Siham Hairoud

Pierre Combeau

Jean-Franc ¸ois Cailbault

Yannis Pousset

Yann Cocheril

Marion Berbineau

Acceleration Technique of Radio Propagation Simulator Based on the Ray Tracing for the Prediction of MIMO Channels in Dynamical Railway Environment

Keywords: radio wave propagation simulator, channel modeling, computation time optimization

The 3D ray tracing method requires a detailed description of the environment and antenna characteristics. Thus, it allows us to consider all the paths between receivers and transmitters for a fixed number of electromagnetic interactions. However, the computational complexity of this method is directly connected to the number of scatterers and the electromagnetic environment interactions. Thus we proposed an optimization method allowing reducing the computation time of a railway tunnel simulation. It consists in modifying the modeling environment, by eliminating non-significant faces. The aim of our method is, in this case, to provide estimated results close to those obtained with a complete environment modeling, but in fewer computation time.

I. INTRODUCTION

Since some years wireless communications systems found their place in the railway transport. Various solutions have been proposed in order to enhance the safety of railway traffic as the use of the transmission of data to locate trains and transmit the information of speed and stop in real time. However, there remain some obstacles to overcome in confined environments as tunnels and metros-stations. Thus, the propagation channel modeling is essential in a digital railway communication. Indeed, it contributes to the development and the optimization of new techniques which allow achieving train-floor high rate transmission systems, robust and safe in radioelectric resources, for confined propagation environments, which present a high variability due to the circulation of trains. Thus, it is fundamental to take into account the dynamical features of such environments not only for the systems conception but also in the final network deployment step.

In the ANR project PREDIT MOCAMIMODYN, we propose an optimization method allowing reducing the computation time of a railway tunnel simulation. It consists in analyzing the impact of the different faces constituting the environment on the wave propagation. The first step of our study consists in identifying the faces that significantly contribute to the wave propagation. The second step consists in lightly modifying the environment modeling, by eliminating non-significant faces. The aim of our method is, in this case, to provide estimated power close to those obtained with a 1 MOdèles de CAnaux MIMO DYNamique complete environment modeling, but in a fewer computation time.

This article is composed of several sections. In Section 2, we present our radio propagation simulator. The section 3 is composed of four sub sections. In sub section 3.A we recall our objectives. The sub section 3.B is dedicated to present the considered scenario and the parameters of the simulator. In the sub section 3.C we define the approach used to meet our objectives and finally, in the sub section 3.D we present our results. In the section 4, we show the impact of the modification of the description of the considered environment on the narrow band parameters. Finally, Section 5 is a brief conclusion where we announce the prospects for this work.

II. PROPAGATION CHANNEL SIMULATION

During the propagation between the transmitter and the receiver, the transmitted signal undergoes some perturbations caused by the electromagnetic interactions. These ones have a direct consequence on the received signal as decreasing the total power received, and on the intrinsic parameters of the waves, as phase and the polarization.

In literature, there are several ways to modelize a complex propagation channel as the statistical, the empirical methods and the deterministic one [START_REF] Iskander | Propagation Prediction Models for Wireless Communication Systems[END_REF]. In our case, the deterministic approach seems very interesting because it uses the complete description of the environment and the antenna characteristics to give a detailed description of the received signal.

In order to predict [START_REF] Escarieu | Outdoor and indoor channel characterization by a 3D simulation software[END_REF], [START_REF] Mora | Optimised Scanning of a Visibility Graph Data Structure for Efficient Ray-Tracing[END_REF] the wireless radio communication channel performances, the XLIM-SIC laboratory has developed a radio propagation simulator whom the synoptic is presented on figure 1. This simulator is based on a 3D ray tracing associated to the geometrical optics laws (GO) and the Uniform Theory of Diffraction (UTD). Input data are the environment description in 3D, the maximum combination of electromagnetic interactions, in addition with antennas features of transmitters and receivers, carrier frequency etc. The output ones are complex impulse responses, received power and all wide-band characteristic parameters. The 3D ray tracing method allows us to consider all the paths between receivers and the transmitter for a fixed number of electromagnetic interactions. However, the computational complexity of this method directly lies on the number of environment scatterers and electromagnetic interactions. Figure 2 gives a screenshot of the radio propagation simulator. Zone 1 allows configuring the transmitter and receivers antennas features like location, orientation, radiation pattern and so on. In zone 2, we state the frequency and the number of electromagnetic interactions. Zone 3 allows us going through the details of each path (as the coordinates of each interaction point, the nature of the interaction and the power associated to this path). Finally, zone 4 is a 3D view of the environment in which appear all the paths between the transmitter and the receiver. This tool is based on a research technique using the method of sources / images based on Snell's formalism (see figure 3(a)) to determine the rays reflected on flat surfaces in addition to the method of folding [START_REF] Kyősti | 1.2 v1.2 WINNER II Channel Models Part I / Part II[END_REF] (see figure 3(b)) based on the generalized Fermat's principle and allows to determine the diffracted rays on the edges. This technique is fast and simple to implement. However, its computation time increases exponentially with the number of obstacles and the number of electromagnetic interactions considered. Thus, the following paragraph provides a method for reducing the computing time by simplifying the geometric modeling environment.

A. Objective

Recall that the aim of this study is to degrade progressively the geometric description of the environment considered by removing all the faces in the shadows areas and do not contribute enough to the energy balance, without compromising the accuracy of simulation results.

In order to determine the impact of each faces of the environment on the waves propagations, we are interested primarily in the received power. However, other characteristics parameters such as delay spread and Doppler effect will be studied in the near future. In addition, we found more appropriate, initially, just focus on the moving elements (the faces of the moving train) rather than those of fixed elements. Especially, the context of our work considers the study of the waves propagations in dynamic environment. Thus, we set the reference power denoted P ref to the power associated to the packet of paths interact with the moving train (see figure 4). We then compare the reference power to the power associated to the packet of paths interact with one face, denoted P f ace . If the difference exceeds a threshold we fix in the subsection 3.C, we can consider this face as a noncontributory because the energy provided by the paths that interact with this face is negligible. The difference between P ref and P f ace is calculated for each position of the train and for each pair of transmitting and receiving antennas in the tunnel. We can then identify all the positions for which the power associated with a face is considered insignificant. In the remainder of this paper, the faces are numbered to facilitate the reading of this document.

