Mohammed Hafsati

Nicolas Epain

Rémi Gribonval

Nancy Bertin

SOUND SOURCE SEPARATION IN THE HIGHER ORDER AMBISONICS DOMAIN

In this article we investigate how the local Gaussian model (LGM) can be applied to separate sound sources in the higherorder ambisonics (HOA) domain. First, we show that in the HOA domain, the mathematical formalism of the local Gaussian model remains the same as in the microphone domain. Second, using an off-the shelf source separation toolbox (FASST) based on the local Gaussian model, we validate the efficiency of the approach in the HOA domain by comparing the performance of toolbox in the HOA domain with its performance in the microphone domain.

To do this we discuss and run some simulations to ensure a fair comparison. Third, we check the efficiency of the local Gaussian model compared to other available source separation techniques in the HOA domain. Simulation results show that separating sources in the HOA domain results in a 1 to 12 dB increase in signal-todistortion ratio, compared to the microphone domain.

Multichannel source separation, local Gaussian model, Wiener filtering, 3D audio, Higher Order Ambisonics (HOA).

INTRODUCTION

There is an increasing interest in new, immersive forms of media, such as 360-degree videos and Virtual Reality (VR) experiences. Producing media experiences of this kind requires new techniques and workflows on both the video and audio sides. In particular, the feeling of immersion that is sought after by VR spectators highly depends on the quality of the audio rendering. In order for the experience to be convincing, sounds must be binauralized according to the spectator's location and orientation relative to the different sound sources.

Among the various 3D-audio technologies available, Higher-Order Ambisonics (HOA) [START_REF] Michael A Gerzon | Periphony: With-height sound reproduction[END_REF][START_REF] Michael A Gerzon | Ambisonics in multichannel broadcasting and video[END_REF][START_REF] Daniel | Ambisonics encoding of other audio formats for multiple listening conditions[END_REF][START_REF] Daniel | Further investigations of High-Order Ambisonics and wavefield synthesis for holophonic sound imaging[END_REF] has become the de facto standard for 360-degree video soundtracks. This is primarily because HOA provides a panoramic representation of the sound field, which can easily be rotated in accordance with the listener's head orientation prior to being played back. Another significant advantage of the HOA representation is that it is straightforward to record HOA sound scenes using relatively compact Spherical Microphone Arrays (SMAs). In contrast to 360-degree videos, however, VR experiences not only allow the spectator to look in any direction, but also to navigate through the virtual environment. This means that the spectator's perspective of the scene may change over time in a manner that cannot be modeled as a simple rotation effect. For instance, the spectator can move toward one of the sound sources, which should translate in this source being louder compared to other sources.

One possible approach to simulate a movement through the scene is to interpolate between several HOA representations corresponding to different points of view [START_REF] Southern | Rendering walk-through auralisations using wave-based acoustical models[END_REF][START_REF] Joseph | Soundfield navigation using an array of Higher-Order Ambisonics microphones[END_REF]. However, in practice this requires to use several SMAs, which may be impossible. Another possibility consists in decomposing one HOA scene into directional components, which typically correspond to sound sources, and changing their directions and gains according to the listener's movements [START_REF] Kronlachner | Spatial transformations for the enhancement of ambisonic recordings[END_REF][START_REF] Allen | Ambisonic soundfield navigation using directional decomposition and path distance estimation[END_REF][START_REF] Hafsati | Editing ambisonic sound scenes[END_REF]. In addition to being more practical than the interpolation method, this approach was shown to yield a better listening experience [START_REF] Allen | Ambisonic soundfield navigation using directional decomposition and path distance estimation[END_REF]. The quality of the navigation effect obtained with the decomposition approach ultimately depends on the accuracy of the sound source separation. In previous work [START_REF] Hafsati | Editing ambisonic sound scenes[END_REF], we showed that sound sources could be separated using a simple beamforming technique, which relies solely on spatial information. In the presence of complex sound scenes, however, the quality of the separation could be improved using multi-channel source separation algorithms, such as those based on the so-called local Gaussian model [START_REF] Duong | Under-determined reverberant audio source separation using a full-rank spatial covariance model[END_REF][START_REF] Ozerov | Multichannel Nonnegative Matrix Factorization in convolutive mixtures for audio source separation[END_REF][START_REF] Vincent | The 2008 Signal separation evaluation campaign: A community-based approach to large-scale evaluation[END_REF].

In this article, we investigate the ability of the local Gaussian model to handle the informed source separation problem (knowing directions of arrival) directly in the HOA domain. In Section. 2, we recall the model in the microphone domain and derive its equivalent in the HOA domain. This allows us to perform experiments with an off-the-shelf source separation toolbox, the Flexible Audio Source Separation Toolbox (FASST) [START_REF] Salaün | The Flexible Audio Source Separation Toolbox Version 2.0[END_REF], presented in Section 3. Experimental results are presented and discussed in Section 5; First, on a small dataset, we investigate the performance of FASST with respect to of the number of channels in the HOA domain and the number of microphones in the microphone domain, and select a number of channels/microphones yielding a fair comparison between the HOA domain and microphone domain. Second, on a large scale dataset, we measure the performance of FASST in both domains, and compare them to each other. Third, we compare the performance of FASST with different types of beamformers. We conclude in Section 6 on the validity of the local Gaussian model and its competitive performance when applied to HOA signals, including in challenging situations such as highly reverberating environments or close source positions.

MIXTURE MODELS

Microphone domain

The source separation problem consists in estimating the contribution cj,t P R I of each source j " 1, ..., J in each microphone i " 1, ..., I and at each time instant t " 1, ..., T . In the absence of noise, the mixture can be written as:

xt " J ÿ j"1 cj,t , (1)
where xt " rx1,t, . . . , xI,ts T P R I are microphone array signals. In a reverberant environment, under the hypothesis of point sources, the source signal sj,t can be related to its contribution cj,t through:

cj,t " rαij ˚sjst , (2)
where ˚denotes the convolution product, αij is the impulse response of the mixing filter between the source j, and the microphone i. Now, under the narrow-band approximation, and assuming the mixing filters are time invariant, the Short-Time Fourier Transform (STFT) of the microphone signals is given by:

x f,n " J ÿ j"1 A j,f s j,f n , (3)
where f and n denote the frequency bin and time-frame index, respectively. A f P C IˆJ contains the frequency responses a ij,f of the filters aijptq, and embeds information on the source directions of arrival (DOA). The source separation problem as defined in Eq.( 1) can be addressed using the multichannel Wiener filtering framework, which will be presented with more details in Sec. 3. This framework requires the selection of a distribution model for the variables to estimate. For simplicity we use the local Gaussian model presented in [START_REF] Vincent | Underdetermined instantaneous audio source separation via Local Gaussian Modeling[END_REF]: @f P r1, F s , n P r1, N s , c j,f,n " Ncp0, Σc j,f,n q , (4) where Σc j,f,n " E " c j,f,n c H j,f,n ‰ is the covariance matrix of the contribution of the j-th source to every microphone at frequency f and time frame n. In line with the literature, this matrix can be further decomposed as the product of a scalar spectral part, v j,f,n , with a time-invariant spatial matrix, Rc j,f , as follows: Σc j,f,n " v j,f,n Rc j,f . Notably, the so-called spatial covariance matrix Rc j,f respects the relation Rc j,f " A j,f A H j,f when the assumptions of Eq. (3) hold.

HOA domain

In the Higher-Order Ambisonic (HOA) framework, the sound field is decomposed over a basis of spherical harmonic functions. The HOA signals, zt, are typically obtained by applying a set of finite impulse response filters, known as encoding filters, to the signals recorded by a spherical microphone array [START_REF] Nicol | Sound spatialization by Higher Order Ambisonics: Encoding and decoding a sound scene in practice from a theoretical point of view[END_REF]. Thus, assuming the encoding filters are short enough, the vector of the HOA signal STFTs z f,n P C M is given by:

z f,n " E f x f,n , (5)
where E f is the matrix of the encoding filter frequency responses. Using Eq. ( 1), we can now model the HOA mixture as follows:

z f,n " J ÿ j"1 E f c j,f,n , (6)
and identify the contribution of the j-th source to the different HOA channels as:

b j,f,n " E f c j,f,n . (7)
As is the case in the microphone domain, in the ambisonic domain source separation consists in estimating the contribution of every source to every channel b j,f,n , which can be solved using a Wiener filtering approach. To this aim we assume the following local Gaussian model:

b j,f,n " Ncp0, Σ b j,f,n q. ( 8
)
Similar to the microphone domain, the covariance Σ b j,f,n can be further decomposed into a spectral part, v j,f,n , and a spatial covariance matrix given by:

R b j,f " E f Rc j,f E H f . ( 9
)

SOURCE SEPARATION WITH FASST

The multi-channel source separation problem can be solved by looking for the filter that minimizes the expected squared error for every source j and every time frequency bin pf, nq:

@j P r1, Js, f P r1, F s and n P r1, N s, W j,f,n " argmin W E " }c j,f,n ´Wj,f,n x f,n } 2 2 ‰ . (10)
The filter W j,f,n is known as the multichannel Wiener filter and is given by:

W j,f,n " Σ pc j,f,n , x f,n q Σ ´1 px f,n , x f,n q , ( 11
)
where the matrices Σ px f,n , x f,n q and Σ pc j,f,n , x f,n q , represent the covariance of the mixture x f,n and the cross-correlation between the vectors c j,f,n and x f,n , respectively. From Eq. ( 4), and assuming the sources are statistically independent, the Wiener filter can be simplified as:

W j,f,n " Σc j,f,n ˜J ÿ j 1 "1 Σc j 1 ,f,n ¸´1 . ( 12
)
Thus, the source separation problem reduces to the problem of estimating the covariance matrices Σc j,f,n or, equivalently in the HOA domain, Σ b j,f,n . Each source contribution is obtained by applying element-wise its corresponding Wiener filter to the mixture:

c j,f n " W j,f n x f n , respectively, b j,f,n " W j,f,n z j,f
,n in the HOA domain. and finally using overlap-add to reconstruct the time-domain signal.

