Julia Monnin
email: monninjulia@yahoo.fr

Hélène Loevenbruck
email: helene.loevenbruck@upmf-grenoble.fr

Anne Vilain
email: anne.vilain@gipsa-lab.grenoble-inp.fr

Consonant and syllable frequency effects in stop and fricative acquisition in preschool Drehu-and French-acquiring children

Keywords: Phonological development, frequency, universal, French, Drehu

Phonological development is often described as reflecting universal constraints, with some consonants or some consonant-vowel sequences assumed to be mastered earlier than others. Differences in order of acquisition are commonly ascribed to universal constraints on production or perception, reflecting biological specificities. Recent studies suggest, however, that language-specific patterns are also observed. To disentangle universal from language-specific constraints, the present study examines Drehu and French acquisition data on stop and fricative consonants (/t/, /d/, /k/, /g/, /s/, /tʃ/), followed by vowels /i, a, u/. Results show that the relative accuracy of stops and fricatives in [i, a, u] contexts in Drehu and French reflect the modulatory influence of language-specific phonotactics on the perceptual and biomechanical constraints claimed to govern the production of consonant and consonant-vowel combinations.

INTRODUCTION

Influential approaches in language development explain the course of phonological acquisition as reflecting either universal phonological principles, e.g. Optimality theory [START_REF] Mccarthy | What Is Optimality Theory?[END_REF], or universal biomechanical constraints, e.g. frame dominance theory [START_REF] Davis | Acquisition of serial complexity in speech production: A comparison of phonetic and phonological approaches to first word production[END_REF], or idiosyncratic trajectories, e.g. templatic theories [START_REF] Vihman | Phonological development: Toward a 'radical' templatic phonology[END_REF]. Inheriting from Jakobson's [START_REF] Jakobson | Kindersprache, Aphasie, und allgemeine Lautgesetze[END_REF] view that a child's early productions are unmarked in structure, phonological approaches suggest that order of phoneme acquisition and evolution of child error patterns can be accounted for by progressive re-ranking of markedness and faithfulness constraints. In this view, unmarked coronals will typically be acquired earlier than marked dorsals, and voiceless stops earlier than voiced stops. This claim is corroborated by data on child acquisition, most of which bearing on the acquisition of English or another (Indo-) European language, showing more accurate production of [t] relative to [k] for instance [START_REF] Locke | Phonological acquisition and change[END_REF][START_REF] Wauquier | Early templates in the acquisition of French phonology[END_REF]. Cross-sectional studies of children acquiring Greek [START_REF] Nicolaidis | Acquisition of lingual obstruents in Greek[END_REF], Japanese [START_REF] Beckman | Language-specific and language-universal aspects of lingual obstruent productions in Japaneseacquiring children[END_REF], and Drehu or French [START_REF] Monnin | Languagespecific influence on phoneme development: French and Drehu data[END_REF], by contrast, suggest that any tendency to produce coronals more accurately than dorsals is modulated by language-specific frequencies. Biomechanical account of child development might argue that this modulation is not due to consonant frequency per se but is related to universal consonant-vowel co-occurrence patterns due to vocal tract constraints: "fronted frames" and "backed frames" constraints for instance [START_REF] Davis | Acquisition of serial complexity in speech production: A comparison of phonetic and phonological approaches to first word production[END_REF]. In other words, consonant accuracy alone is not informative, and should be compared across different vowel contexts. To disentangle universal from language-specific constraints, the present study examines Drehu and French data on stop and fricative consonants in different vowel contexts. Drehu is an Austronesian language, with around 15 000 speakers, mostly spoken on Lifou Island in New Caledonia. Although French is present in the environment of Drehu-acquiring children, Drehu is the dominant language at home and at school on Lifou Island [START_REF] Moyse-Faure | Le drehu, Langue de Lifou (Iles Loyauté)[END_REF]. Rather than observing which consonants are spontaneously produced, this study examines consonant accuracy in a word repetition task. This method aims at better describing the way children generalize over their preferred motor schemes, as described in [START_REF] Vihman | Phonological development: Toward a 'radical' templatic phonology[END_REF], to produce the ambient language words. This study tests the following two predictions that stem from universal constraint approaches, first, that marked phonemes (such as dorsals, fricatives or voiced consonants) are acquired later, and, secondly, that frame-dominant (such as [ti, di, si] or [ku, gu]) are mastered before non-frame dominant sequences (such as [tu, du, su] or [ki, gi]). These predictions are compared with frequency accounts, according to which consonant or syllable frequency might influence consonant accuracy.