B. Scenario and configuration of the simulator

The study was conducted in the case of a dynamic environment associated with a 4x4 MIMO channel. We have modeled a bi-ways concrete tunnel (ǫ = 10 F m -1 and σ = 0.1 Sm -1 ) that contains two metallics trains (ǫ = 1 F m -1 and σ = 56 000 Sm -1 ). One is fixed and placed below the network of transmitting antennas (see figure 4) and the other is mobile and placed below the network of receiving antennas (see figure 4). The mobile train moves with a step set at λ/ 3, ie every 1.7 cm with a frequency of 5.8 GHz. We have chosen a small sampling step in order to take into account all the variations of the received power can appear from one position to another. The transmit power is 1 mW, the dimensions of the tunnel are 500 x 4.5 x 4.5 m 3. The two trains have the same size, 120 x 3 x 4 m 3. The emitters antennas are placed at a height of 4.3 m (depending on the z axis) from the floor of the tunnel either at 30 cm from the roof of the train, the four transmitters are spaced by 50 cm. The receivers antennas used here are dipoles that radiate evenly in the azimuthal plan (omnidirectional). The receivers are placed at a height of 4.1 m (depending on the Z axis) of the tunnel floor either at 10 cm from the roof of the moving train and spaced by 50 cm. According to several previous studies carried out in railway environment [START_REF] Mariage | Theoretical and Experimental Approach of the Propagation of High Frequency Waves in Road Tunnels[END_REF], we have limited the number of reflexions and diffractions to six and one respectively which is sufficient to garanty the power convergence. Obviously no transmission is needed due the width of tunnel walls.

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Fig. 4. Studied environment

The sixteen faces of our environment are numbered from 0 to 15 (see Figure 4

C. Method of identifying influential faces

For each position, denoted i, to the train and the receivers antennas (network of mobile antennas) in the tunnel, we calculate the instantaneous gap (1), noted err inst,i (see figure 5) such as: With a threshold that we arbitrarily fixed, we can identify three categories of faces. Thus: -If err inst,i is less than or equal to 10 dB, the face belongs to the category of contributory Faces (CF); -Otherwise, if err inst,i is between 10 dB (excluded) and 20 dB (inclusive), the face belongs to the category of Medium Contributory Faces (MCF); -Finally, if err inst,i is strictly greater than 20 dB, then the face belongs to the category of Non-Contributory Faces (NCF).

To quantify the impact of each face on the wave propagation in tunnel, we take the maximun of three values obtained by counting the number of positions, where a face is classified as FC, FMC and finally FNC: max(sum(FC), sum(FMC), sum(FNC))

We consider a threshold of 10 dB which is very pessimistic because an error of 10 dB is relatively consistent. Thus, we avoid to discriminate faces whose can be important to the waves propagations. Although crude, this arbitrary value can, however, to lay the foundations of our optimization method. Consequently , a further study will be conducted to refine the threshold.

D. Results

In this subsection, we propose to apply the method mentioned in sub section 3.C, taking as reference power, the power P ref . This will allow us to initially identify the faces of the mobile train were hit by the rays do not contribute sufficiently to P ref . Thus, we can count the number of times a face belongs to the category CF, MCF and NCF. We then classify these faces of the more energetic to the least energy. We first present some results obtained for the link E1R1. Table III-D allows us to classify the six faces of the moving train by category of importance. We observe that the face 4 is the most contributory because 79.60 % of the received powers represent a gap less than 10 dB compared to P ref,i . This is followed by the face 3 (66.98%), face 1 (28.30%), face 5 (20.64%), face 2 (5.97%) and finally the face 0 ( 0.19%) (see figure 6). To generalize those observations to the overall results, we have plotted a diagram (see figure 7) which represents all those percentages values for the 16 sub-channels MIMO. For each face we have three bars. Each bar bears a label to refer to the category to which this face belongs, and the number of times a face belongs to a category. We can notice that each bar contains several pads which we have assigned by different colors. Each color represents a link. The height of each pad will increase or decrease according to the percentage of power received by the corresponding face. When the percentage of the received power tends to zero, the higher of the associated pad tends to a thread. With this representation, we found that regardless of the association considered the face 4 is still the highest contribution followed by the face 3, face 1, face 5, face 2 and finally face 0. This conclusion is supported by the geometric description of our environment as carefully watching our scenario (see figure 4), we can see that the body of the moving train is behind the network of receiving antennas except for the face 4 that is in the same plane as the receivers antennas. Knowing that the number of diffractions is limited to one and the number of reflection to six, we can imagine that the number of rays that interact with the face 0, face 1, face 2 and finally the face 5 are very few and non-energitics. This suggests to us that we can start the degradation of the geometry of our environment by removing some faces of the moving train, ie the faces 0, 1, 2 and 5.

IV. VALIDATION RESULTS

The removal of certain faces of the environment aims to save us considerable time. To validate our findings and ensure that the degradation of the geometric description of our environment does not affect the accuracy of our results obtained from a full environment (see figure 4), we will launch a new campaign of simulations where we model only the face 3 and 4 of the moving train. We define the second scenario (see figure 8) where the faces (0, 1, 2 and 5) will not be modeled. This is almost similar to the first scenario. cording to this new environment representation, in comparison with a complete one (train with 6 faces). Here, we notice that obtained curves are very similar. This investigation allowed us to reduce the simulation time of a dynamic environment associated with a 4x4 MIMO channel at a frequency of 5.8 GHz. The results are potentially interesting. We have shown that it was possible to degrade the geometric description of the environment without compromising the accuracy the total power obtained with the full environment description. However, we would verify this by taking other characteristics parameters as the delay spread and Doppler effect. In addition, we would refine the threshold that we set ourselves in the subsection 3.C by adopting thresholds of 5 dB or 3 dB. With this parametric study related to this thresholding technique we can make a gradual degradation of our environment. We will launch other simulations to validate our results.

Fig. 1 .Fig. 2 .

 12 Fig. 1. Synoptic of the radio propagation simulator

Fig. 3 .

 3 Fig. 3. Research methods of the reflected and the diffracted paths exist between transmitter and receiver

  ) as: -The face 0 corresponds to the floor of the moving train -The face 11 corresponds to the floor of the fixed train -The face 12 corresponds to the tunnel floor -The face 13 corresponds to the tunnel ceiling -The face 14 corresponds to the tunnel wall side the moving train -The face 15 corresponds to the side wall of the tunnel the fixed train

Fig. 5 .

 5 Fig. 5. The computation of the gap between P ref and P f ace

Fig. 6 .

 6 Fig. 6. Identification of contributory faces , medium contributory faces and non-contributory faces for the link E1R1

  faces de la rame mobile

Fig. 7 .

 7 Fig. 7. Identification of contributory faces , medium contributory faces and non-contributory faces

Figures 9 ,

 9 illustrates the evolution of received power ac-

Fig. 8 .

 8 Fig. 8. Modeling a part of the train moving (face 4, face 3)

Fig. 9 .

 9 Fig. 9. Evolution of total received power for different environments modeling

TABLE II THE

 II COMPARISON OF THE SIMULATIONS TIMES

	Scenarii	6 faces dynamics	2 faces dynamics	Gain 6f/2f
	Simulations times (s)	40 d 12 h 38 min	8 d 1 h 13 min	5


Using asymptotic methods to compute diffracted pressure by curved surfaces.

