17)
From γ and R, we can derive X, the total number of bits transferred, in average, by a user during its whole sojourn in the cell:

X = γ R. (18)
Of course, X ≤ E(Σ).

We finally estimate the global handover probability H, i.e., the probability that an active user leaves the cell before completing its transfers, whatever the number of zones he has visited meanwhile. H is calculated as the ratio between the average number of active users leaving the cell by unit of time (from zone J) and the total number of new active users that appear in the cell by unit of time:

H = QJ P J0 E(Θ J ) λ . (19)

D. Performance bounds

The performance of the system we consider with mobile users physically moving with some speed V , is bounded by that of two systems. In the first one, users are static (V = 0), and in the second one, users have an infinite speed (V = ∞).

The model corresponding to the first system where users are static, is the classical multi-class PS queue with parameters λ j = Λ j and µ j = Cj E(Σ) . It is known to be equivalent to a single-class PS queue with an arrival rate λ = J j=1 λ j , and a service rate µ equal to the harmonic mean of each service rate:

1 µ S = J j=1 pj µj , p j being the probability that a new PS Λ 2 μ 1 μ 2 Λ 1 λ 1 λ 2 h 2 P 21 1-h 2 P 21 h 1 1-h 1 Λ p 1
1 C S = J j=1 p j C j . ( 20
)
In this system, static users obtain an average throughput γ S during their transfer, given by:

γ S = C S (1 -ρ), (21)
where ρ = λ µ S , as long as the stability condition λ < µ S is satisfied. Figure 3 illustrates this first bound in red.

As shown in [START_REF] Abbas | Opportunistic gains of mobility in cellular data networks[END_REF], the second system where users have an infinite speed, is equivalent to a single-class PS queue with an arrival rate λ = J j=1 λ j , and a service rate µ ∞ equal to the arithmetic mean of each service rate: µ ∞ = J j=1 α j µ j , where µ j = Cj E(Σ) and α j is the probability that a user is physically in zone j (active or not) and is given by:

α j = j-1 i=1 E(Θ i+1 )P i i+1 E(Θ i )P i+1 i α 1 , j = 2, ..., J, with J i=1 α i = 1.
(22) Figure 3 illustrates this second bound in blue, corresponding to a PS queue with an equivalent capacity C ∞ and users' average throughput γ ∞ given by:

C ∞ = J j=1 α j C j , and γ∞ = C ∞ (1 - ΛE(Σ) C ∞ ). ( 23
)
The performance of the system where users have a given speed V , e.g., the throughput obtained by users during their sojourn in the cell, lies in between these two bounds. As illustrated in Figure 3, the throughput starts from a value C V when the load is very low, and converges to the limit C∞ E(Σ) (the same for any speed V ) when the load is very high. In order to calculate the value of C V , we consider the Continuous-Time Markov Chain (CTMC) represented in Figure 4 (for J = 4 zones) and modeling the behavior of the system submitted to a very low load, i.e., when there is at most one customer in the cell. States j of this CTMC corresponds to one customer in zone j, and state 0 to an empty system.

Λ V = 0 V = ∞ V γ C V C ∞ C S C ∞ Ε(Σ) C S Ε(Σ)
Fig. 3: Bounds on the throughput obtained by users in the cell Fig. 4: CTMC modeling the cell at very low load C V can be expressed from the stationary probability π i of this CTMC as in relation 24, and corresponds to the arithmetic mean of capacities C j weighted by the probabilities β j that an active user (supposed to be alone in the cell) is in zone j:

C V = J j=1 β j C j with β j = π j J i=1 π i , j = 1, ..., J. (24)
Note that this result does not depend on rate δ appearing in this CTMC. Finally, one can easily check that for any speed

V of users, C S < C V < C ∞ , C V → C S when V → 0, and C V → C ∞ when V → ∞.
Last but not least, we will see in Section IV that our model respects all of these bounds.

E. End to end performance of users

We still consider the case of a network of statistically equivalent cells, and derive the end to end performance of mobile users, possibly making intra-cell and inter-cell handovers. Because the size Σ of data to be transferred by users is supposed to be exponentially distributed, when a user makes a handover and leaves the cell without having completed its transfer, the data remaining to be transferred has the same distribution as the original one (memoryless property of the exponential distribution). As a result, the number n h of handovers a user has to make and the number n c of cells a user has to visit, before the completion of its transfer, are both geometrically distributed with parameter H (starting from 0 for the first one and from 1 for the second one), with means given by:

nh = H 1 -H and nc = 1 1 -H . (25)
We denote by t h the duration of a handover procedure, i.e, the duration of the service interruption when a user change cell. We can estimate with relation 26, the average end to end transfer time T of user, defined as the average time for a user to complete a full transfer, whatever the number of cells and the number of zones in each cell the user has visited during its transfer:

T = nc R + nh t h . (26)
Finally, we obtain the average end to end throughput Γ a user obtains during its full transfer, as:

Γ = E(Σ) T . (27)

IV. PERFORMANCE RESULTS

We have compared the results provided by the models to those delivered by a home-made discrete-event simulator developed in Matlab. We reproduced in simulation the traffic assumptions and the mobility model described in Sections II-B. We assumed that the cell uses a number of 100 Ressource Blocks for the downlink channel and offers to users four MCS (28,23,16,[START_REF] Bonald | Flowlevel performance and capacity of wireless networks with user mobility[END_REF]. This results in four transmission zones with corresponding capacity C 1 = 75 Mbit/s, C 2 = 51 Mbit/s, C 3 = 31 Mbit/s and C 4 = 10 Mbit/s [START_REF][END_REF]. We set the constant K = 1 √ π and we take E(Σ) = 10 MB for the mean data volume to be transferred by all users. We used the following radius corresponding to the concentric circles model of the cell: R 1 = 100 m, R 2 = 150 m, R 3 = 200 m and R 4 = 250 m, R 4 corresponds approximately to the operating range of LTE antenna in urban environment. The mean sojourn time in each zone E(Θ j ) is given by equation 2. According to the estimations given in Section II-C, resulting probability are reported in Table I.