METHODS

Children recording

A total of 158 monolingual French-acquiring and 48 Drehu-acquiring children (in bilingual Drehu-French environment) have been recorded (French: 41 2year, 41 3-year, 40 4-year, 40 5-year olds; Drehu: 14 3-year, 16 4-year and 18 5-year olds). Word-initial consonant-vowel sequences were elicited in a picture-prompted word-repetition task, as in [START_REF] Monnin | Languagespecific influence on phoneme development: French and Drehu data[END_REF]. In this study, we focus on the following 6 target consonants: /t/, /d/, /k/, /g/, /s/, /tʃ/, followed by vowels /i, a, u/. Data are missing for /du, gi, tʃi/ in French and for /di, du/ in Drehu. Supplementary data are available for /ʃa, ʃi, ʃu/ in French. The words were chosen to be as frequent as possible in each language. Each syllable occurred in three different words. Target sounds were transcribed by native speakers. Parts of the corpora were retranscribed by two independent transcribers. In (rare) cases of disagreement, the data were not included.

Frequency data

Type frequency data in content words were obtained from recordings of adults speaking to 2-year olds in French and in Drehu. More information on these corpora and their analyses is given in [START_REF] Monnin | The influence of frequency on word-initial obstruent acquisition in Hexagonal French[END_REF]. Frequencies were calculated for consonants (C) as well as for consonant-vowel syllables (CV). Note that /tʃ/ is rare in French and has a low frequency in adult lexicons [START_REF] New | Une base de données lexicales du français contemporain sur internet: LEXIQUE[END_REF]. In the French CDS corpus, /tʃ/ was in fact absent. Syllable [gu] was also absent. Similarly, [ti] and [da] were absent from the Drehu CDS corpus. Therefore [tʃi, tʃa, tʃu, gu] in French and [ti, da] in Drehu are not taken into account in the correlation analyses below (Table 1).

RESULTS

Overall consonant accuracy in Drehu and French

Mean consonant accuracy for Drehu and French are presented in Fig. 1, for all vowel contexts, and for 3 to 5 year olds grouped together (data not available for Drehu-acquiring 2-year olds). Note that /ʃ/ accuracy is added to the graph for French, although it is not taken into account in the statistical analyses, as Drehu children did not produce this consonant. Mean accuracy across the 6 target consonants for all 3-to 5-year olds is similar in the two languages (74.13% (sd: 29.45) in Drehu vs. 73.72% (sd: 33.48) in French). Individual consonant accuracies show language-specific trends, however. A linear mixed model analysis was carried out, with Language as a between-subject factor and Consonant as a withinsubject factor, for children aged 3 to 5 years. Consonant and Language x Consonant interaction are significant (Consonant: F(5, 2348) = 149.77, p> 0.001; interaction: F(5, 2348) = 34.51, p < 0.001). These data suggest that the order of consonant acquisition is the following in Drehu: /k, t/ > /s/ > /tʃ, g/ > /d/ (all contrasts significant). In French, a different order is observed: /t, k, s/ > /d/ > /g/ > /tʃ/ (all contrasts significant). Further Student tests show that for /d/, French-acquiring children are significantly more accurate than Drehu-acquiring children, whereas for /tʃ/ the reverse pattern is observed. In summary, Drehu dorsal /k/ is not mastered later than its unmarked counterpart /t/. In French, marked dorsal /k/ and sibilant /s/ are not mastered later than unmarked /t/. Some fricatives are mastered earlier than some stops: in Drehu, /s/ and /tʃ/ before /d/; in French, /s/ before /d/ and /g/.