In this work we use the flexible audio source separation toolbox (FASST) [START_REF] Salaün | The Flexible Audio Source Separation Toolbox Version 2.0[END_REF][START_REF] Ozerov | A general flexible framework for the handling of prior information in audio source separation[END_REF], a software toolbox which allows the estimation of these parameters and apply the subsequent Wiener filter. In FASST the parameters are estimated by maximizing the loglikelihood of the observations with an Expectation-Maximization (EM) algorithm, and a multi-channel non negative matrix factorization (NMF) model can be enforced on the source covariances Σc j,f,n [START_REF] Ozerov | Multichannel Nonnegative Matrix Factorization in convolutive mixtures for audio source separation[END_REF].

EXPERIMENTAL PROTOCOL

Dataset

In order to evaluate the source separation performance, we built a dataset as follows. First, fifty songs were picked randomly from the Mixing Secret Dataset (MSD100) 1 . In the MSD100 database, each song consists of four sound sources (voice, bass, drums and "others") provided as separate tracks.

In this work, microphone array recordings were then simulated using MCRoomSim [START_REF] Wabnitz | Room acoustics simulation for multichannel microphone arrays[END_REF], a room acoustics simulation software. This software calculates impulse responses modeling the acoustic propagation between acoustic sources and sensors in reverberant environments. A total of 16 simulations were run, corresponding to four rooms and four source configurations. The eight rooms had the same dimensions, 10 m ˆ8 m ˆ3 m, but different wall absorption coefficients, which resulted in the following reverberation times: 0 s, 0.2 s, 0.4 s and 0.7 s. The four source configurations are illustrated in Fig. 1. In every simulation the microphone array was modeled to match the characteristics of the Eigenmike 2 and was located at the same position in the room. In order to calculate the microphone mixtures, for each song and each of the 32 conditions the separate source tracks were then convolved with the simulated impulse responses and summed with each other.

We then built two different inputs for source separation: a microphone mixture x obtained using every sensor of the Eigenmike (32 channels), and a fourth order ambisonic mixture b (25 channels) obtained by applying encoding filters to the microphone mixture x.

90°1 0°1 0°9 0°9 0°A ) B) D) C) 10°F
ig. 1: The four sound source configurations considered in our simulations. Note: stars represent sound source locations. All the sources and the spherical microphone array are at the same height. Across all the examples, the position of sources with similar type was the same.

Evaluation criteria

In order to validate the adaptability of FASST in the HOA domain, we propose to compare its performance to that obtained by applying FASST in the microphone domain. A fair comparison requires to compute the chosen performance measures in the same domain.

However, some information is lost when converting microphone signals to HOA signals, therefore it is impossible to convert the separated HOA signals back to the microphone domain for comparison. To alleviate this issue, instead of computing the evaluation measures in terms of the contribution of each source in each channel/microphone (FASST's outputs), we propose to compute them in terms of sound objects.

We obtain sound objects by applying beamforming to the signals separated by FASST. For simplicity, we use the beamforming technique known as the matched filter for both the microphone and HOA domains. For the j-th sound source, the sound object is calculated by projecting the separated signals onto the steering vector corresponding to a plane wave incoming from the source direction,pθj, φjq. In other words, the estimated source object j in the microphone and HOA domains are calculated as follows:

ŝMic j,f,n " a H j,f ||a j,f || 2 ĉj,f,n (13)
ŝHOA j,f,n " y H j,f ||y j,f || 2 bj,f,n (14)
where yj corresponds to the spherical harmonic vector evaluated at the direction of arrival of the source j. We then compare the estimated sound object signals to reference signals, which we define as the sound objects obtained by applying the same beamforming to the actual (i.e. oracle) sound source contributions. In other words for the source j the reference signal in the microphone and HOA domains are given by:

s Mic j,f,n " a H j,f ||a j,f || 2 c j,f,n (15)
s HOA j,f,n " y H j,f ||y j,f || 2 b j,f,n (16)
Lastly, we assess the source separation performance by comparing the signals given by Eq. ( 13), and Eq. ( 14) to the ones given by Eq. ( 15) and Eq. ( 16), respectively, using the following performance measures [START_REF] Vincent | Performance measurement in blind audio source separation[END_REF]: Signal to Distortion Ratio (SDR), Signal to Artifact Ratio (SAR), and Signal to Interference Ratio (SIR). These measures are then calculated with the BSS-eval toolbox3 [START_REF] Févotte | BSS_EVAL toolbox user guide-revision 2[END_REF].

Evaluated methods

The first method we examine consists in applying FASST to the HOA mixtures (see Sec. 3.) We compare it to the equivalent microphonedomain mixtures with the same number of channels. We also compare the performance obtained by using FASST in the HOA domain with that of two different beamformers.

The first beamformer has already been introduced in Section. 4.2. It is the matched filter beamformer(PWD), but this time applied directly to the HOA mixture, which is given by:

sHOA j,f,n " y H j,f ||y j,f || 2 z f,n . (17)
The second beam former, which we refer as the pseudo-inverse beamfomer, consists in multiplying the HOA signals with the pseudoinverse of the matrix containing the steering vectors for the directions of the sources. The resulting beamformer is a particular case of the Linearly-Constrained Minimum-Variance beamformer (LCMV). It is given by:

sHOA j,f,n " pY H Yq ´1Y H z f,n , (18)
where the matrix Y contains the spherical harmonic vectors of the sources direction of arrivals.

FASST parametrization and initialization

The FASST toolbox requires choosing configuration parameters, as well as providing initial values for the covariance matrices Σc j,f,n in Eq. ( 12). Tab. 1 summarizes the parameters used for all experiments. Further, in order to match the scene configuration, the number of sources was fixed to 4 in the anechoic condition and 5 in reverberant conditions, where we observed that it was beneficial to add a source accounting for diffuse noise or late reverberation.

Transform type STFT

Sampling frequency 44100 Hz

Window length 69 ms (3072 samples)

NMF rank 16

Stopping criterion 150 iterations

Table 1: FASST parameters

In FASST, covariances are decomposed into a spectral part and a spatial part, and the spectral part further modeled by NMF, as proposed by [START_REF] Ozerov | A general flexible framework for the handling of prior information in audio source separation[END_REF]. For each of the first 4 sources, the spatial covariance was initialized to the rank-1 matrices R HOA f " yjy H j and R Mic f " a j,f a H j,f for the HOA and microphone domain, respectively. In the microphone domain, the steering vectors a j,f were derived from the microphone array characteristics and source locations. In the HOA domain, the steering vectors yj were derived as the vector of the first nine spherical harmonic functions evaluated in the source directions. In the reverberant case, the fifth source was assumed to have a full-rank spatial covariance, which was was initialized to the identity matrix. Lastly, regarding the spectral part of the covariance, NMF factors were initialized as random numbers.

VALIDATION OF THE APPROACH

As explained before the main goal is to validate experimentally the local Gaussian model assumption for source separation in the HOA domain. To this aim the performances obtained using FASST in the HOA domain are compared to that obtained in the microphone domain. However, we first need to select a number of microphone channels and HOA signals that ensures a fair comparison between the two methods. Thus, we first investigate the influence of the number of microphone channels and HOA signals on the source separation performance for a fraction of the dataset. The results of this study, presented in Sec. 5.1, indicate that it is fair to compare the results obtained with 9 HOA signals (order 2 HOA signals) with that obtained using 9 microphone channels. In Sec. 5.2 we present further experimental results obtained using the selected number of channels for the entire dataset.

Selection of the number of microphones/channels

The computational cost of FASST depends primarily on the square of the number of channels of the mixture, and considering the size of our dataset, it is important to spare time and resources in the main experiment that will soon be described. A naive approach would be to adopt a lower HOA order L ă 4, and consider on the one hand HOA mixtures with M " pL `1q 2 channels, and on the other one, the same mixtures given by a sub-antenna of the Eigenmike, where the number of the chosen capsules is I " pL `1q 2 .

However, one could argue that while HOA mixtures are obtained by considering all of the capsules of the Eigenmike, the microphone mixtures are given by only M " pL `1q 2 selected microphones, and therefore, the comparison could be considered unfair. To clarify this point, we begin our experiments by measuring the source separation performance in both domains when varying respectively the number of channels and the number of microphones. This preliminary experiment is done on a small proportion of the created dataset (see below).

First, in the microphone domain we considered different sub antennas from the Eigenmike where the capsules were selected in order to be distributed regularly on the sphere. The considered numbers of microphones are I " 4, 9, 12, 16, 25, 32 (the numbers 4, 9, 16, 25 were chosen to match the number of possible channels in the HOA domain, the number 12 is considered because the chosen capsules can be regularly distributed in the best way to cover the sphere). Second, HOA signals make sense if they are grouped by order L, each order L corresponding to a number of channels M " pL `1q 2 . We have already at our disposal the 4th order signals (25 channels) by encoding the information provided by the 32 capsules of the Eigenmike. In order to have the first, the second, and the third order we have to simply truncate respectively the 25 HOA signals to the first M " 4, 9, 16 channels. Considering the selected capsules in the microphone domain and the truncation of the signals in the HOA domain, from our data set we considered randomly 160 mixtures, all the listed time reverberations and sound source configurations were considered.

We applied FASST to the different mixtures, considering the sources DOA known, the initialization and the parametrization of the toolbox are given in Sec. 4.4. The results in terms of SDR, SIR and SAR are given in Fig. 2.

In the microphone domain, we observe that the SIR tends to improve by 0.07 dB in average when increasing the number of microphones, the SAR tends to decrease, when it comes to the SDR we observe that it increases slightly by 0.02 dB in average until 9 microphones and drops after. In the HOA domain, we observe an improvement of all performance measures when increasing the number of channels. We can clearly see that adding more microphones doesn't improve the source separation performance in the microphone domain. As a conclusion it is unnecessary to add more microphones in the microphone domain, and therefore the comparison of FASST's performance between the HOA domain and the microphone domain is fair if the number of channels/microphones is equal to I " M " 9.

Extensive experiments with 9 microphones/channels

In the following the considered number of channels is equal to the considered number of microphones I " M " 9. In the microphone domain the selected capsules are given in Table . 2. More Table 2: Elevation (θ) and azimuth (φ), in degrees, of the selected Eigenmike microphone capsules. The radius of the microphone is 4 cm. The origin of space is the center of the Eigenmike.