A. Validation

This subsection Aims to validating the proposed model; to do this we consider one cell with J = 4 zones (with above parameters). considered cell as a function of the total arrival rate of new connection demands Λ. We chose different values of speed V . We observe that the predicted model is very close to simulation results. The average relative error between performances predicted by the model and those obtained by simulation is about 8% in average and this error remains less than 22% in the worst case.

As explained in section III-D, mobile users' throughput (Figure 5) is bounded by the throughput obtained when all users are static (lower bound) and the throughput obtained with infinite speed (upper bound). As expected, users mobility improve users' sojourn time (see Figure 7), capacity C V , users' throughput and stability which is Λ < 0.24 for the system with static users and Λ < 0.30 when users are mobile. In this latter case, observe that the stability condition does not depend on users speed. The reason of this improvement is that, gain from users moving from a poor channel conditions to a good ones outweighs the loss from users moving in the reverse direction. Note that users in a favorable conditions have a better chance to complete their transfers before moving to a worse location.

It is important to observe that, of course, this improvement comes with an increase in the handover rate (see Figure 6). The handover H starts from initial value that depends on speed and increases with Λ. H converging to 1 when Λ → 0.30.

B. End to end performance

We now investigate the end to end users' performance in the system.

Figure 8 show the number of visited cells in the network as a function of the total arrival rate of new connection demands Λ for different values of users speed. As expected and in view of equation 25 and Figure 6; the number of visited cells is strongly impacted by the users' speed and is an increasing function of the arrival rate of new connection demands. Since in LTE only hard handover is supported, the time required to complete a transfer must take account of service interruption due to handover procedure.

Figure 9 depicts users' sojourn time in the network (the time a user needs to complete a transfer) still as a function of the total arrival rate of new connection demands Λ. We chose three values for the duration of a handover procedure t h (1s, 2s, 3s) and 100 km/h for users speed.

Unsurprisingly, the time a user takes to complete a transfer increases with total arrival rate Λ. This time is also affected by the duration of the handover procedure.

C. Impact of users mobility

We now discuss the impact of users speed in performances. Figure 10 present, for a total arrival rate of new connexion demands Λ = 0.15, the throughput as a function of users' speed. Note from this curve the significant gain of mobility compare to the scenario where all users are static. In particular, the throughput of mobile users is increasing function of the speed, bounded by static users throughput and the throughput obtain if users had infinite speed.

Figure 11 is the analog of Figure 10 with consideration of the duration of service interruption due to handover procedure which is set to 2s. This curve highlights the impact of handover on throughput gain. Specially, for speed less than 420 km/h, mobility leads to a throughput gain; and above this value, we observe a loss of throughput compared to scenario with only static users.

This numerical results show how handover rate counterbalance the flow-level performance improvement and prove that, mobility gain is a non-monotonic function of users' speed.

V. CONCLUSION

We have developed a PS queue model for performance evaluation of data cellular networks with a round-robin policy, taking into account intra-and inter-cell mobility of users. We have shown that this model is consistent with known analytical bounds corresponding to static users or infinite speed, and provides a very good accuracy for more general speeds. Our model confirms that mobility may improve performance of users in a given cell, and enables to quantify the gain. It also provides end-to-end performance of users among a network of statistically equivalent cells, and shows that performance is not anymore a monotonic function of the speed. We believe that the approach used in this paper consisting in decoupling the mobility model from the cell model, can be applied to many other models or systems, e.g., coupled PS queues models for networks of interacting base stations, or models for more general scheduling policies.

p 2 ENDFig. 2 :

 22 Fig. 2: Illustration of handovers between zones and from the outside

Fig. 5 :

 5 Fig. 5: Throughput obtained by active users as a function of the total arrival rate of new connection demands.

Fig. 6 :

 6 Fig. 6: Handover probability as a function of the total arrival rate of new connection demands.

Figure 5 ,Fig. 7 :

 57 Figure 5, 6 and 7 respectively show comparison of users' throughput, handover probability and users' sojourn time in the

Fig. 8 :Fig. 9 :

 89 Fig. 8: Average number of visited cells as a function of the total arrival rate of new connection demands.

Fig. 10 :Fig. 11 :

 1011 Fig. 10: Average Throughput obtain by active users as a function of users' speed when Λ = 0.15.

TABLE I :

 I Cells parameters

	parameters	zone 1	zone 2	zone 3	zone 4
	new connection	0.16	0.20	0.28	0.36
	probabilities pj
	moving probabilities	P12 = 1	P21 = 0.33	P32 = 0.37	P43 = 0.40
	Pij		P23 = 0.67	P34 = 0.63	P40 = 0.60

* This work has been carried out in the framework of IDEFIX project, funded by the ANR under the contract number ANR-13-INFR-0006.