Evolution of mean consonant accuracy with age

The evolution of consonant accuracy is shown in Fig. 2, from 2 to 5 years in French and from 3 to 5 years in Drehu. First, overall accuracy in all 6 target consonants increases with age (French: from 48.52% in 2 year olds to 79.71% in 5 year olds, Drehu: from 65.91% in 3 year olds to 80.37% in 5 year olds). Secondly, the language specific pattern described above is maintained during development.

Thirdly, only [t] and [k] in Drehu and [t], [k], and [s] in

French are fully mastered (>75% accuracy) in 3 year olds. Finally, in 5 year olds, accuracy is above 75% in all consonants but [d] in Drehu and [tʃ] in French. In both languages, consonant accuracy is modulated by vowel context. In Drehu-acquiring children, fronted frames (as referred to in [START_REF] Davis | Acquisition of serial complexity in speech production: A comparison of phonetic and phonological approaches to first word production[END_REF]) are not more accurate than their non frame-dominant counterparts ([ti] not significantly higher than [tu]; [si] not significantly higher than [su]). In French-acquiring children, some fronted or backed frames are not more accurate than non frame-dominant counterparts ([ti] is significantly lower than [tu]; [gu] is not significantly higher than [ga]). A linear mixed model analysis was carried out for children aged 3 to 5 years. Syllable (F(13, 2119) = 71.22, p > 0.001) and Language x Syllable (F(13, 2119) = 13.21, p > 0.001) are significant. Significant differences are observed, with [gu], [tʃi], [tʃa] more accurate in Drehu and [da] more accurate in French.

Vowel context effect

Frequency effects

The relationship between consonant accuracy in each CV and CV frequencies in each age group is shown in Fig. 3. Accuracies are plotted for /ti, da/ in Drehu and /tʃi, tʃa, tʃu, gu/ in French, but they were not taken into account in the correlation analyses (see 2.4): correlation lines include these data, but reported r coefficients and p values do not. Table 1 provides correlation between consonant accuracy in each CV and available CV frequency as well as between C accuracy and available C frequency in each language, for each year. In Drehu, correlations with CV frequency are significant (or near significant) in each age, whereas correlations with C frequency are not. In French, all correlations, except with C frequency at 5 years, are significant. In summary, in both Drehu and French, the results show that the more frequent a CV syllable is, the more accurate the consonant is produced in that vowel context, whatever the age. In younger Frenchacquiring children, the correlation with consonant frequency itself is significant and higher than the correlation with C frequency. These correlations suggest that the lower frequency of

DISCUSSION

These results first show that overall accuracy for the 6 stops and fricatives of interest is similar in French and Drehu-acquiring 3-to 5-year olds. The order of consonant acquisition differs in the two languages, however. Some consonants are mastered earlier in one language than the other. The earlier-acquired consonants are not always the unmarked ones. For instance, marked consonant /k/ is not mastered later than its unmarked counterpart /t/ in both Drehu and French. In addition some fricatives are mastered earlier than some stops. This suggests that phoneme markedness alone cannot account for these data. It has been argued that vowel context should be taken into account when examining consonant accuracy, with some intrasyllabic co-occurrences being more favored than others [START_REF] Davis | Acquisition of serial complexity in speech production: A comparison of phonetic and phonological approaches to first word production[END_REF]. These cooccurrences are claimed to be reflective of constraints emerging from mandibular oscillation cycles. Consonant accuracy was therefore examined taking account of frame dominance. The results reveal that, in both Drehu and French, some fronted or backed frames are not more accurate than their non frame-dominant counterparts. Our data therefore suggest that frame-dominance constraints do not fully explain French and Drehu acquisition. Correlation analyses suggest that the relative accuracy of stops and fricatives in [i, a, u] contexts increases with ambient language CV frequency. This suggests that language-specific phonotactics play an influential role in phonological development and modulate the phonological and biomechanical constraints claimed to govern children production of consonant and consonant-vowel combinations. In younger French-acquiring children, the correlation with C frequency is higher than that with CV frequency. This could mean that consonant accuracy is better explained by C frequency than by CV frequency in French. Ongoing analyses of error patterns examine how place, manner and voicing are affected in different vowel contexts. This could shed further light on the relative influences of consonant vs. consonant-vowel frequency.