In the following, we consider the whole dataset described in (Sec. 4.1). The results of the comparison are given in Fig. 3. As expected performance decrease as reverberation and scene complexity increase, regardless of the signal domain. However, in most configurations, separating the sources in the HOA domain resulted in better performance measures compared to the microphone domain. We can clearly see a gain of 7 to 12 dB for the least challenging sound source configuration, and a gain of 1 to 6 dB for the most challenging one. Tab. 3 summarizes the difference in SDR values between the HOA domain and the microphone domain for configurations (A) and (D): the SDR is almost always higher in the HOA domain, regardless of the reverberation or song. As well, the gap between the performance obtained in the two domains reduces as the complexity of the scenario increases, with a more prominent influence of reverberation time. Separating sources in the HOA domain results in a 1 to 12 dB increase in signal-to-distortion ratio, compared to the microphone domain. One reason may explain these results. Indeed, in FASST's EM algorithm, the empirical covariance matrix is inverted while estimating the first Wiener filter [START_REF] Salaün | The Flexible Audio Source Separation Toolbox Version 2.0[END_REF] and the numerical stability of this inversion differ in the two signal domains. We calculated the condition number of the empirical covariance matrix in both domains for a random example picked from the dataset. It appeared that, for frequencies below 2 kHz, the condition number was generally higher in the microphone domain than in the HOA domain, and could be about 1000 times greater for some frequency values. Therefore, the conversion of the microphone signals into HOA signals seems to act as a pre-conditioning for the EM algorithm.

Having established the interest of performing the source separation in the HOA domain with FASST, we now compare it with the reference methods. Results are presented in Fig. 4. FASST clearly outperforms the reference methods. This is because, contrary to the reference methods which are solely based on spatial cues, FASST also exploits spectral cues. This gives FASST an advantage when sources are close to each other and spatial information is more ambiguous. Although this fact has already been observed in microphone domain source separation [START_REF] Vincent | Contributions To Audio Source Separation And Content Description[END_REF], we confirm it here also on HOA-domain source separation.

Surprisingly, FASST outperforms the PIV method even in anechoic environment where the PIV method could have been expected to give the best results in terms of performance. Indeed, 9 signals should be enough to form a beam toward one source and cancel 3 interfering sources at the same time. This can be explained with the fact that encoded HOA signals don't match perfectly the theoretical signals. This imperfection is mainly caused by the physical limitations of the microphone array. Indeed, the capsules of the Eigenmike are relatively close to each other, which results in spatial aliasing and a loss of lower frequencies [START_REF] Moreau | 3D sound field recording with higher order ambisonics-Objective measurements and validation of a 4th order spherical microphone[END_REF].

CONCLUSION

In this paper we investigated for the first time the ability of the local Gaussian model to handle the source separation problem in the HOA domain. To this aim we have established the model's equations in the HOA domain and run numerical experiments. Our simulation results show that applying a local Gaussian model-based source separation method in the HOA domain typically results in the SDR increasing by 1 to 12 dB, compared to the microphone domain with the same number of microphones/channels I " M " 9, including in challenging situations such as reverberant environments and complex source configurations. In future work we will explore using proximity microphones in order to guide the source separation and improve its performance, and finally employ this method to allow navigation through HOA sound scenes.

Fig. 2 :

 2 Fig. 2: Comparing FASST performance in regards of number the used microphones/channels.

Fig. 3 :Fig. 4 :

 34 Fig. 3: Comparing FASST's performance in the HOA domain to FASST's performance in the microphone domain, I " M " 9

Table 3 :

 3 ∆SDR " SDRHOA ´SDRMIC , in dB, for scenarios A and D.

		RT60psq 0	0.2	0.4	0.7
		max	21	14.35 11.9 12
	A	median	12.43 7.69	7	6.84
		min	4.3	2.52	2.5	2.9
		max	10.17 6.6	6.7	6.32
	D	median	6.05	0.83	1.52 2.45
		min	-1.84 -6	-5	-4

https://sisec.inria.fr/sisec-2015/

2015-professionally-produced-music-recordings/ 2 https://mhacoustics.com/products

BSS-eval version 3.0 for Matlab, http://bass-db.gforge. inria.fr/bss_eval/


Patrick Maillé
email: patrick.maille@telecom-bretagne.eu

Bruno Tuffin
email: btuffin@irisa.fr

Analysis of Price Competition in a Slotted Resource Allocation Game

Pricing telecommunication networks has become a highly regarded topic during the last decade, in order to cope with congestion by controlling demand, or to yield proper incentives for a fair sharing of resources. On the other hand, another important factor has to be brought in: there is a rise of competition between service providers in telecommunication networks such as for instance the Internet, and the impact of this competition has to be carefully analyzed. The present paper pertains to this recent stream of works. We consider a slotted resource allocation game with several providers, each of them having a fixed capacity during each time slot, and a fixed access price. Each provider serves its demand up to its capacity, demand in excess being dropped. Total user demand is therefore split among providers according to Wardrop's principle, depending on price and loss probability. Using the characterization of the resulting equilibrium, we prove, under mild conditions, the existence and uniqueness of a Nash equilibrium in the pricing game between providers. We also show that, remarkably, this equilibrium actually corresponds to the socially optimal situation obtained when both users and providers cooperate to maximize the sum of all utilities, this even if providers have the opportunity to artificially reduce their capacity.

I. INTRODUCTION

The Internet, and more generally telecommunication networks, have progressively switched from an academic or monopolistic network to a commercial one with competitive service providers. In order to get a return on investment, each provider has to define a pricing strategy to charge users for the service they experience. Pricing has at the beginning been seen as a way to cope with congestion, to control demand, to deal with and satisfy heterogeneous applications with different quality of service (QoS). It has also been regarded as a way to introduce fairness among users with respect to the traditional flat-rate pricing where light consumers pay as much as big ones. Therefore, there have been many proposals for new pricing schemes motivated by different objectives: the network planner may want to elicit users to efficiently share the scarce network resources in order to maximize social welfare (see, among others, [START_REF] Key | User policies in a network implementing congestion pricing[END_REF]- [START_REF] Yaïche | A game theoretic framework for bandwidth allocation and pricing in broadband networks[END_REF]), to guarantee fairness among users [START_REF] Kelly | Rate control in communication networks: shadow prices, proportional fairness and stability[END_REF], [START_REF] La | Window-Based Control with Heterogeneous Users[END_REF], or to maximize revenue [START_REF] Bas | Revenue-maximizing pricing and capacity expansion in a many-users regime[END_REF]- [START_REF] Paschalidis | Congestion-Dependent Pricing of Network Services[END_REF]; the typical modeling tool being that of noncooperative game theory [START_REF] Osborne | A Course on Game Theory[END_REF]. For surveys on pricing in telecommunication networks, the reader is advised to look at [START_REF] Courcoubetis | Pricing Communication Networks-Economics, Technology and Modelling[END_REF]- [START_REF] Tuffin | Charging the Internet without bandwidth reservation: an overview and bibliography of mathematical approaches[END_REF].

A very large proportion of papers deal with the monopolistic case, where there is only one provider. Though, telecommunication networks have become highly competitive and it seems primordial to us to deal with that competition in pricing models when defining the optimal prices, since competition may highly affect the results of price determination (while pricing in a monopolistic context generally means a single level of game between users, competition actually introduces an additional level of game, between providers, resulting in a so-called Stackelberg game [START_REF] Osborne | A Course on Game Theory[END_REF]). Some typical illustrations of competition are:

• for wired access, DSL users can choose among several competing providers to connect to the Internet; • the case of wireless access is more flexible. For example a user wishing to connect to a WiFi hotspot may be located in a zone covered by several wireless access providers, and can choose which provider to use for the time of his connection.

• The same user can/will even be able to choose between different and competitive transmission platforms: WiFi, WiMAX, 3G, Wired operators, with a possible combination of all those ones (the so-called multihoming).

Our goal in this paper is to introduce a pricing model dealing with competition. In our model, time is discretized, divided into slots, and each provider has the capability to serve a given number of packets per slot. We assume that providers do not share a common limited amount of capacity/bandwidth, but instead each provider has its own service capacity: it can model for instance competition between 3G, WiFi and WiMAX providers for instance. Note that it does not correspond to competition at a WiFi hotspot if providers share the same bandwidth, but rather to the case where providers are being operated on different frequency channels and using different PHY modes. In our model, as soon as demand exceeds capacity at a provider, the excess demand is lost (packets are selected randomly). We consider a pricing scheme inspired by the (monopolist) one introduced in [START_REF] Marbach | Analysis of a static pricing scheme for priority services[END_REF], where users are charged for the number of packets they submit regardless of their being treated or lost, in order to incentivize them to limit their demand. We show existence and uniqueness of a Wardrop equilibrium for demand at each provider for every combination of (fixed) prices, such that each user chooses the provider with the least perceived cost, where perceived cost is expressed in terms of charge and drop probability. Remark that this kind of modeling behavior is also of interest in the case of multihoming, when users are able to split their traffic between providers in order to keep the lowest overall cost. From this Wardrop equilibrium, we show, under mild conditions, that there is a unique Nash equilibrium among competing providers for the game consisting of setting prices, and we characterize that equilibrium. Those prices represent a point where no provider can increase its benefits by changing unilaterally its price. An important property is that the resulting prices correspond to the socially optimal situation, where the sum of utilities of all agents in the system -including users and providers-is maximized. This is a very desirable property, in favor of the application of such pricing strategies, since usually noncooperation leads to a loss of efficiency, quantified by the so-called price of anarchy. Another issue that we address is the interest for a provider to declare or use only a part of its real capacity. Indeed, it may happen that, due to congestion, serving less users, therefore at a higher market price, results in a larger revenue. We show here that in this context of competition, it cannot occur if demand is sufficiently elastic.