Mikael Vermet, Nicolas Noé, Rodolphe Vauzelle, Yannis Pousset, Pierre Combeau 27 août 2009

1 Introduction.

This article presents an original and efficient method to compute acoustic pressure diffracted by curved surfaces. Our approach is perfectly suited to be integrated into ray of beam tracing softwares. First we will remind how diffraction by curved surfaces can be handled in asymptotic methods. Keller [START_REF] Keller | Geometrical theory of diffraction[END_REF], stating on generalized Fermat' principle, established that the diffraction of an acoustic wave by a surface could be solved using creeping rays (fig 1). These creeping rays must hit the surface so as to be tangent with the surface (attachement point) and must leave it the same way (ejection point). Between these two points the ray must follow a geodesic on the surface (according to Fermat's principle). All these points are illustrated on figures 1(a) and 1(b). Nevertheless integrating creeping waves computation in a ray or beam tracing code in order to solve complex problems is difficult. Indeed, the integration of creeping rays computation inside a global ray calculation can be very cumbersome. Moreover, almost all the complex surfaces available are originally discretized. As a consequence we decided to use a discretized approach, i.e. meshing our curved surfaces into triangles or quadrangles and to consider the creeping wave problem as a succession of diffractions by the straight edges separating these triangles or quadrangles (fig 2). Geodesics are then substituted by the intersection of the Keller cone with the meshed surface, at each diffraction edge. The reflection of rays on the curved surfaces is handled with a smoothed mesh of the surfaces, using linear interpolation of normals [2]. 2 Implementation of the chosen method.

In order to handle successive diffractions in a ray-tracing environment we have to use asymptotic formula of diffraction coefficients. In [START_REF] Pathak | A uniform geometrical theory of diffraction for an edge in a perfectly conducting surface[END_REF] Kouyoumjian et Pathak treat multiple diffractions by multiplying the diffraction coefficients associated to each edge, and then apply a 1/2 factor for each edge on which incident waves grazes on the surface (for our case every edge will be applied this 1/2 factor). Unfortunately this approach fails, even for the simple case of a double diffraction by a wedge. It gives unsatisfying results in the transition region of the second edge and when edges are closer than 10 lambda. These two conditions (see figures 3(a) and 3(b)) are mandatory in our application to tesselated curved surfaces. We then chose to use Albani and Capolino formula [START_REF] Capolino | Double diffraction at pair coplanar skewed edges[END_REF][START_REF] Capolino | Diffraction at thick screen including corrugations top face[END_REF] that breaks through these limitations. Our approach is to handle successive edges two by two and to compute on each couple Albani and Capolino coefficients. When the number of diffractions is odd, the last one is computed using [START_REF] Pathak | A uniform geometrical theory of diffraction for an edge in a perfectly conducting surface[END_REF]. This is correct since Albani and Capolino formula enforces pressure continuity when the receiver lies in the transition region of the following even edge.

Application to the case of a 2D cylinder.

We expose in this section the computation of acoustic diffraction by a cylinder with our method and compare it to reference results. The test configuration is displayed on fig. 5. The receivers are situated inside the rectangle delimited by the red lines. The cylinder is tesselated into 16 rectangular patches as illustrated in 2(b). Thus, the distance between two edges is about 1 wavelength at a 1 kHz frequency and 1/2 wavelength at a 2 kHz frequency. So this tesselation is sufficiently small because Albani and Capolino's formula gives good results above a 1/4 wavelength distance. The computation is performed in 2D, that is to say that the cylinder is infinite along the y axis. Moreover the wave is cylindric (lineic source). Our method is checked against a 2D BEM numerical [START_REF] Jean | A variationnal approach for outdoor sound propagation and application to railway noise[END_REF] exact method. Indeed, the 2D BEM Method developed inside CSTB is able to treat cylinders without discretization. Moreover this configuration can be solved exactly with analytic asymptotic method (and so can be other analytic curved surfaces). Thus, we compare on figures 6 and 7 2D BEM results with asymptotic analytic solution of creeping waves found in [START_REF] Pathak | An asymptotic analysis of the scattering of plane waves by a smooth convex cylinder[END_REF], for a 1 kHz frequency. Thanks to this comparison we emphasize the efficiency of 2D BEM in treating exact cylinders. Indeed, we observe a good matching between the two results espescially in deep shadowing region where error is lower than 1 dB. Then we present on fig. 8 and9 comparisons between 2D BEM and our method using beam-tracing [2] on a tesselated cylinder to find diffrated rays on the surface. Results are shown for 1 kHz and 2 kHz. There is very good matching between 2D BEM results and our method, espescially in deep shadow region where error is lower than 1 dB. Fig. 9-Error between 2D BEM results and our method using beam-tracing on a tesselated cylinder at 2 kHz.

Open cylinder.

An open cylinder configuration is presented on figure 10. The cylinder is cut on ten degrees on each side of the x axis. Consequently the incident field from source S on points D and D' is first diffracted par straight edges on z axis at these points. These two points become then secondary sources on the cylinder. The analytical asymptotic solution of pressure at R is [START_REF] Pathak | A uniform GTD analysis of the diffraction of electromagnetic waves by a smooth convex surface[END_REF][START_REF] Kouyoumjian | Uniform GTD for diffraction by edges vertices convex surfaces, extrait de "Theoretical methods for determining the interaction of electromagnetic waves with structures[END_REF] :

P (R)= e -jks ′ s ′ • D GT D (φ) • H e -jks d √ s d ( 1
)
Where H is the Fock function.

We now present results from this analytical asymptotic solution compared to our beam-tracing computation result on a tesselated surface. We do not use 2D BEM here since an open cylinder is not a closed surface [START_REF] Jean | A variationnal approach for outdoor sound propagation and application to railway noise[END_REF] (we would have to add thickness to the shape, leading to incoherent comparisons) that's why all the following validations are made with asymptotic analytical results. As for the full cylinder we tesselated the shape into 16 rectangular patches. Results are displayed on figures 11, 12 (1 kHz) and 13, 14 (2 kHz). We can observe on these figures a very good agreement between asymptotic analytical results and our beam-tracing results. The differences between acoustical levels calculated by these 2 methods at 1 kHz and 2 kHz are inferior to 1 dB on a large part of the receivers area. (a) Error between our method and analytical asymptotic method.

(b) Scale (in dB)

Fig. 12 -Error between our method and analytical asymptotic method, 1 kHz.

(a) Results with our method on a tesselated cylinder.

(b) Results with analytical asymptotic method.

(c) scale (in dB). (a) Error between our method and analytical asymptotic method.

(b) Scale (in dB).

Fig. 14 -Error between our method and analytical asymptotic method, 2 kHz.

A complete 3D test case.

We present in this section results for a complete 3D test case (figures 15) which is a part of a cylinder. This enables us to show that, when added with all the other contributions (diffractions by straight and ciruclar edges, reflexions,...), creeping waves contributions gives non negligible improvements to results accuracy and better agreement with exact numerical results. The incident wave is now a spherical wave and the geometry is now limited in the y direction : hence part of circular edges that bound the surface will now diffract. All interactions (reflections on the source side, diffractions by straight and curved edges, creeping waves) are taken into account. Figure 16

Conclusion

We have proposed through this paper an original method which is able to calculate creeping waves on tesselated curved surfaces. This last aspect is very important because in beam tracing applications, surfaces are very often discretized. Results presented in this paper show a very good agreement between our approach and exact numerical methods. Moreover we emphasize the improvements provided by the creeping waves contributions concerning the pressure calculation accuracy when a complete 3D configuration has to be considered.