Figure 1 :

 1 Figure 1: Mean consonant accuracy for Drehuand French-acquiring children, aged 3 to 5 years

Fig. 3

 3 Fig. 3 displays mean accuracy for the target consonants in each vowel context, as a function of CV frequency in each age group.

  [ti] relative to [tu] (in both languages) or of [gu] relative to [ga] (in French) might explain the lower accuracy of [t] in the front context or of [g] in the back context. Similarly, lower frequencies of [gu], [tʃi] and [tʃa] in French relative to Drehu, might explain their lower accuracy in French relative to Drehu. Finally, the lower accuracy of [da] in Drehu relative to French can be related to its lower frequency in Drehu.

Table 1 .

 1 Correlation between consonant accuracy and C or CV frequency in Drehu and French at all ages

Figure 2 .Figure 3 . 5

 235 Figure 2. Growth in mean consonant accuracy for Drehu-and French-acquiring children


Samba Traoré
email: samba.traore@supelec.fr

Babar Aziz
email: babar.aziz@supelec.fr

Daniel Le Guennec
email: daniel.leguennec@supelec.fr

Yves Louet
email: yves.louet@supelec.fr

Adaptive non-uniform sampling of sparse signals for Green Cognitive Radio

Keywords: Non-Uniform sub-Nyquist sampling, Software Radio, Cognitive radio, Non-Uniform spectrum sensing. 5

Based on previous results on periodic non-uniform sampling (Multi-Coset) and using the well known Non-Uniform Fourier Transform through Bartlett's method for Power Spectral Density estimation, we propose a new smart sampling scheme named the Dynamic Single Branch Non-uniform Sampler. The idea of our scheme is to reduce the average sampling frequency, the number of samples collected, and consequently the power consumption of the Analog to Digital Converter. In addition to that our proposed method detects the location of the bands in order to adapt the sampling rate. In this paper, through we show simulation results that compared to classical uniform sampler or existing multicoset based samplers, our proposed sampler, in certain conditions, provides superior performance, in terms of sampling rate or energy consumption. It is not constrained by the inexibility of hardware circuitry and is easily recongurable. We also show the eect of the false detection of active bands on the average sampling rate of our new adaptive non-uniform sub-Nyquist sampler scheme.