A. Related work on competition

Studying the impact of provider competition on pricing schemes is a quite new topic which is receiving increasing attention in the networking community. The importance of this field has been highlighted in [START_REF] Gibbens | Internet service classes under competition[END_REF], where it was shown that the very promising Paris Metro Pricing (PMP) scheme, which just consists in separating the network into disjoint networks served in the same manner but with different access prices, does not allow service differentiation under competition at equilibrium. In [START_REF] El-Azouzi | Telecommunications network equilibrium with price and quality-of-service characteristics[END_REF], a model where competitive providers play both on price and on a QoS parameter is used, and demand at each provider is driven by an arbitrary function depending on the parameters of all competitors. We feel that using a unique total demand function that is split between providers thanks to Wardrop's principle, i.e. all users choosing the provider with cheapest perceived cost, is more relevant. [START_REF] Liu | Pricing information services in a competitive market: avoiding price wars[END_REF] studies competition for e-services, with also a kind of Wardrop equilibrium, but where QoS does not depend on demand. In other specific contexts, competition between wireless operators in the case of a shared spectrum (more flexible and leading to a more efficient management of the spectrum) has been studied in [START_REF] Ileri | Demand responsive pricing and competitive spectrum allocation via a spectrum server[END_REF], [START_REF] Felegyhazi | Wireless operators in a shared spectrum[END_REF]. In [START_REF] Ileri | Demand responsive pricing and competitive spectrum allocation via a spectrum server[END_REF], operators are charged by a central entity for the amount of bandwidth they use, and therefore try to design proper service offers for users. Competition is shown to increase users' acceptance probability for offered service. Our analysis is based on a less specific network modeling (and without competition for capacity between providers). In [START_REF] Felegyhazi | Wireless operators in a shared spectrum[END_REF], the authors discuss a similar competition problem, but operators only play with the power of the pilot signals of their base stations, and no pricing is considered. On the other hand, note that there is an increasing bunch of works looking at independent and selfish providers on a path, that forward traffic of competitors to ensure end-to-end delivery [START_REF] He | Pricing internet services with multiple providers[END_REF]- [START_REF] Shakkottai | Economics of network pricing with multiple ISPs[END_REF], but do not consider a direct competition for users between providers, a different perspective that we adopt.

The closest works to ours are that in [START_REF] Acemoglu | Competition and efficiency in congested markets[END_REF], [START_REF] Hayrapetyan | A network pricing game for selfish traffic[END_REF], where providers are represented by parallel links and the quality of service is delay. The price of anarchy, measuring the loss of efficiency due to competition with respect to cooperation, is determined, for fixed demand in [START_REF] Acemoglu | Competition and efficiency in congested markets[END_REF] and random demand but linear delay in [START_REF] Hayrapetyan | A network pricing game for selfish traffic[END_REF]. This is extended to the case of parallel-serial links in [START_REF] Acemoglu | Price competition in communication networks[END_REF], for fixed demand. Those models consider that users are sensitive to the sum of the price charged and a congestion-dependent delay, and providers are only sensitive to their revenue. Our model is different since we assume that the total cost perceived by a user is the price charged multiplied by a congestion factor, the externality being losses here (which can be considered closer to some wireless environments). Moreover we consider that providers experience managing costs that increase with demand, those costs not being perceived by users.

Remark that the type of slotted and capacity-based model we are dealing with can be related to the one in [START_REF] Marbach | Analysis of a static pricing scheme for priority services[END_REF], where a similar slotted capacity model is used, but with several priority traffic classes, and in the case of a monopoly instead of an oligopoly here. We therefore have a different goal: study price war between providers instead of price discrimination among users.

B. Organization

The paper is organized as follows. In Section II, we present our basic model and the required assumptions. Section III describes the socially optimal allocation when all providers cooperate, this allocation being used later on to investigate the price of anarchy for non collaborating providers. Section IV investigates the demand repartition among providers for fixed prices, following Wardrop's principle. Section V is devoted to the Nash equilibrium for the pricing game between providers; the quite long proof of the main result (existence ands uniqueness) is deferred to the appendix. Finally, in Section VI, we consider the interest for providers to artificially decrease their capacity, and provide our conclusions and directions for future research in Section VII.

II. GENERAL MODEL

Consider a model where time is discretized, divided into slots. Assume that there is a set I := {1, . . . , I} of providers in competition at an access point (I ≥ 2), provider i (i ∈ I) having the capacity of serving C i packets (or units, seen as a continuous number) per slot and asking a price p i per packet. If demand exceeds capacity at a given provider, demand in excess is lost. We assume that lost packets are chosen uniformly over the set of submitted ones. If d i is the total demand at provider i, the number of served packets is actually min(d i , C i ), meaning that packets are actually served with probability min(C i /d i , 1).

Users are assumed to be charged for each submitted packet instead of each served one in order to incentivize them to limit demand. The average perceived price per served traffic unit at provider i is therefore

p = p i / min(C i /d i , 1) = p i max(d i /C i , 1).
Indeed, the total income of provider i is d i p i and the total service "rate" is d i min(C i /d i , 1), giving the unit perceived price just above by a direct ratio. In that situation, it can be seen that the negative externality of congestion is expressed in terms of losses experienced by users.

We assume that total user demand is a function D(•) of the perceived price p, and that D is continuous and strictly decreasing with p on its support [0, p max ), with eventually p max = +∞. We moreover assume that D(0) > P i∈I C i , i.e. there is some congestion: the total resource available is not sufficient to satisfy the maximum demand level. Remark that the demand function can be interpreted in two ways (or as a combination of those two effects):

• it can stem from user heterogeneity in their willingnessto-pay for the service: consider a continuum of infinitesimally small users with fixed demand, whose willingnessto-pay (in terms of unit price) is distributed according to a given distribution. Then for a given unit price p ≥ 0, D(p) is the amount of flow generated by users with valuation larger than p and is naturally decreasing. • Likewise, the decreasingness of the total demand function D can also stem from the decreasingness of individual functions. We can also define the function v : q 7 → inf{p : D(p) ≤ q} (with the convention inf ∅ = 0). Since D is continuous, strictly decreasing for p < p max and null for p ≥ p max for a given p max then we simply have

v(q) = ⎧ ⎨ ⎩ D -1 (q) if q ∈ (0, D(0)) p max if q = 0 0 if q ≥ D(0). (1)
The quantity v(q) is then the unit price that one has to impose on users in order to ensure that total demand will be q (for q ≤ D(0)). The function v is called marginal valuation function: v(q) is indeed the maximum unit price that can be charged for the q th unit of demand without making the demand decrease: it is a nonincreasing function since D is nonincreasing. We finally define V (q) as the sum of the marginal valuations of the q units of users with largest willingness-to-pay, i.e.

V (q) := Z q x=0 v(x)dx.
As an example, if all users experience a fixed unit price p, then only the q = D(p) units with valuation larger that p will subscribe to the service, and the sum of their marginal valuations will be V (q). We refer to this function V as the valuation function: V (q) corresponds to the total price that the q units of demand with highest marginal valuation are willing to pay to be served (remark that the function V is nondecreasing and concave). For a fixed unit price p, the user surplus then equals V (D(p)) -pD(p).

A first step of our work will be to study how, for fixed prices p i (i ∈ I), total demand is split among providers. This is described and characterized in Section IV in terms of a Wardrop equilibrium.

Knowing a priori the distribution d := (d 1 , . . . , d I ) of demand that will result from a price profile, the goal of each provider is, by playing on its unit price p i , to maximize its net benefit

R i (p 1 , . . . p I ) := p i d i -i (d i ),
where p i d i is the money earned directly from demand, and i (d i ) represents the cost for provider i of managing a demand level d i . We assume that for all i, i is nondecreasing, differentiable and convex. Notice that contrary to the model considered in [START_REF] Acemoglu | Competition and efficiency in congested markets[END_REF], this cost is not reflected here in the quality of service experienced by users, but is only perceived by providers.

The price chosen by a provider has an impact on demands, and therefore on benefits, of other providers; consequently our model induces a game between providers. Since we consider competitive providers, the framework is that of noncooperative game theory. We are going to investigate in Section V, under mild conditions, the existence and uniqueness of a Nash equilibrium for the price game, that is a price vector p := (p 1 , . . . , p I ) such that no provider can increase its own benefit by unilaterally changing its access price, i.e., ∀i ∈ I, ∀p ≥ 0,

R i (p 1 , . . . , p I ) ≥ R i (p 1 , . . . , p i-1 , p, p i+1 , . . . , p I ).
Most of our results are valid under the following assumption.

Assumption A: The marginal cost of every provider when its demand equals its capacity is lower than the global marginal valuation of the sum of all provider capacities. In other terms, ∀i ∈ I,

0 i (C i ) ≤ v ⎛ ⎝ X j∈I C j ⎞ ⎠ , or equivalently ∀i ∈ I, D( 0 i (C i )) ≥ X j∈I C j ,
where 0 i is the derivative function of i . We will sometimes need a slightly more restrictive assumption.

Assumption B: For each provider i ∈ I, the following inequality holds:

0 i (C i ) ≤ Ã 1 - C i P j6 =i C j ! v ⎛ ⎝ X j∈I C j ⎞ ⎠ . (2)
With respect to Assumption A, Assumption B is more restrictive especially if a provider has a significant "power" in terms of capacity, i.e. if Ci P j6 =i Cj is not negligible. In the extreme case of a large number of providers with the same capacity C i = C, Assumption B simply resumes to Assumption A. In general, it is reasonable to consider that the managing costs are small, if not negligible with respect to the price users are willing to pay for the service. Therefore we expect Assumption B to hold in networks where competition occurs.

III. SOCIALLY OPTIMAL SITUATION

In this section, we define social welfare as the sum of the utilities of all agents in the system (i.e. users and providers), and study the maximal value of that criterion. It is well-known in Game Theory [START_REF] Fudenberg | Game Theory[END_REF] that agent selfishness does not lead in general to a socially efficient situation. The loss of efficiency due to the divergence of user interests, often referred to as the Price of Anarchy [START_REF] Koutsoupias | Worst-case equilibria[END_REF], is an interesting performance measure for a game: if social welfare at an equilibrium of the game is close to its maximal value, then letting agents choose their actions selfishly can be preferable to introducing costly control or incentive schemes.