4 Annexe A : coefficient de double diffraction de Albani et Capolino.

We sumarize here Albani and Capolino formula used into our creeping rays diffraction method. A 1/2 factor is added to the original formula since the incident ray on edge E3 is grazing. We define respectively distances between edges E2 and E3, between E3 and E4, E4 and receiver R as : r 1 , l, r 2 . n 1 π and n 2 π are the external wedges opening angles associated to the edges E3 and E4.φ 1 which is the incidence angle on edge E3 is equal to n 1 π, while φ 2 corresponds to the angle between the diffracted ray and the face common to edges E3 and E4. Formula giving pressure

E1 E2 E3 E4 E5 S R φ1 φ2
Fig. 18 -Creeping rays needed to compute diffraction by the surface between source S and receiver R.

at R is given by :

P (R)= e -jkr1 r 1 • A • D h 12 (2)
D h 12 is the double diffraction coefficient adaptated to edges E3 et E4.

A = r 1 lr 2 (r 1 + l + r 2 )
e -jk(l+r2) (3)

D h 12 = 1 4πjk 2 p,q=0 (-1) p+q n 1 n 2 cot Φ p 1 2n 1 cot Φ q 2 2n 2 T (a p ,b q ,w) (4)
T is the transition function defined here.

T (a,b,w)= 2πjab √ 1 -w 2 G a, b + wa √ 1 -w 2 + G b, a + wb √ 1 -w 2 +G a, b -wa √ 1 -w 2 + G b, a -wb √ 1 -w 2 (5)
G(x,y)= y 2π e jt 2 ∞ 0 e -jt 2 t 2 + y 2 dt (6)

An asymptotical formula of this function is given in [START_REF] Capolino | Simplified closed form expressions for computing Fresnel integrals their application vertex diffraction[END_REF]. Formulas [START_REF] Pathak | An asymptotic analysis of the scattering of plane waves by a smooth convex cylinder[END_REF], ( 8), ( 9), [START_REF] Capolino | Simplified closed form expressions for computing Fresnel integrals their application vertex diffraction[END_REF] and ( 11) give us all the parameters involved in transition function T.

a p = 2k r 1 l r 1 + l sin Φ p 1 -2n 1 N p π 2 (7)
b q = 2k r 2 l r 2 + l sin Φ q 2 -2n 2 N q π 2 (8)

w = r 1 r 2 (r 1 + l)(r 2 + l) (9)
Φ p 1 = φ 1 +(-1) p π (10)

Φ q 2 = φ 2 +(-1) q π (11)

N p and N q are the integers which verify the more closely the following relations :

2πn 1 N p -Φ p 1 = 0 (12) 2πn 2 N q -Φ p 2 = 0 (13)
Références

  Top view of rays creeping around a cylinder.

Fig. 1 -

 1 Fig. 1-Creeping rays needed to compute diffraction by the surface between source S and receiver R.

  (a) Cross section view of a tesselated cylinder.

  Top view of a tesselated cylinder.

Fig. 2 -

 2 Fig. 2-Tesselation of the surface into rectangular patches.

  The receiver is in the transition region of the second wedge case of failure of Pathak and Kouyoumjian formula.

Fig. 3 -

 3 Fig. 3-Situations in which Pathak and Kouyoumjian formula fails.

Fig. 4 -

 4 Fig. 4-Overview of our method to compute diffraction by a tesselated curved surface.

Fig. 5 -

 5 Fig. 5-First test case : 2D cylinder, cross section view.

Fig. 6 -

 6 Fig.6-"Exact" methods results for a 2D cylinder.

Fig. 7 -

 7 Fig. 7-Error between "exact" methods for a 2D cylinder.

  (a) Results with our method on a tesselated cylinder. (b) Error with 2D BEM results. (c) Scale (in dB).

Fig. 8 -

 8 Fig.8-Error between 2D BEM results and our method using beam-tracing on a tesselated cylinder at 1 kHz.

  (a) Results with our method on a tesselated cylinder. (b) Error with 2D BEM results. (c) Scale (in dB).

Fig. 10 -

 10 Fig. 10 -Second test cas, 2D open cylinder.

Fig. 11 -

 11 Fig. 11 -Results with our method and analytical asymptotic method on an open cylinder, 1 kHz.

Fig. 13 -

 13 Fig. 13 -Results with our method and analytical asymptotic method on an open cylinder, 2 kHz.

  is a 3D view of the geometry, source and receivers. Source position is [-1.4;0.7;-1.7] and is the red spot. Receivers are in a rectangle delimited by y = 0, x in [-1:4], z in [-3:3] and are the green dots.

Fig. 15 -

 15 Fig. 15 -Overview of the 3D case.

Figure 17

 17 Figure 17 emphasizes the importance of the creeping waves in this particular test case. Figure 17(a) corresponds to a computation without creeping waves; reflections, diffractions by both vertical straight edges and diffractions by both top and bottom circular edges are taken into account. On figure 17(b) creeping waves are taken into account. Results shown on figure 17(c) has been obtained by 3D BEM. We

Fig. 16 -

 16 Fig. 16 -3D test case with source and receivers .

  (a) Results without creeping rays. (b) Results with creeping rays.(c) 3D BEM results.

Fig. 17 -

 17 Fig. 17 -Comparison of beam-tracing results to 3D BEM reference on a part of cylinder: importance of creeping waves in the interference pattern.


Oana Goga
email: oana.goga@lip6.fr

Patrick Loiseau
email: ploiseau@soe.ucsc.edu

Paulo Gonc
email: paulo.goncalves@ens-lyon.fr

On the Impact of the Flow-Size Distribution's Tail Index on Network Performance with TCP Connections

Keywords: Heavy-tailed distribution, network performance, TCP

In this paper, we study the impact of the flow-size distribution on network performance in the case of a single bottleneck with finite buffer. To tackle the case where flows are transmitted with the TCP protocol, we use real experiments and ns-2 simulations. Our preliminary results show that the distribution's tail index impacts the performance in a more complex way than what is reported in existing literature. In particular, we exhibit situations where a heavier tail gives better performance for certain metrics. We argue that a main cause of our observed results is the transient behavior at the beginning of each flow.