Introduction

Radio Frequency (RF) allows modulation of narrow band signals with a high carrier frequency. The radio signals of human origin are often sparse. In other words, they are composed of a relatively small number of narrow band transmissions spread across a wide spectral region. A practical description of as bandwidth and spectrum. Moreover, the current trends in wireless technology have increased the complexity of the receiver, more specically its Analog to Digital Converter (ADC).According to Shannon-Nyquist theorem, a signal whose spectral support is limited tofnyq 2 and fnyq 2 can be perfectly reconstructed by sampling at f nyq . To sample a wide band signal with Nyquist rate will require a high sampling rate ADC which consumes a lot of energy. To reduce the sampling rate, and in turn the energy consumption, several researchers have studied the possibility of sub-Nyquist sampling. In [? ] , a sub-Nyquist sampling method is proposed for sparse multi-band signals, called Modulated Wideband Converter (MWC). MWC consists of several stages and each stage uses a dierent mixing function followed by a low pass lter and a low uniform sampling rate. This sampling technique shows that perfect reconstruction is possible when the band locations are known. In [? ] [? ], authors have studied the blind case, where the location of the bands is unknown with an inexible and sub optimal sampler scheme in term of sampling rate. Over the recent years, multi-coset sampling [? ] [? ] [? ] has gained fair popularity and several methods of implementing the multi-coset sampling have been proposed. The most famous architecture is composed of several parallel branches, each with a time shift followed by a uniform sampler operating at a sampling rate lower than the Nyquist rate. Recently, a dierent approach for implementing multi-coset sampling has been proposed in [? ], which uses uniform samplers operating at dierent rates and is known as the Synchronous Mutlirate Sampling. In[? ], a Dual-sampling architecture is presented for multi-coset sampling which is basically a subset of the Synchronous Multi rate Sampling and uses only two uniform samplers. But for optimal reconstruction, all these methods assume that the number of bands and the maximum bandwidth, a band can have, is known. In [? ], authors have dened some requirements that need to be fullled for a sampling system to be ecient. These requirements are stated as follows: the sampling rate (average) should be as low as possible, the system has no prior knowledge of the band locations, and the system can be designed with existing devices. In this paper, we propose a blind sampler based on multicoset sampling scheme which respects the above stated denition of an ecient sampler. Our scheme estimates the spectral support, using our non-uniform spectrum sensing model proposed in [? ? ], to minimize the average sampling rate, thereby reducing the number of samples as well as energy consumption. We show that our proposed spectrum sensing model provides accurate results using less data samples. Its performance is examined at low rate SNR values with less data samples. In addition, its power consumption is compared with regular ADC when the input signal is very sparse and it is found to produce satisfactory results.

This article is organized as follows. In Section 2, we present the signal model along with an overview of multi-coset sampling. In Section 3, we present our proposed sampling system and explain all its blocks. Then the non-uniform spectrum sensing model is presented and the functionality of each block is explained. Numerical results are presented in Section 4 followed by a conclusion in the end.

M(B) = x(t) ∈ L 2 (R) : X(f ) = 0 ∀ f / ∈ B (1)
where

B = [-fnyq 2 , fnyq 2 ] and X(f ) = +∞ -∞ x(t)e -j2πf
t dt is the Fourier transform of the signal x(t).

Let F represent the spectral support of the signal dened by

F = N i=1 [a i , b i ] (2)
where F ⊂ B, a i and b i represent, respectively, the lower and upper bound of each bands, and N is the number of bands in B.

Multi-Coset Sampling

Multi-Coset (MC) sampling is a periodic non-uniform sub-nyquist sampling technique which samples a signal x(t) at a lower rate than the Nyquist rate, thereby capturing only the amount of information required for an accurate reconstruction of the signal [? ? ]. In short, the process of MC sampling can be viewed as rst sampling the input signal at a uniform rate with period T and then selecting only p non-uniform samples from L equidistant uniform samples. The process is repeated for consecutive segments of L uniform samples such that the p selected samples have a sampling period L. The set C = {c i } p-1 i=0 species the p samples that remain in each segment of length L such that

0 ≤ c 0 < c 1 < ... < c p-1 ≤ L -1.
MC sampler is usually implemented by placing p ADCs in parallel as shown in Fig. 1. Each ADC operates uniformly at a period T s = LT . The ∆ i = c i T represents the time shifts in sampling instants introduced in each branch. It should be noted that a good choice of the sampling pattern C reduces the margin of error due to spectral aliasing and sensitivity to noise in the reconstruction process [? ]. It is quite evident from Fig. 1 that once the sampling parameters (such as p) are selected, architecture of the MC sampler will remain unchanged irrespective of the input signal characteristics. In other words, once designed, the sampler in Fig. 1 cannot be changed because of hardware limitations. If the input signal changes the MC sampler does not adapt, which results in suboptimal sampling of the signal, as will be show later. This motivated us to look for an exible system which conforms with the spectrum of the input signal. In the next section, we explain the functionality of our new sampling scheme with and show that it is more exible compared to the MC sampler in Fig. 1.