Definition 1: For a demand configuration d := (d 1 , ..., d I ), we call social welfare the quantity

SW := V Ã X i∈I min(d i , C i ) ! - X i∈I i (d i ). (3)
Social welfare SW accounts for the utilities of users (the quantity P i∈I min(d i , C i ) is the total effective user rate, and the first term of ( 3) is total user valuation) and of providers (the second term accounts for the costs associated to the demand configuration). Remark that no monetary exchanges appear in (3): this is due to the fact that such exchanges would be added to the providers utility and subtracted from the users utility, which would not affect the sum of all agent utilities.

We compute here the maximal value of social welfare, that will be used as a reference in the next sections.

Proposition 1: Under assumption A, the maximum value of social welfare is reached when d i = C i for each provider i.

Proof: The social welfare maximization problem is expressed by

max d∈R I + V Ã X i∈I min(d i , C i ) ! - X i∈I i (d i ). (4)
Notice that since i is strictly increasing for all i, the objective function is strictly decreasing in d i for all i ∈ I when d i ≥ C i , therefore our optimization problem is equivalent to

max d∈R I + V Ã X i∈I d i ! - X i∈I i (d i ) subject to d i ≤ C i ∀i ∈ I.
Since V is a concave function and the provider cost functions ( i ) i∈I are convex, this last problem is a classical convex problem (maximization of a concave function over a convex set), that can be solved by the Lagrangian method. Denoting by λ i ≥ 0 the Lagrange multiplier relative to the constraint d i ≤ C i , the first order conditions imply that for a demand configuration (d * 1 , . . . , d * I ) maximizing social welfare, we have ∀i ∈ I,

λ i + 0 i (d * i ) = p * , (5)
with

p * := v ¡P i∈I d * i ¢ (i.e. P i∈I d * i = D(p * )).
Then the complementary slackness conditions yield

∀i ∈ I, min(λ i , C i -d * i ) = 0. (6)
Let us define I 0 := {i ∈ I :

d * i = C i } and I u := {i ∈ I : d * i < C i },
and assume that I u 6 = ∅. Then for an i 0 ∈ I u , (6) implies that λ i0 = 0, and we have

X i∈I C i > X i∈I u d * i + X i∈I 0 C i = X i∈I d * i = D(p * ) = D( 0 i0 (d * i0 )) ≥ D( 0 i0 (C i0 )) ≥ X i∈I C i ,
where the second line comes from ( 5), the third one from the convexity of i0 and the nonincreasingness of D, and the last line stems from Assumption A. We reach a contradiction, which means that I u = ∅, and establishes the proposition.

IV. WARDROP EQUILIBRIUM FOR USERS

In this section, we investigate how demand is split among providers when the price p i per unit of sent traffic is fixed by each provider i ∈ I. We assume that users are infinitely small and therefore their choices do not individually affect the demand levels (and therefore the perceived costs) of the different providers: such users are said to be price-takers since they individually have no influence on prices. The outcome resulting from such user interactions is described by Wardrop's principle [START_REF] Wardrop | Some theoretical aspects of road traffic research[END_REF]: demand is distributed in such a way that all users choose one of the cheapest providers. As a result, the user perceived price is the same for all providers with positive demand, and is lower than the unit price p i of providers i with no demand. Since we considered the case of elastic demand, the total demand level must also correspond to the common perceived price on all providers that receive some demand. Those properties are summarized in the next definition.

Definition 2: For given capacity C := (C 1 , . . . , C I ) and price p = (p 1 , . . . , p I ) configurations, a user equilibrium is a demand configuration d = (d 1 , . . . , d I ) such that for all i, j ∈ I,

d i > 0 ⇒ p i max(1, d i /C i ) ≤ p j max(1, d j /C j ), ( 7
)
d i > 0 ⇒ p i max(1, d i /C i ) = v à X k∈I d k ! . ( 8
)
According to Condition [START_REF] Bas | Revenue-maximizing pricing and capacity expansion in a many-users regime[END_REF], all providers with positive demand have the same perceived unit price, otherwise part of the demand will have interest in changing providers. Condition [START_REF] Campos-Náñez | On-line tuning of prices for network services[END_REF] states that total demand corresponds to that common value of the perceived unit price via the demand function D.

Remark that this definition corresponds to the one provided in [START_REF] Hayrapetyan | A network pricing game for selfish traffic[END_REF], but for a different pricing scheme and different cost functions. The following proposition characterizes the user equilibria corresponding to fixed capacities and prices.

Proposition 2: For any capacity and price configuration, there exist a (possibly not unique) user equilibrium demand configuration.

Moreover, at a user equilibrium d, the common perceived unit price p of providers i with d i > 0 is unique and equals

p = inf{p : D(p) ≤ X i∈I f i (p)}, (9)
where

f i (p) := 1l {p≥p i } C i p i p. ( 10
)
Proof: Since the perceived unit price functions for each provider are continuous and nondecreasing, for any value of the total demand d there exist a demand configuration d such that P i d i = d and condition ( 7) is satisfied. Moreover the common value p c (d) of the perceived unit price at providers with strictly positive demand is unique when total demand is fixed (this was established in [START_REF] Beckmann | Studies in the economics of transportation[END_REF]) when perceived unit prices are continuous and nondecreasing functions of demand. It is easy to see that p c (d) is a continuous and nondecreasing function of total demand d.

The second property, given by condition [START_REF] Campos-Náñez | On-line tuning of prices for network services[END_REF], that a user equilibrium must satisfy can be expressed as p c (d) = v(d). Since v is a continuous nonincreasing function and p c is continuous, nondecreasing and tends to infinity as d becomes large (p c (0) = min i p i and for d sufficiently large p c is a strictly increasing linear function of d), then

• if v(0) ≥ min i p i then there exists a d such that p c (d) = v(d).
The corresponding value p := v(d) is unique due to the respective monotonicity properties of p c and v. We now prove that p verifies (9). Since we assumed that v(0) = p max ≥ min i p i , we necessarily have d = D(p) from (1). Moreover [START_REF] Bas | Revenue-maximizing pricing and capacity expansion in a many-users regime[END_REF] implies that d i = f i (p), so D(p) = P i∈I f i (p). Relation (9) then comes from p ≤ p max and the strict decreasingness of D on (0, p max ).

• if v(0) < min i p i then there is no intersection point between p c and v, since no user is interested in subscribing to the service at a unit price min i p i . Therefore the configuration d = (0, . . . , 0) is the unique user equilibrium.

Figure 1 displays the demand function D and the function P i∈I f i for a given price configuration, and illustrates the existence and uniqueness of the Wardrop equilibrium perceived price p. Function f i basically represents the corresponding share of demand 1 that provider i can get at a given perceived price p.

Remark 1: From this proposition, we are able to characterize the unique perceived price p. Total demand is therefore D(p). For all providers with price p i 6 = p, demand d i is then d i = f i (p). All providers such that p i = p (if any) share the remaining demand D(p) -P i:pi<p d i , all possible sharing providing a Wardrop equilibrium. In this sense, there 1 f i is indeed the generalized inverse of the perceived unit price function

d i 7 → p i max(1, d i /C i ), i.e. f i (p) = sup{d i : p i max(1, d i /C i ) ≤ p}. D(p) p 1 p 2 p 4 d 1 d 2 d 3 C 3 C 4 C 2 C 1 p 3 = unit price p p quantities ∑ i f i (p) 0 Fig. 1.
Wardrop equilibrium for three providers and a given price configuration: the common perceived price at each provider with positive demand (i.e. providers 1, 2, 3) is p = p3.

is not always uniqueness for the Wardrop equilibrium and the corresponding revenues for each provider are not necessarily unique. Note nonetheless that the resulting total revenue is always the same. Moreover, we will see in the following that when providers are at a Nash equilibrium of the pricing game, then the corresponding user Wardrop equilibrium is unique.

V. PRICE COMPETITION AMONG PROVIDERS

In this paper, we consider that providers setting their prices is the first stage of a two-level game, where the second stage corresponds to users reacting according to the Wardrop equilibrium described in Definition 2. We assume that providers are aware of their advantage of playing first, i.e. they take into account users' reaction when determining their price. This common knowledge complicates the competition among providers, and is the purpose of the analysis in this section.

Before analysing the pricing game between providers, we first prove a lemma that establishes some monotonicity results of Wardrop equilibria with respect to the price configuration p. In the following, we often compare two situations from the point of view of one provider to study the consequences of its price decisions. We therefore use the superscript " n " to refer to the values (price, demand, benefit) corresponding to a new situation in contrast to the reference situation.

Lemma 1: Consider a price configuration p = (p 1 , . . . , p I ), and i ∈ I. If provider i raises its price, i.e. chooses p n i > p i while all other providers j 6 = i keep their price to p j , then

• the common perceived price (for providers with positive demand) increases: p n ≥ p, • if d i > 0 then the demand of provider i strictly decreases: [START_REF] Paschalidis | Congestion-Dependent Pricing of Network Services[END_REF] we have the functional inequality f n i ≤ f i . Therefore, [START_REF] Hayel | Less-than-Best-Effort Services: Pricing and Scheduling[END_REF] implies that for

d n i < d i . Proof: Since p n i > p i then from
all ε, 0 < ε ≤ p, D(p-ε) > D(p) ≥ X j∈I f j (p) ≥ X j∈I f j (p-ε) ≥ X j∈I f n j (p-ε),
which yields p n ≥ p. Now assume that d i > 0 and that d n i ≥ d i . The equality p = p i max(d i /C i , 1) implies that p n > p. Therefore the demand on all links j 6 = i should strictly increase to reach the same perceived price p n . This would mean that total demand increases, which is in contradiction with p n > p and the nonincreasingness of total demand D.

We can now establish our main result. Proposition 3: Assume that ∀i ∈ I, i is strictly increasing and convex. Under Assumption B, there exists a unique Nash equilibrium on price war among providers, given by ∀i ∈ I,

½ d i = C i p i = p * , where p * = v ³ P j∈I C j ´, that is X i∈I C i = D(p * ). ( 11
)
The proof is quite technical and is left to the appendix. The proposition basically states that the only equilibrium is such that demand matches capacity for all providers, all of them setting the same unit price.

Remark 2: Proposition 3 establishes that the demand configuration at the user Wardrop equilibrium corresponding to the competitive Nash equilibrium of the provider pricing game is exactly the demand pointed out in Proposition 1. In other terms, letting providers setting selfishly their prices and users choosing selfishly their provider yields the same social welfare as a perfect coordination mechanism would have given. This suggest that when congestion and competition are sufficient (in order for Assumption B to hold), no regulation schemes are needed since the market itself determines the right prices and allocations.