INTRODUCTION 1.1 Motivations and related work

It is well established that the distribution of flow sizes in the Internet is heavy-tailed [START_REF] Crovella | Self-similarity in World Wide Web traffic: Evidence and possible causes[END_REF]. Following up this important discovery, in the last 15 years or so, significant research efforts have been devoted to understanding the impact of this property on network performance, mainly in the case of a single bottleneck. Experimental and theoretical works using an open-loop approach have concluded that heavy tails degrade performance for large buffers, whereas performance is insensitive to the tail for small buffers [START_REF] Erramilli | Experimental queueing analysis with long-range dependent packet traffic[END_REF][START_REF] Grossglauser | On the relevance of long-range dependence in network traffic[END_REF][START_REF] Mandjes | The shape of the loss curve and the impact of long-range dependence on network performance[END_REF]. However, most of today's Internet traffic uses the TCP protocol, which is based on a closed-loop mechanism. In this case, the simulation-based study of [START_REF] Park | On the effect of traffic self-similarity on network performances[END_REF] showed that, with their particular setting, heavier tails (i.e., smaller tail indices) degrade the performance for all metrics (loss, delay, throughput). Most of the rest of the literature concluded, based on various models, that first-order performance metrics (e.g., mean flow-rate) are insensitive to the flow-size distribution [START_REF] Arvidsson | On traffic models for TCP/IP[END_REF][START_REF] Ben Fredj | Statistical bandwidth sharing: a study of congestion at flow level[END_REF]. However, these models use simplifying assumptions, notably on bandwidth sharing, that do not account for the transient behavior at the beginning of each TCP flow. Finally, numerical methods based on more complex models have been proposed to evaluate performance metrics given the input conditions [START_REF] Garetto | An efficient technique to analyze the impact of bursty TCP traffic in wide-area networks[END_REF]; but they do not provide qualitative insight to understand the impact of the flow-size distribution.

Contributions

In this paper, we study the impact of the flow-size distribution's tail index in the case of a single bottleneck of finite buffer, with the TCP protocol. To grasp the full complexity of the problem, we use experiments on a real (but fully controlled) network testbed [START_REF] Loiseau | Metroflux: A high performance system for analyzing flow at very fine-grain[END_REF] rather than models based on simplifying assumptions. We also use ns-2 simulations to complement our study. We report preliminary results that contrast with existing literature by showing that: -the impact of the tail index on performance depends on many parameters and not only on the buffer size; and -there exists situations where a heavier tail gives better performance for certain metrics.

To the best of our knowledge, this last effect has not been reported in previous literature. We argue that an important element to interpret these results is the transient behavior, i.e., the flow's behavior during slow-start, which is not taken into account in the aforementioned models.

SETTING

We use a butterfly topology with the same number Nsrc of sources and destinations. Experiments on a real network and ns-2 simulations use the same setting with Nsrc = 45 and 500 sources respectively. Each source behaves as an ON/OFF source: it alternates between flow transmission using the TCP protocol (with Reno variant) and idle (OFF) period. To avoid multiple congestion points, each source always sends to the same destination and there is no bottleneck on the return path (for the ACKs transmission). The minimal RTT (excluding queueing delay) is 10 ms for each pair source-destination. The bottleneck queue uses a Drop Tail policy with finite buffer of size B which is 96 or 896 pkts in the experiments and 100 or 1, 000 pkts in the simulations. These values correspond to maximal queueing delays of 1.15, 10.75, 1.2 and 12 ms, respectively. The capacity of both the source links and the bottleneck link is 1 Gb/s. Each experiment lasts 2 hours and each simulation 1 hour.

We use a Pareto distribution for the flow-size distribution, of mean 1, 000 pkts and of tail index α varying across experiments to assess its impact. In all the experiments/simulations, we use an exponential OFF-time distribution of mean 0.2 s. Note that due to the closed-loop mechanism of TCP, the load may vary between the experiments [START_REF] Schroeder | Open versus closed: A cautionary tale[END_REF].

An important parameter in our study is the TCP tuning parameter which controls whether the ssthresh parameter is cached from one TCP connection to the next connection with the same source and destination. The ssthresh parameter determines the maximal congestion window of the slow-start. When it is reached, the connection goes to the congestion avoidance phase, whereas if a loss occurs before it is reached, then the congestion window at this loss event becomes the new ssthresh. If it is cached, then it can only decrease from one flow to the next flow. In our experiments, there is a non-negligible loss rate. Then, the ssthresh rapidly stabilizes around 10 pkts for each source-destination pair, which significantly limits the slow-start phase. We refer to this situation where the ssthresh is cached as "without slow-start". It is the default situation in our linux implementation. We also perform experiments with the caching manually disabled (i.e., "with slow-start"). Both situations may represent different real-life cases. Ns-2 simulations are performed with disabled caching.

For the experiments on a real network, the performance results are derived from the analysis of synchronized packetlevel captures at the input and at the output of the limiting buffer. For the ns-2 simulations, we use monitoring directly in the code. Error bars displayed correspond to the standard deviation on the estimation of the mean, estimated via bootstrap techniques on each experiment/simulation. Many are so small that they are not visible on the plots. Note however that for α close to one, the results across different experiments may have variability not included in these error bars, due to the natural difficulty to impose the mean of the flow-size distribution.

RESULTS

Fig. 1 presents the results obtained from the experiments on a real network with Nsrc = 45 sources. The top plots show the sensitivity of buffer-level performance metrics (loss rate, mean delay and throughput) to the tail index α. We observe that the impact of α not only depends on the buffer size (as shown in previous literature), but also on the presence/absence of the slow-start phase. Moreover, the impact varies from metric to metric.

More precisely, in the case "without slow-start", the performance in terms of loss rate and mean delay degrades with heavier tails, whereas the mean throughput increases with heavier tails which corresponds to a performance improvement. In the case "with slow-start", most results are inverted, except for the throughput which appears almost insensitive to the tail index for a small buffer (B = 96 pkts). In particular, in terms of loss rate and mean delay, a heavier tail turns out to give better performance. This non-intuitive result has not been reported in previous literature.

To interpret the results, the bottom plots of Fig. 1 show the mean rate achieved by a flow, as a function of its size (in loglog scale). In both cases, it is a strongly non-flat function, which stabilizes only for very large flow sizes. This is due to the effect of the transient behavior of each flow, which has an important impact on the mean rate even for large flows (see also [START_REF] Prasad | Router buffer sizing revisited: the role of the output/input capacity ratio[END_REF]). This transient behavior is longer and smoother "without slow-start"; it is faster and sharper "with slow-start" which explains the form of the bottom plots of Fig. 1. In the case "with slow-start", due to lower loss rate and mean delay, large flows also have a higher mean rate for smaller α's. Then, the performance results are mainly driven by an intricate combination of the shape of the mean flowrate curve and of the flow-size distribution; which imposes the load. This load impacts the loss rate and delay, which in turn impacts the mean flow-rate curve, in a closed-loop manner. It appears from this interpretation that the transient behavior of each flow is largely responsible for the impact of the flow-size distribution observed on performance. "without slow-start" "with slow-start" loss rate (%) necessary, the curves corresponding to the different B's have been split and the scales adapted to improve readability. Flow rates (bottom plot) are presented for B = 896 pkts.