System Model

With the exponential growth in the means of communications, modifying radio devices easily and cost-eectively has become business critical. Softwaredened radio (SDR) technology brings the exibility, cost eciency and power to drive communications forward, with wide-reaching benets realized by service providers and product developers through to end users [? ]. One of the main objectives of SDR is to propose new technologies to design radio terminals and wireless infrastructure able to support hardware-independent, multi-service operations and which are remotely recongurable. Furthermore, Cognitive radio that not only adapts to the changes of the input signal but is also remotely recongurable and is, therefore, not constrained by the inexibility of hardwired circuitry. Our proposed non-uniform sampler is shown in Fig. 2. We call it the Dynamic Single Branch Non-Uniform Sampler (DSB-NUS) or simply the DSB sampler [? ]. DSB sampler is based on the principle of MC sampling except that instead of p input branches in case of MC sampling, the DSB sampler has only one. Furthermore, our DSB sampler has an adaptation loop which allows to measure the spectral support of the input signal periodically, and adapt the sampling rate according to the spectrum of the input signal. The regular MC sampling lacks this adaptive and feedback mechanism and is rather a static, hardwired system. The DSB sampler operates in blind mode, without any knowledge of the input signal's spectral support F and the number of bands N , and eciently reduces the number of samples while ensuring optimal reconstruction with minimum error.

The DSB-NUS operates in two phases:

1. Adaptation phase: When the switch is in position 1, the sampler adapts the sampling parameters according to the spectral support of the input signal x(t) through a feedback loop. 2. Reconstruction Phase: When the switch is in position 2, the DSB-NUS sampler performs Multi-Coset reconstruction.

As this will be shown shortly, the average sampling rate of our system depends on the signal sparsity in the frequency domain. Next we present the operation of each block of the DSB sampler.

Non-Uniform Sampler block (NUS)

In this paper, we design the NUS of the DSB sampler as a recongurable Additive Pseudo-Random Sampler (APRS) [? ] in conjunction with MC sampling. Thus, unlike the regular MC sampler with several branches, the input of the DSB-NUS is composed of a single branch non-uniform sampler. The N sampling instants of the mono-branch DSB sampler are dened as:

t m = t m-1 + α m = t 0 + m i=1 α i (3) where 1 ≤ m ≤ N , E[t m ] = mT and var[t m ] = mσ 2 ∀ N ≥ 1.
Note that {α m } is a set of i.i.d random variables with probability density p 1 (τ ), mean T and with variance σ 2 . Although, the input components of the DSB sampler have changed compared to the regular MC sampler but the signal at the output of the NUS remains the same as in the case of MC sampling [? ]. As a result, for each L uniformly spaced samples in case of Nyquist sampling, we get p non-uniform samples (see Fig. 3).

Let T = {τ i } p-1 i=0 be the set of the durations between p adjacent samples, where The set C = {c i } p-1 i=0 contains p distinct integers from {0, 1, ..., L -1}. By taking t 0 = c 0 T , (3) can be written as

τ i-1 = c i -c i-1 for 1 ≤ i ≤ p -1 and τ p-1 = (L + c 0 ) -c p-1 (see Fig. 3).
tm = c0T + T m i=1 τi-1 1 ≤ m ≤ p. (4)
The set of sampling instants {t n } n∈Z is non-uniform and periodic with period Following the approach of the MC sampling presented in Fig. 1, we express the non-uniform samples x(t n ) at the output of the NUS in terms of periodic sequences x i [m] = x(t mp+i ), where 0 ≤ i ≤ p-1. The Fourier transform, X i (f ), of the sampled sequence x i [n] is related to the Fourier transform X(f ) of the unknown signal x(t) by the following equation [? ] :