Remark 3: If Assumption B is not verified, for example if one provider i has a strong power in the sense that C i > P j6 =i C j , then Proposition 3 does not hold. In that case it is possible that the dominating provider take benefit of its position to improve its revenue by setting a high price p i > p * .

VI. GAME ON DECLARED CAPACITIES

In this section, we assume that a provider i ∈ I may voluntarily declare a false value C n i ≤ C i of its capacity C i . Only the declared values C n i ≤ C i are feasible: whereas provider i can easily degrade artificially its service rate, it cannot increase it above its real capacity C i , and a false declaration aimed at increasing one's demand to get a larger benefit would consequently be detected.

From the point of view of a provider's net benefit, there are two opposite effects of lowering one's capacity: on the one hand the unit selling price at equilibrium increases and the managing cost decreases because the quantity sold decreases, whereas on the other hand less quantity sold means less revenue. The effect that overcomes the other depends on the elasticity of demand, i.e. the extent to which total demand is affected by variations of unit price. Recall that when the demand function D is differentiable, the elasticity of demand at a unit selling price p is defined by

pD 0 (p) D(p)
.

Remark that for all p, the elasticity at p is a negative number due to the nonincreasingness of D.

Since the demand is the inverse function of the marginal valuation v, a small demand elasticity (in absolute value) means that v decreases quickly with the unit price. In such a case, a small decrease of total demand corresponds to a large price increase, and the positive effect on revenue of underdeclaring one's capacity exceeds the negative one. The next proposition gives a sufficient condition on demand elasticity for that situation not to occur: if demand is sufficiently elastic (i.e. absolute value of elasticity larger than 1), then providers have interest in truthfully declaring their capacity.

Proposition 4: Under Assumption B, if the absolute value of demand elasticity is larger than 1 for p ≥ p * , i.e.

∀p ≥ p * , -pD 0 (p) D(p) ≥ 1, (12)
then no provider can increase its revenue by artificially lowering its capacity.

Proof: Without loss of generality, it is sufficient to prove that the net benefit of provider 1 always decreases when it underdeclares its capacity. For any value C n 1 ≤ C 1 of the declared capacity of provider 1, Assumption B still holds with declared capacities, and the equilibrium of the price competition game is therefore given by Proposition 3. If we define C -1 := P i6 =1 C i , the unit price p * n at the price competition equilibrium is then

p * n = v(C n 1 + C -1 ), (13)
each provider i 6 = 1 gets demand C i , and provider 1 obtains demand C n 1 and gets total benefit

R n 1 = C n 1 p * n -1 (C n 1 )
. Notice that p * n ≥ p * due to the nonincreasingness of the marginal valuation function v.

From ( 13), the total demand should therefore verify

C n 1 + C -1 = D(p * n ) = D(p * ) + Z p * n p * D 0 (p)dp ≤ D(p * ) - Z p * n p * D(p) p dp ≤ D(p * ) -C -1 Z p * n p * 1 p dp,
where the second line comes from [START_REF] Courcoubetis | Pricing Communication Networks-Economics, Technology and Modelling[END_REF], and the third one from the nonincreasingness of total demand and (13) which imply that D(p) ≥ C -1 for all p ∈ [p * , p * n ].

We then obtain

C n 1 ≤ C 1 -C -1 log(p * n /p * ) := d1 . (14)
From the convexity of 1 and Assumption A (implied by Assumption B), we have

R n 1 ≤ p * n d1 -1 ( d1 ).
By truthfully declaring C 1 , provider 1 would get a net benefit

R 1 = p * C 1 -1 (C 1 )
. The gain of underdeclaring one's capacity can thus be upperbounded:

R n 1 -R 1 ≤ p * n d1 -p * C 1 + 1 (C 1 ) -1 ( d1 ) ≤ p * n d1 -p * C 1 + (1 -C 1 /C -1 )p * (C 1 -d1 ) = C 1 (p * n -p * ) + log(p * n /p * ) (p * (C -1 -C 1 ) -p * n C -1 ) .
where the second line stems from the convexity of 1 and Assumption B, and the third one from [START_REF] Tuffin | Charging the Internet without bandwidth reservation: an overview and bibliography of mathematical approaches[END_REF]. Define x := p * n /p * ≥ 1. We can thus write

R n 1 -R 1 ≤ p * g(x), (15)
where g :

x 7 → p * C 1 (x-1)+p * log(x) ((C -1 -C 1 ) -xC -1 ).
It is then straightforward to check that

• g is concave,

• g(1) = 0,

• g is derivable and g 0 (1) = 0. Those three points imply that g(x) ≤ 0 for all x ≥ 1, which from [START_REF] Marbach | Analysis of a static pricing scheme for priority services[END_REF] means that R n 1 ≤ R 1 and gives the proposition.

VII. CONCLUSION

In this paper, we have analyzed a pricing game among service providers with fixed capacities, such that each provider has an access price it can play with. We have characterized how demand will be naturally split between those providers, following Wardrop's principle, and determined the existence and uniqueness of the Nash equilibrium of the pricing game. We have shown that this equilibrium corresponds to the socially optimal point (meaning that the price of anarchy is 1), and discussed the interest for providers to voluntarily reduce their capacity.

There are different ways to extend the results we have obtained. First of all, we could investigate the case where providers share, at least partially, their capacities: does it lead to a price war? Another interesting issue concerns the capacity expansion game. Indeed, capacity could also be an important parameter providers can play with, at the same time as prices: what would be the resulting equilibrium? Finally, it would be of interest to extend the game to a multiclass system with different priority levels at each provider similarly to what was done in [START_REF] Marbach | Analysis of a static pricing scheme for priority services[END_REF] for the monopolistic game.

ACKNOWLEDGMENT

The authors would like to thank New-Zealand ISAT program for igniting this research activity, and acknowledge the support of Euro-FGI Network of Excellence through the specific research project CAP (Competition Among Providers), as well as WINEM ANR project for the second author.

APPENDIX

We prove here Proposition 3.

Proof: The proof can be decomposed into two steps:

1) We first show that the point such that d i = C i and p i = p * ∀i, with p * = v ¡P i∈I C i ¢ defines a Nash equilibrium; 2) then we prove that no other point can be a Nash equilibrium.

Step 1: p i = p * , ∀i is a Nash equilibrium. Note first that in order to have R i ≥ 0, i.e., p i d ii (d i ) ≥ 0 at the equilibrium point, means that pC ii (C i ) ≥ 0, i.e. here

i (C i ) C i ≤ v ⎛ ⎝ X j∈I C j ⎞ ⎠ .
This last inequality indeed holds under Assumption A in the realistic case when i (0) = 0, due to the convexity of i for all i.

We begin by establishing that the price configuration with p i = p * for all i where p * is given by ( 11) is indeed a Nash equilibrium of the pricing game. First notice that in this situation, the common perceived price for users is p * , and the demand of each provider is d i = C i . We need to prove that in such a price configuration, no provider can improve its revenue by unilaterally changing its suggested price. Without loss of generality, consider a possible move of provider 1 from p * to p n 1 6 = p * . We distinguish two cases.

• If p n 1 < p * then applying Lemma 1 to a change from p * to p n 1 we get that -the new perceived price p n at the Wardrop equilibrium is lower than the original perceived price: p n ≤ p * , because of a smaller function f 1 while the others remain unchanged. -the new allocation d n 1 at provider 1 is strictly positive, therefore from Lemma 1 it is strictly above the original one: [START_REF] Campos-Náñez | On-line tuning of prices for network services[END_REF]. Consequently, the revenue gain of provider 1 for lowering its price is

d n 1 > C 1 , which implies that d n 1 = C 1 p n /p n 1 from (
R n 1 -R 1 = p n 1 d n 1 -p * C 1 + 1 (C 1 ) -1 (d n 1 ) = (p n -p * ) | {z } <0 C 1 + 1 (C 1 ) -1 (d n 1 ) | {z } <0 < 0. • If p n 1 > p * then
d n 1 = D(p n 1 ) -p n 1 X j6 =1 C j /p * ≤ D(p * ) -p n 1 X j6 =1 C j /p * , (16)
where the first line second line comes from [START_REF] Osborne | A Course on Game Theory[END_REF] and the nonincreasingness of the demand function. This is illustrated in Figure 3. In that case, the revenue gain for provider 1 is

R n 1 -R 1 = p n 1 d n 1 -p * C 1 + 1 (C 1 ) -1 (d n 1 ) ≤ p n 1 d n 1 -p * C 1 + 0 1 (C 1 )(C 1 -d n 1 ) ≤ d n 1 Ã p n 1 -p * Ã 1 - C 1 P j6 =1 C j !! | {z } >0 -p * C 2 1 P j6 =1 C j ≤ - P j6 =1 C j p * (p n 1 -p * ) 2 < 0,
where the second and third line respectively stem from the convexity of 1 and Assumption B, and the fourth line is obtained after upper-bounding d n 1 by the expression in [START_REF] Gibbens | Internet service classes under competition[END_REF] and performing some simplifications.

Step 2: uniqueness of the Nash equilibrium. Now knowing that there exists a Nash equilibrium under Assumption B, we establish that this equilibrium is unique. Indeed, the proof of the uniqueness only needs Assumption A to hold, as we see below.

To prove uniqueness, consider a Nash equilibrium of the pricing game, i.e. a price configuration, and decompose the set of providers I into three disjoint subsets:

I = I s ∪ I 0 ∪ I u , p quantities ∑ i C i unit price ∑ i =1 C i d n 1 D(p) ∑ i f i (p) p * p = p n 1 0 Fig. 3. Wardrop equilibrium if player 1 switches from p * to p n 1 < inf{p : D(p) ≤ p P j6 =1 C j /p * }.
where

I s := {i ∈ I : d i > C i }, (17)
I 0 := {i ∈ I : d i = C i }, (18)
I u := {i ∈ I : d i < C i }. (19)
It is sufficient to show that I s and I u are empty sets, since only the price configuration p = (p, ..., p) can lead to the demand configuration with d i = C i for all i.