To corroborate this interpretation, Fig. 2 presents the results obtained with simulations using Nsrc = 500 sources, in the case "with slow-start". In the case where the buffer is small (B = 100 pkts), the transient regime is very limited. Therefore the mean flow rate appears almost flat (bottomleft plot), and the buffer-level performance metrics are roughly insensitive to the tail index of the flow-size distribution (except for α = 1.1, due to the difficulty to impose the mean flow size for such a small α). However, with a larger buffer (B = 1, 000 pkts), the effect of the longer transient regimes becomes important (bottom-right plot, compare also to the right-side of Fig. 1 corresponding to the same case with less sources), and performance becomes sensitive to α. Interestingly though, the evolution is different from the case of Fig. 1-(right) with Nsrc = 45. As mentioned previously, we believe that it is the result of a complex combination between the mean rate as a function of the flow size, and the flow-size distribution. In particular, it appears that the most "aggressive" flows in terms of average rate are not the largest ones, but the ones of "medium size". This suggests that a key to fully understand the impact of the flow-size distribution in complex situations involving TCP connections with transient regimes is to look at the impact of the distribution as a whole, rather than the impact of its tail index only. We leave such a complete study as future work. loss rate (%)

CONCLUSION

In this work, we presented experimental and simulation results on the impact of the flow-size distribution's tail index on the performance of a single bottleneck with TCP connections. Our preliminary results show that the impact of heavy tails on performance is more complex than what described in previous literature: it can be negative, negligible, or even positive depending on many parameters such as buffer size, metric considered (loss rate, mean delay or throughput) and TCP tuning parameters (ssthresh).

We argue that a main element to understand these results lies in the transient behavior of each flow, which relates to the slow-start mechanism. Therefore, our work identifies the non-stationarity of a flow's transmission as an important feature (which is absent from all tractable models proposed up to now) to understand the performance of TCP connections.

Though our study shows a clear impact of the flow-size distribution's tail index on performance, a more complete study is required to confirm our interpretations and to find meaningful parameters to intuitively understand better the impact of this distribution. We believe that a key toward this goal is to consider the impact of the whole distribution rather than its tail index only.

Figure 1 :

 1 Figure 1: Performance metrics results as a function of α, from the experiments with Nsrc = 45 sources.necessary, the curves corresponding to the different B's have been split and the scales adapted to improve readability. Flow rates (bottom plot) are presented for B = 896 pkts.

Figure 2 :

 2 Figure 2: Performance metrics results as a function of α, from the simulations with Nsrc = 500 sources. The simulations are performed in the situation "with slow-start". Whenever necessary, the curves corresponding to the different B's have been split and the scales adapted to improve readability.

ACKNOWLEDGMENTS

We are grateful to the ALADDIN-G5K initiative whose engineers provided us with full support to perform experiments on the Grid'5000 infrastructure. We thank C. Dovrolis and R. Prasad for sharing some of their ns-2 code with us. Part of this work was done while the first two authors were with INRIA and Université de Lyon / ENS Lyon, France.


Bernard Ycart
email: bernard.ycart@imag.fr

Keywords: Leon Battista Alberti, history of statistics, statistical linguistics MSC 2010: 68T50, 01A40

Four centuries before modern statistical linguistics was born, Leon Battista Alberti (1404-1472) compared the frequency of vowels in Latin poems and orations, making the first quantified observation of a stylistic difference ever. Using a corpus of 20 Latin texts (over 5 million letters), Alberti's observations are statistically assessed. Letter counts prove that poets used significantly more a's, e's, and y's, whereas orators used more of the other vowels. The sample sizes needed to justify the assertions are studied, and proved to be within reach for Alberti's scholarship.

Introduction

August 18, 1851 is the date of birth generally agreed upon for statistical linguistics [Williams 1956, Bailey 1969]. On that day, Augustus de Morgan (1806-1871) sent a letter to a friend, suggesting statistical counts to settle authorship disputes [START_REF] De Morgan | Memoir of Augustus de Morgan[END_REF]. This later inspired the first statistical study of the kind by T. C. Mendenhall (1841Mendenhall ( -1924) ) [START_REF] Mendenhall | The characteristic curves of composition[END_REF]].

Four centuries before, Leon Battista Alberti (1404Alberti ( -1472)), had made a curious remark on the frequency of vowels among poets and orators. The first observation of the kind ever, it has remained unnoticed since. That remark, and its statistical verification, are the subject of this note. 1 Alberti, the well known polymath of Italian Renaissance, has been hailed as a pioneer in quite different domains [START_REF] Grafton | master builder of Italian Renaissance[END_REF]]: architecture, grammar, mathematics, etc. His "De componendis cifris", written in 1466 or 1467, stands out as the first western text in the history of cryptology [Kahn 1973].

The latin text is available online [Alberti 1466]; several translations have been published, among which we shall mainly quote [Alberti 2010]; we have also used an early Italian translation by C. [START_REF] Bartoli | Opuscoli morali di Leon Battista Alberti Gentil'huomo firentino[END_REF]], and M. Furno's French translation [Alberti 2000]. Our focus is on the following passage of section IV.

Sic enim adnotasse videor apud poetas vocales a consonantibus numero superari non amplius quam ex octava; apud rhetores vero non excedere consonantes ferme ex proportione quam sesquitertiam nuncupant. Nam si fuerint quidem connumeratae in unumque collectae omnes istius generis paginae vocales numero puta tricentarum, reliquarum omnium consonantium numerus una coadiunctus erit fere quadringentarum.

Here is K. Williams' translation [Alberti 2010, p. 173].

From my calculations, it turns out that in the case of poetry, the number of consonants exceeds the number of vowels by no more than an octave, while in the case of prose the consonants do not usually exceed the vowels by a ratio greater than a sesquialtera. If in fact we add up all the vowels on a page, let's say there are three hundred, the overall sum of the consonants will be four hundred.

The interpretation of Alberti's assertions in this passage, has been debated. Contrarily to note 3 in [Alberti 2010, p. 173], we believe that the correct translation of "ex octava" in the first sentence is indeed "one eighth", as in several other translations. For convenience, the proportion of vowels for poets will be denoted by P , being aware that what precisely is meant by "poets" and "vowels", is far from clear at this point. We believe that Alberti's assertion can be mathematically translated into (1 -P ) -P < 1/8, or else P > 7/16. Alberti opposes poets to "rhetores", i.e. orators, for which the proportion of vowels against consonants is said to be "sesquitertia", translated by [START_REF] Bartoli | Opuscoli morali di Leon Battista Alberti Gentil'huomo firentino[END_REF]] as "del terzo piu", or else four against three. If we denote by R the proportion of vowels for orators (with the same precautions as before), what is stated is (1 -R)/R < 4/3, or else R > 3/7. The difference between 7/16 = 43.75% and 3/7 = 42.86% is smaller than 1%: how did Alberti observe and justify such a small difference? The last sentence of the passage contains a hint. At first glance, saying that for 300 vowels, 400 consonants are counted, seems redundant, once the proportion has been set to 3:4. Yet "De componendis cifris" is a rather short and concentrated text, and no sentence is superfluous. Mentioning a page of 700 letters might have been Alberti's way of indicating that his observations were supported by counts on large enough sets of letters, i.e. the modern notion of sample size.

Admittedly, Alberti is far from being the first to have counted letters. Six centuries before, Al Kindī (ca. 801-873) had written a "Manuscript on deciphering cryptographic messages" [Al-Kadi 1992, Mrayati et al. 2002]. Remarkably similar in its organization and contents to Alberti's "De componendis cifris", Al Kindī's treatise contains a table of letter frequencies, much more detailed than Alberti's observations. It also contains quite accurate linguistic observations on poetry and prose. Similar counts and remarks can also be found in later Arab treatises, in particular that of ibn Dunaynīr (1187-1229) [Al-Kadi 1992, Mrayati et al. 2005]. Nevertheless, all Arab treatises, as well as all western writings after Alberti until the middle of the 19 th century, viewed letter frequencies as a characteristic of a language. Alberti was the first to assert that a stylistic difference (poetry vs. oration) in the same language, could yield a quantifiable difference in letter frequencies.

Our first objective was to investigate whether Alberti's observations could be justified, by modern statistical standards. We gathered a corpus of classical latin texts, from 10 different poets and as many different orators, totalling over 5 million letters. The letter frequency analysis encountered the difficulty of distinguishing the uses of u and i, as vowel or consonant. A counting algorithm was proposed, based on classical Latin grammar. Using that grammatical algorithm, the vowel counts did show a clear difference between poets and orators: poets use consistently more a's, e's and y's, whereas orators use more of the other vowels. However, the total percentages of vowels among poets (42.92%) and among orators (43.14%) were not significantly different, the percentage among orators being even slightly higher. Another counting algorithm did meet Alberti's observations: it consisted in counting as consonants all i 's before another vowel. Even though it cannot be asserted that our counting algorithm matches his, we believe that our results support Alberti's observations, and that he had indeed detected the quantitative difference between poets and orators.

Our second objective was to calculate how many pages of 700 letters would have been necessary to statistically assess Alberti's observations. We translated them into 3 statistical tests, and computed the number of pages necessary to raise the power of the tests to 95%, at threshold 5%. Detailed results will be reported for the 3 tests, using both the classical Bernoulli model, and a Monte-Carlo study on the corpus. One of our conclusions is that, randomly selecting 2 pages in each of the 20 texts of our corpus would suffice to ensure a power of 95% for the test of P = R against P > R.

Alberti is known as a sholar of immense culture, a lover of litterature, grammar and alphabets: see [Patota 1999] and in particular note 127 p. XLIII of [Alberti 2003]. He might well have counted letters in 40 pages of poems and orations: he could definitely have proved his assertions. . . had he known statistics. As he says in section III [Alberti 2010, p. 172]: I went about this with not indifferent industry or care, reflecting again and again on the elements of writing, and investigating intensely until I had clear in mind some fundamental concepts, which now the most brilliant minds will acknowledge as having contributed to understanding the whole question of ciphers.

Two sections follow; in the first one, our corpus of texts will be presented, and different ways of counting vowels will be discussed. The vowel frequency difference between poets and orators will be statistically assessed. The sample size problem is treated in section 3. Three tests will be defined, and their powers computed using both the Bernoulli model, and random pages extracted from our corpus. The free statistical software R was used for letter counts and simulation experiments ( [START_REF][END_REF][R , Gries 2009]. The corpus, the R script of functions, and a user manual have been made available online as a compressed file1 .

Corpus and vowel counts

Alberti's encyclopaedic knowledge makes it a hopeless task to guess from which texts he could have made his letter counts. We selected ten of the most famous classic Latin poets and orators, and copied their texts from "The Latin Library"2 . Each text was edited to remove non ascii characters, and also Roman numerals that could have biased letter counts. The texts were given a two letter code for further reference. Even though "De Bello Gallico" is not explicitly written as a speech, it can be argued that Caesar's style owed much to his expertise as a politician. Not being known as an orator, Vitruvius was included because his "De Architectura" was an important source of inspiration for Alberti [START_REF] Grafton | master builder of Italian Renaissance[END_REF]].

Statistical evidence showed that both texts behaved similarly to those of other orators regarding letter counts.

If vowels are counted on the basis of the occurrence of characters a, e, i, o, u, y, Alberti's assertions can hardly be justified: the vowel proportions are far from the announced values (45.76% for poets and 45.15% for orators instead of 43.75% and 42.86%). Moreover, the vowel percentage is not consistently higher for poets. However, it is a well known particularity of Latin grammar that the letters i and u can be used as vowels or as consonants. Alberti was perfectly aware of that; here are some quotations from sections IV, VI and VIII of [Alberti 2010].

After the vowel O is sometimes found the I and also the U, but this is rather rare, while more frequently the O is followed by the U used as a consonant, as in 'ovem'.

[. . . ] In fact, neither I or V used as consonants ever follow the vowels in a syllable, and neither does Q.

[. . . ] In monosyllabic words the vowels can be followed by all the consonants except F, G, P, Q, I, and V.

[. . . ] In contrast, within a word, there are some consonants that never appear in the next syllable after V and the consonant I.

The usage in rendering vocalic [i], [u] and consonantal [j],

[w] into letters has varied, even for classical Latin. In our corpus, the letter i (lower or upper case) always denoted either sounds [i] and [j]. Lower case v 's were rather rare, and upper case U 's never appeared. In order to define what might be called a "grammatical count", we have used both Alberti's own remarks, together with the rules that can be found in most Latin grammars, such as [START_REF] Tafel | Latin pronunciation and the Latin alphabet[END_REF][START_REF] Blair | Latin pronunciation[END_REF], Allen 1978]. Here are the counting rules that were implemented.

• I or i was counted as a consonant:

at the beginning of a word before a vowel, -after prefixes ad, ab, conj, ex, and before another vowel, -between two vowels.

• U or u was counted as a consonant:

at the beginning of a word before a vowel, -after q and g, and before a vowel, -between two vowels,

• V or v was counted as a vowel before a consonant.

Our choice to count qu as two consonants instead of one is disputable: see [Tafel 1860, p. 20], [Blair 1874, p. 44], [Allen 1978, p. 16]. We believe it is supported by the following remark in section VI, [Alberti 2010, p. 175]:

On the other hand, when it was established that the U is combined with the Q, it doesn't appear to have been taken into account that the U itself is implicit in this letter Q, so it sounds like KU.

Many exceptions to the above rules are recorded in grammars; for instance, i often has the consonantal sound [j] after a consonant, before a vowel other than i. No implementation of a count including exceptions could be attempted. Vowel percentages in our grammatical count are reported in Table 1. The difference between poets and orators is clear: poets use more a's, e's, and y's and less of the other vowels. In order to assess statistical significance, Student's T-test was applied to paired lines of Table 1: for each letter, the 10 proportions among poets were compared to the 10 proportions among orators, and the one-sided p-value was returned.

• percentage of A: larger for poets, p-value = 4.12 × 10 -7 ,

• percentage of E : larger for poets, p-value = 2.10 × 10 -3 ,

• percentage of I : smaller for poets, p-value = 1.98 × 10 -9 ,

• percentage of O: smaller for poets, p-value = 3.01 × 10 -2 ,

• percentage of U : smaller for poets, p-value = 1.95 × 10 -2 ,

• percentage of Y : larger for poets, p-value = 6.51 × 10 -5 . A 8.75 8.02 8.11 8.68 8.39 8.28 8.05 8.28 9.08 8.29 8.28 E 11.50 11.79 11.56 11.43 11.64 11.37 11.58 11.27 11.32 11.72 11.51 I 10.85 10.99 11.31 10.20 11.23 11.05 11.43 11.77 11.45 10.95 11.22 O 5.35 5.72 5.68 5.36 5.05 5.27 5.44 5.59 5.30 5.77 5.52 U 6.63 6.51 6.54 6.77 6.87 6.74 6.30 6.11 6.59 6.96 6.51 Y 0.07 0.00 0.02 0.05 0.04 0.02 0.01 0.10 0.02 0.02 0.04 43.21 43.17 43.34 42.52 43.30 42.82 42.86 43.18 43.88 43.68 43.14 Table 1: Percentage of vowels in the texts of our corpus, counting i 's, u's, and v 's according to grammatical rules.