We first prove that I s = ∅. Assume it is not the case, and consider i s ∈ I s . We study the influence of provider i s increasing its unit price p is to p n is = p is + ε for ε > 0, all other providers keeping their price unchanged. From the continuity of the perceived unit price at Wardrop equilibrium in terms of the price configuration, there exists ε > 0 such that p n > p n is , which implies that d n is = C i s p n /p n is from the relation between demand and perceived price. For this ε, the net benefit of provider i is R n is = p n C isis (d n is ). Its gain in net benefit with respect to the initial situation is therefore

R n is -R is = C is (p n -p) + ¡ is (d is ) -is (d n is ) ¢ > 0,
where the strict positivity comes from Lemma 1 and the strict increasingness of is . We have established that any provider in the set I s can strictly increase its net benefit by increasing its price, which implies that at a Nash equilibrium,

I s = ∅. ( 20
)
Remark that I s = ∅ implies that d i ≤ C i for all i ∈ I, thus d = D(p) ≤ P i C i and consequently p ≥ v( P i∈I C i ). Now we assume that I u 6 = ∅, and prove that at least a provider i u ∈ I u can strictly increase its net benefit by decreasing its price. We still consider a Nash equilibrium price configuration, and denote by p the user perceived price for that price configuration. The total demand should therefore be D(p). Since we previously proved that I s = ∅, then D(p) <

When there are at least two providers, at a Nash equilibrium with I u 6 = ∅ then for all i we have d i ≤ min(C i , D(p)), since I s = ∅ and total demand equals D(p). Moreover, there necessarily exists a provider i u for which d i < D(p): indeed there exists at least a provider in I u , and if this provider gets all the demand then it implies that all the other providers have demand 0. Thus consider i u such that d iu < min(C iu , D(p)).

(

) 22
We now prove that provider i u can strictly improve its benefit by changing its price from p iu ≥ p to p ε iu := p -ε for a sufficiently small ε > 0. We distinguish two cases.

• If C iu ≤ D(p), then we easily see from ( 9) that the new perceived price p ε verifies

p ε iu = p -ε < p ε ≤ p.
By changing its price to p -ε, provider i u is the only provider with the lowest declared unit price, therefore from Definition 2 its demand d ε iu equals C iu p ε p-ε , which tends to C iu when ε tends to 0. where the first and second lines come from the convexity of iu and y ≤ C iu , the third one from Assumption A, and the last line stems from [START_REF] He | Pricing internet services with multiple providers[END_REF]. Consequently, provider i u can strictly improve its net benefit by unilaterally changing its declared price, which contradicts the Nash equilibrium condition and establishes that we necessarily have at a Nash equilibrium,

I u = ∅. (23)
Relations ( 20) and ( 23) together with the demand relation P i∈I d i = D(p) imply the uniqueness of the Nash equilibrium: a Nash equilibrium is necessarily such that each provider i declares unit price p i = p * .

  P i∈I C i = D(p * ), with p * = v ¡P i∈I C i ¢ , therefore p > p * .

  • If C iu > D(p) then for ε sufficiently small (such that D(p -ε) ≤ C iu ), provider i u gets all the demand, i.e. d ε iu = D(p -ε). When ε tends to 0, that demand tends to D(p) because of the continuity of the demand function. Consequently, for a sufficiently small ε, the demand for provider i u of switching from price p iu to price p -ε can be arbitrarily close to y := min(C iu , D(p)) > d iu , and the corresponding revenue gain can then be arbitrarily close to p(y -d iu )iu (y) + iu (d iu ) ≥ (p -0 iu (y)) (y -d iu ) | {z } >0 ≥ (p -0 iu (C iu ))(y -d iu ) ≥ (p -p * )(y -d iu ) > 0,

  from Lemma 1 we have p n ≥ p * andd n 1 < C 1 . First notice that if p n 1 ≥ inf{p : D(p) ≤ p P j6 =1 C j /p * } (i.e. the situation displayed in Figure2), then d n i = 0 and provider 1 gets a null profit, that is lower than the profit yielded by playing p * as noticed at the beginning of the proof. Therefore only the case when d n

			∑ i f i (p)
			D(p)
		∑ i C i
	quantities	∑ i =1 C i	C 1
		0	p * p p n 1	p
			unit price
	Fig. 2. Wardrop equilibrium if player 1 switches from p * to p n 1 ≥ inf{p : D(p) ≤ p P j6 =1 C j /p * }.
		∀i 6 = 1, d n i = C i p n 1 /p

1 > 0 needs to be proved. Actually since 0 < d n 1 < C 1 then we have p = p n 1 > p * , which from

[START_REF] Campos-Náñez | On-line tuning of prices for network services[END_REF]

implies that * , and


Meiyi Wu

Karine Le Guen

Jean-Michel André

Philippe Jonnard

Ian Vickridge

Didier Schmaus

Emrick Briand

Philippe Walter

Qiushi Huang

Zhanshan Wang

Kossel diffraction observed with X-ray color camera during PIXE of nano-scale periodic multilayer

Keywords: PIXE, periodic multilayer, Kossel diffraction

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

The Kossel process refers to the interference of characteristic X-rays generated within and scattered by the atoms present within a periodic micro-structure. Typical diffraction patterns are formed around the Bragg angle corresponding to the wavelength (or energy) of the emission and the period of the lattice. In 1935, Walther Kossel experimentally demonstrated such interferences for a crystalline structure [START_REF] Kossel | Die Richtungsverteilung der in einem Kristall entstandenen charakteristischen Röntgenstrahlung[END_REF]. The phenomenon was then named after him as "Kossel effect". Its observation requires two essential factors: first, the ionization of a core level in order to produce the characteristic emission; second, a periodic structure to diffract the emitted X-ray. The source for the ionization can vary. It can be X-ray provided either by an X-ray tube [START_REF] Chauvineau | Analyse angulaire de la fluorescence du fer dans une multicouche périodique Fe/C[END_REF][START_REF] Chauvineau | Lignes de Kossel observées avec des multicouches périodiques Fe/C[END_REF][START_REF] Chuev | Direct observation of anomalous Kossel lines[END_REF] or by synchrotron radiation [START_REF] Marchesini | Standing waves and Kossel line patterns in structure determination[END_REF][START_REF] Schetelich | Laue and Kossel diffraction on quasicrystals by means of synchrotron radiation[END_REF][START_REF] Jonnard | Spontaneous soft X-ray fluorescence from a superlattice under Kossel diffraction conditions[END_REF][START_REF] Tu | X-ray fluorescence induced by standing waves in the grazing-incidence and grazing-exit modes: study of the Mg-Co-Zr system[END_REF], energetic electrons [START_REF] Lider | X-ray divergent-beam (Kossel) technique: a review[END_REF][START_REF] Jonnard | Modulation of X-ray line intensity emitted by a periodic structure under electron excitation[END_REF][START_REF] Jonnard | Soft-X-ray Kossel structures from W/C multilayers under various electron ionization conditions[END_REF][START_REF] André | Radiation emitted by an oscillating dipole embedded in a periodic stratified structure: a direct matrix analysis[END_REF] or energetic charged particles (protons or ions) [START_REF] Geist | The influence of the crystal lattice on the angular distribution of X-rays emitted from a GaP single crystal by fast proton bombardment[END_REF][START_REF] Roberto | Positive ion induced Kossel lines in copper[END_REF][START_REF] Geist | Kossel interferences of proton-induced Al Ka radiation[END_REF][START_REF] Geist | Investigation of lattice deformation in proton bombarded GaP and ZnSiP2 by means of the proton-induced kossel effect[END_REF][START_REF] Geist | Investigation of GaN heteroepitaxial layers by means of the kossel effect technique[END_REF][START_REF] Rickards | Crystallography using the diffraction of proton induced X-rays[END_REF][START_REF] Schetelich | Recording of Kossel patterns using monochromatic synchrotron radiation[END_REF] as we report in this paper.

Studies of both crystals and interferential multilayers using various techniques have been widely

reported, but it was not until our previous paper [START_REF] Wu | Kossel interferences of proton-induced X-ray emission lines in periodic multilayers[END_REF] that the study of periodic multilayer by Kossel diffraction of X-rays generated by proton beam was first reported. We successfully observed the Kossel curves of Cr Kα and Sc Kα emissions of a Cr/B 4 C/Sc periodic multilayer irradiated by 2 MeV protons.

However due to the very small solid angle of the highly collimated detector the counting statistics were not ideal even for long acquisition times, and we mentioned in our conclusion that it might be improved by using an X-ray color camera, that is a CCD camera in which each pixel can provide an energy-dispersed X-ray emission spectrum. This allows us to avoid the angular scans in our measurements, thus reducing the overall acquisition time while allowing an improvement of statistics.

The work presented in this paper focuses on the Pd/Y based multilayers, which are designed as reflecting mirrors to work in the 7.5-11 nm wavelength range. Derivative systems are also designed with the motivation of improving the optical performance of the reflecting mirror owing to more abrupt interfaces.

Experimental details

Samples are deposited onto sliced and polished Si (100) wafers by using DC magnetron sputtering.

The originally designed structure is B 4 C(2.5 nm)/[Pd(2 nm)/Y(2 nm)] ×40 /Si. The B 4 C capping layer is added to protect the samples against oxidation. A series of samples is fabricated with 0-6% of nitrogen gas N 2 in the sputtering gas instead of pure argon because we believe that the formation of YN will help improve the thermal stability of the sample and reduce the interdiffusion between Pd and Y layers [START_REF] Xu | Enhancement of soft X-ray reflectivity and interface stability in nitridated Pd/Y multilayer mirrors[END_REF][START_REF] Wu | Study of Pd/Y based multilayers using high energy photoemission spectroscopy combined with x-ray standing waves[END_REF]. Another series of samples has B 4 C barrier layers inserted at Pd-on-Y, Y-on-Pd or both interfaces in order to prevent or reduce the interdiffusion between the Pd and Y layers [START_REF] Prasciolu | Thermal stability studies of short period Sc/Cr and Sc/B 4 C/Cr multilayers[END_REF]. The thickness of the barrier layers is 1 nm. The physical structures of the samples, including thickness and roughness of each layer are determined by grazing incident X-ray reflectometry immediately following the deposition process.