Poets

Figure 1 shows a two-dimensional representation of the same data, with percentages of A,E,Y on the x-axis, percentages of I,U,O on the y-axis. The difference between poets and orators is quite visible. On the same graphic, the two lines x+y = 7/16 (red) and x+y = 3/7 (blue) correspond to Alberti's assertions. Orators are indeed above the red line on average, but poets are below the blue line. No clear difference on vowel counts can be seen, and the global percentage of vowels is even slightly smaller for orators than for poets: the grammatical count does not match Alberti's observations. 45.15 44.28 43.62 43.86 43.82 44.35 44.37 45.20 44.63 45.32 44.41 Orators AM CG CP HS LP PP QD SC SO TO All 43.43 43.15 43.49 43.69 42.79 43.38 42.97 42.79 44.15 43.24 43.19 Table 2: Percentage of vowels in the texts of our corpus, counting i as consonant before a,e,o,u.

The difference between Alberti's and grammatical counts obviously comes from the letter i a vowel, which remains too frequent among orators. Another count was proposed, that consisted in counting as consonant all i 's before a,e,o,u, counting all u's as vowels and v 's as consonants. Table 2 shows the total vowel counts, and Figure 2 the graphical representation, with the same conventions as Figure 1. This time, Alberti's observations are verified: poets are above the blue line, and orators above the red one, on average. The following comparisons are statistically significant, using again Student's T-test, and denoting by P and R the proportions of vowels among poets and orators respectively.

• P > 7/16: p-value = 1.16 × 10 -3

• R > 3/7: p-value = 2.24 × 10 -3

• P > R: p-value = 1.79 × 10 -5
Of course, verifying Alberti's conclusions with our second counting algorithm, does not prove that his counts would have matched ours. Alberti's way of counting will always remain a mystery. Yet we believe that our results support his observations, and that he had indeed detected the quantitative difference between poets and orators.

Sample sizes

Our focus in this section is on the last sentence of Alberti's assertions, about counting vowels in a page of 700 letters: could that sample size be sufficient to support Alberti's statistical conclusions? If not, how many pages should be counted? We shall first answer the question theoretically on the classical Bernoulli model, then show that the minimal sample sizes are somewhat lower using random pages from the actual corpus. All statistical techniques used here are quite classical: see [Gries 2009] as a general reference.

We rely upon the observations made on our corpus using the second counting algorithm, that matches Alberti's observations (Table 2 and Figure 2). Denoting by P and R the respective proportions of vowels among poets and among orators, we consider the following tests, were H 0 and H 1 denote as usual the null hypothesis and the alternative.

[T1 ] H 0 : P = 7/16 against H 1 : P > 7/16,

[T2 ] H 0 : R = 3/7 against H 1 : R > 3/7,

[T3 ] H 0 : P = R against H 1 : P > R.

The threshold of all three tests is fixed at 5%. Throughout this section, we call "page" a set of 700 consecutive letters extracted from a text in our corpus.

A random page is made of 700 hundred consecutive letters starting at some letter whose rank is chosen uniformly between 1 and N -699, if there are N letters in the text. We are interested in the number of pages, needed to raise the probability of rejecting H 0 , i.e. the power of the tests, above 95%.

Assume first a Bernoulli model for the alternance of vowels and consonants.

The model states that each letter is a vowel with probability p, or a consonant with probability 1 -p, independently of the others. The probability distribution of the number of vowels in a sample of size n is binomial with parameters n and p. The determination of n such that the power of the test is higher than 95% is a standard calculation. Using the data of Table 2, the minimal values of n were computed. Dividing them by 700, the following values are found, for the number of pages necessary to raise the power of the test above 95%.

[T1 ] at least 88 pages,

[T2 ] at least 343 pages,

[T3 ] at least 52 pages.

The Bernoulli model is quite unrealistic: vowels and consonants do not alternate independently in a text. A consequence is that the actual standard deviation of letter counts on a random page is smaller than the theoretical standard deviations from the Bernoulli model. To illustrate this, we have extracted 10000 random pages of each text in our corpus, counted vowels on each page, and returned the mean and standard deviations of the 10000 proportions. Table 3 shows the results, for the 20 texts: estimated standard deviations are about 50% of what they would be, if vowel occurrences were independent. The 10000 proportions counted on each text are represented as boxplots on Figure 3. Alberti's observations are indeed verified on average. Yet, in spite of the rather small standard deviation, the vowel counts on one given page may have important variations: counting one single page of poetry and one of orations at random, might easily lead to the conclusion that Alberti was wrong.

The following experimental setting was chosen. Selecting k random pages from each text of our corpus, the proportion of vowels was computed for each page. The 3 tests T1, T2, T3, were then applied to the 10 × k proportions among poets and the 10 × k proportions among orators, and the 3 p-values were returned. This was repeated 10000 times. The number of pages in each text needed to raise the power above 95% is k = 3 for T1 (P > 7/16), k = 6 for T2 (R > 3/7), and k = 2 for T3 (P > R).

As expected, these values are smaller than those predicted by the Bernoulli model.

Figure 1 :

 1 Figure1: Vowel proportions in grammatical counts. Each text is represented by its code, color blue for poets, red for orators. The abscissa of a text is the cumulated percentage of a's, e's, and y's the ordinate is the cumulated percentage of the other vowels. The results on the global corpuses of poets and orators are represented by a blue and a red cross. Poets use significantly more a,e,y, and less of the other vowels. The two lines correspond to Alberti's assertion: x + y = 7/16 (in blue for poets), x + y = 3/7 (in red for orators).

Figure 2 :

 2 Figure 2: Vowel proportions, counting i as consonant before a,e,o,u, with the same conventions as in Figure 1. Alberti's assertions are verified: poets are above the blue line on average, orators above the red line.

Table 3 :

 3 For each test, the proportion Estimated standard deviations for the proportions of vowels in 10000 random pages of each text in our corpus. Under the Bernoulli model, these standard deviations should be close to 0.44(1 -0.44)/700 = 0.0188. Boxplots for the proportions of vowels in 10000 random pages of each text. The horizontal lines correspond to Aberti's observations: 7/16 for poets, 3/7 for orators. of the 10000 p-values below 5%, estimates the power of the test. Table4reports the estimated powers for the 3 tests, with k = 1, . . . , 6 random pages extracted from each text. The first values of k such that the estimated power is greater than 95% are:

	Poets

Table 4 :

 4 Estimated powers (in percents) of tests T1, T2, T3 with k random pages extracted from each of the 20 texts.

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