The proton beam is generated by the Van de Graaff accelerator of the SAFIR (Système d'Analyse par Faisceaux d'Ions Rapides) platform. The proton energy is set at 2.0 MeV, which is suitable to ionize the Pd atoms in their L shell uniformly through the whole multilayer. The energy loss of the proton beam is only about 0.5% after penetrating the multilayer whose total thickness is approximately 200 nm. The X-ray color camera is an iKon-M from Andor Technologies equipped with a 1024×1024 sensor array with 13×13 µm pixels. Instead of using the native spatial resolution, we select a 4×4 binning leading to a resolution of 256×256. Such selection improves the energy resolution by minimizing the charge sharing effect between adjacent pixels while still providing adequate angular resolution. It also simplifies the data treatment by reducing the size of data files. The camera spans an angular range of 2.7° corresponding to 256 horizontal pixels in Fig. 1.

For a given X-ray emission, we measure its intensity as a function of the detection angle with the X-ray color camera while irradiating the sample with the proton beam. The experimental setup is schematically presented in Fig. 1 where the X-ray color camera is placed perpendicular to the incident proton beam with a 200 µm beryllium film to filter the scattered protons. The experimentally obtained image of particle induced X-ray emission (PIXE) presented in Fig. 1 is the spatial distribution of total X-ray intensity on the detected area for B 4 C(2.5 nm)/[Pd(2 nm)/B 4 C(2 nm)/Y(2 nm)] ×40 /Si multilayer obtained after a 2 hour irradiation with about 35 nA of 2 MeV protons in a 2 mm beam spot. This corresponds to about 5×10 16 protons cm -2 . The Kossel feature cannot be observed on this image yet. It requires further treatment, which will be explained later, to be extracted from this image. Note that a complete X-ray spectrum is recorded by each pixel of the X-ray color camera. To identify the emission peaks, we sum the spectra recorded by all pixels in order to get more reliable statistics. An overview (intensity in log scale) of the summed X-ray spectrum of the B 4 C(2.5 nm)/[Pd(2 nm)/B 4 C(2 nm)/Y(2 nm)] ×40 /Si sample as well as the zoomed range of interest (intensity in linear scale) are presented in Fig. 2. The Y Kα emission can be observed on the spectrum at photon energy near 15 keV, but due to its low intensity and small estimated angular range for Kossel detection (about 0.5° grazing exit angle, in the total reflection zone), it would not be possible to observe its Kossel oscillation.

The energy calibration of the spectrum is carried out with Pd Lα (2.838 keV) and Y Kα (14.958 keV) peaks. Y Lα and Si Kα emissions are not considered because their energies are close to the low energy limit of the camera and their detection is not as efficient as Pd Lα emission. Pd Lβ emission is not discussed either because its intensity is much less than that of the of Pd Lα emission and would not bring supplementary information. The Pd Lα1 and Lα2 emissions are not resolved, the difference in their energies being less than the detector resolution of about 150 eV at Mn Kα (5.9 keV). A region of interest (ROI) is placed on the spectrum in order to distinguish the Pd Lα emission from other photons, see Fig. 3

Results and discussion

In Fig. 6 we present the Kossel curves for Pd Lα emission of all measured samples. To facilitate the comparison of the shapes of the curves, for each sample the origin of the detection angle is set to the Bragg angle, which corresponds to the center of its Kossel oscillation. Fig. 6(a) shows the Kossel curves of the Pd/Y multilayers deposited with a proportion of N 2 within the argon sputtering gas ranging from 0 to 6%. For the Pd/Y multilayer grown in pure argon we observe no Kossel feature at all. This indicates that the intermixing is so severe that even the periodicity of the multilayer is compromised. As the proportion of N 2 increases, we start to observe the Kossel feature and its contrast (defined as the difference between the maximal and minimal intensity values as the detection angle crosses the Bragg angle) increases as well. As expected [START_REF] Xu | Enhancement of soft X-ray reflectivity and interface stability in nitridated Pd/Y multilayer mirrors[END_REF][START_REF] Wu | Study of Pd/Y based multilayers using high energy photoemission spectroscopy combined with x-ray standing waves[END_REF], the nitrogen has a positive effect in preventing or reducing the intermixing at the interfaces of the multilayer. The first one exhibits an intensity maximum at +0.05° while the latter shows rather flattened feature at the same angular position. When inserting 1 nm B 4 C at both interfaces (the sample labeled as B4C/Pd/B4C/Y_1212), the intensity maximum at +0.05° become even more obvious, however at -0.05° the intensity decrease of the curve is more flattened than the two samples mentioned above.

Here we do not present the sample mentioned in the previous section with 2 nm B 4 C barrier layer because it is considered expendable for the long-time non-damaging test. Thus it is unsuitable to be put in the comparison series. given interface or at both interfaces. For this latter series, the name of the samples refers both to the order of the layers within the period and to the thickness (in nm) of the individual layers.

To simulate the PIXE-Kossel curve, we build model of the multilayer using the structural parameters (thickness and roughness of the layers) determined by grazing incident X-ray reflectivity.

The simulation method is originally designed for X-ray fluorescence calculation under Kossel effect [START_REF] Chauvineau | Analyse angulaire de la fluorescence du fer dans une multicouche périodique Fe/C[END_REF].

It calculates the electric field generated by the emitted X-ray within the multilayer under X-ray irradiation. We make an approach in order to adapt it to the PIXE-Kossel calculation. Since the ionization is uniform through the whole multilayer, we consider the incident proton beam as an photon beam with neither attenuation nor refraction. Such approach actually simplifies the calculation. An example of the simulation is given in Fig. 7 where both experimental and simulated the PIXE-Kossel curves are presented in the near Bragg angle zone. The agreement is fairly good showing that the PIXE-Kossel curve can be used to determine the structure of multilayer stack.

Conclusion

We have shown the feasibility of using proton induced X-ray emission combined with Kossel diffraction to characterize the structure of nanometric multilayers. We have measured the Kossel diffraction of proton induced Pd L emission in Pd/Y based multilayers. Compared to our previous experiment [START_REF] Wu | Kossel interferences of proton-induced X-ray emission lines in periodic multilayers[END_REF], we have significantly improved the data acquisition with the X-ray color camera. We also demonstrate here that the shape of the Kossel curve is very sensitive to the detailed structure of the multilayer and its interface. In particular we confirm that nitridation during the multilayer deposition process has a positive effect in preventing or reducing the interdiffusion between Pd and Y layers, and we further show that Pd/Y based multilayers with B 4 C barrier layers at different interfaces can be distinguished by the shape of their Kossel curves.

Fig. 1 .

 1 Fig. 1. Geometry of the experimental setup. θ is the detection angle. The presented experimental image is the spatial distribution of total X-ray intensity on the detector array for B 4 C(2.5 nm)/[Pd(2 nm)/B 4 C(2 nm)/Y(2 nm)] ×40 /Si multilayer irradiated as described in the text.

Fig. 2 .

 2 Fig. 2. Spectrum of B 4 C(2.5 nm)/[Pd(2 nm)/B 4 C(2 nm)/Y(2 nm)] ×40 /Si multilayer: a) overview, intensity in log scale; b) zoomed range of interest, intensity in linear scale.

  (a) where the intensity of Pd Lα emission is obtained by integrating the shadowed area. The ROI width of 100 eV is based on a compromise between obtaining good statistics and limiting the influence of the Pd Lβ emission. An image of the spatial distribution of the Pd Lα emission intensity (Fig. 3(b)) is obtained after applying the energy filtering ROI of Fig. 3(a) to the total intensity image presented in Fig. 1. While it is not possible to observe the Kossel feature in Fig. 1, now it can be clearly seen in Fig. 3(b). Finally we integrate this filtered image along the vertical pixels in order to obtain the Kossel curve which is presented in Fig 3(c).

Fig. 3 .

 3 Fig. 3. a) Selection of the ROI for Pd Lα emission; for the case already depicted in Fig. 1: b) image of the intensity of X-ray emission, of which the energy is in the ROI; c) Kossel curve depicted by integrating the X-ray emission intensity along the vertical pixel.To check the stability of the measurement and possible sample damage from the proton irradiation, we consider the sample B 4 C(2.5 nm)/[Pd(2 nm)/B 4 C(2 nm)/Y(2 nm)] ×40 /Si and continue measuring for over 7 hours. The result is presented in Fig.4where the curve intensities are normalized by the proton beam current and the acquisition time. We divide this series of measurements into 4-time slices and depict separately the related Kossel curves of Pd Lα emission to see if any deformation of the Kossel feature happens. The oscillation maintains its shape, indicating that the structure of the multilayer is not changed during the measurement and the proton beam causes no discernible change to the sample structure. We can safely say that the whole experiment is non-damaging because the standard acquisition time for each sample is about 2 hours.

Fig. 4 .

 4 Fig. 4. Non-damaging test of the sample B 4 C(2.5 nm)/[Pd(2 nm)/B 4 C(2 nm)/Y(2 nm)] ×40 /Si.

Fig. 5 .

 5 Fig. 5. Calibration of the detection angle using the comparison between a) experimental and b) simulated Kossel curves.

Fig. 6 (

 6 b) presents the Kossel curves of the multilayers with 1 nm B 4 C diffusion barriers at one given interface or at both interfaces.The Kossel feature of each sample can be clearly observed. For the two samples with 1 nm B 4 C barrier layer inserted at either Y-on-Pd or Pd-on-Y interfaces (noted as B4C/Pd/Y_122 and Pd/B4C/Y_212 respectively), the Kossel features are different when the detection angle is higher than the Bragg angle.

Fig. 6 .

 6 Fig. 6. Kossel curves for Pd Lα emission of all measured samples. a) series of samples fabricated with different concentration of N 2 in the sputtering gas; b) series of samples with B 4 C barrier layers at one

Fig. 7 .

 7 Fig. 7. Comparison between experimental result and simulation of the PIXE-Kossel curves of the sample B 4 C(2.5 nm)/[B 4 C(1 nm)/Pd(2 nm)/B 4 C(1 nm)/Y(2 nm)] ×20 /Si.