X i (f ) = 1 LT exp(j2πfc i T ) L 2 -1 n=-L 2 X(f + n LT )exp(j2π c i L n) (5)
where

0 ≤ i ≤ p -1, and f ∈ B 0 = [0, 1 LT ]. Note that x(t) ∈ M(B).
The expression in (5) can be written in a matrix form

y(f ) = A C s(f ), (6)
where f ∈ B 0 and y(f ) is a vector of size p × 1 whose i th element is given by

y i (f ) = X i (f )exp(-j2πf c i T ). ( 7
)
A C is a matrix of size p × L whose (i, l) th element is given by [? ]

[A C ] il = 1 LT exp(j2π c i L (l - L 2 )), (8)
where 0 ≤ l ≤ L -1. Note that s(f ) in ( 6) represents the unknown vector of size L × 1 whose l th element is given by

s l (f ) = X(f + l -L 2 LT
).

(

) (9) indicates that, if the spectrum X(f ) is divided into L cells indexed from -L 2 to L 2 -1, 9
then each cell corresponds to a line in the vector s(f ), for f ∈ B 0 .

Non-Uniform Spectrum Sensing Block (NUSSB) 140

The reconstruction of x(t) requires to solve [START_REF] Dominguez-Jitnenez | Analysis and design of multirate synchronous sampling schemes for sparse multiband signals[END_REF]. Using the fact that x(t) is sparse in the frequency domain and that there is no energy in certain cells, the number of unknowns in ( 6) can be reduced. The set K consists of the spectral indexes of the cells that contain signal which are called the active cells. But in order to nd K dierent methods have been proposed in literature [? ? ? ]. But the limitation of the methods proposed is that they require complete knowledge of the matrix A C , the number of bands N and the maximum bandwidth. Since we assume that we do not have any information regarding the spectrum of the signal x(t), therefore, the matrix A C , along with other parameters, cannot be known. In order to overcome this hurdle, we treat this scenario as a missing data problem. In [? ] we have proposed to use the Lomb-Scargle's method [? ] to estimate the power spectral density (PSD) of the non-uniformly sampled signal, in spectrum sensing. In this paper, rather than Lomb-Scargle periodogram, we propose to use the well know Non-Uniform Fourier Transform through Bartlett's method [? ]. The PSD obtained for N s = KpM samples x(t n ) taken at times t n , n = 0, ..., N s -1, is dened by [START_REF] Palicot | Radio engineering: From software radio to cognitive radio[END_REF] :

PBart (ω) = 1 N s K-1 i=0 pM -1 m=0 x(t m + iM LT )exp(-jωt m ) 2 (10)
where

ω=2πf with f ∈ [0, 1 2T ], M > 0, K > 0 and 0 < p ≤ L.
Once the PSD is estimated, the threshold value, η, is selected dynamically as a function of P SD max i.e. η = P SD max -β , where β is a xed value and * is the oor function. With reference to η, the number of bands N and the spectral support F are computed. The block diagram of our non-uniform spectrum sensing model is presented in (Fig. 4). Once the support F and N are found, the set of spectral indexes

K = {k r } q-1 r=0 , where k r ∈ {-L 2 , -L 2 -1, . . . , L 2 -1},
can be calculated using [START_REF] Ben Romdhane | Echantillonnage non uniforme appliqué à la numérisation des signaux radio multistandard[END_REF] as follows where 1 ≤ i ≤ N and * is the oor function. Once all the κ i are calculated for each band, the set of spectral indexes is given by :

a i LT ≤ κ i ≤ b i LT (11)
K = N i=1 κ i . ( 12
)
The process is illustrated in Fig. 5 for a signal with N = 5.

The set K, thus, is sent to the optimal average sampling rate search and the 150 multi-coset reconstruction blocks, as shown in (Fig. 2). The performance of our Non-Uniform Spectrum Sensing Block is evaluated by computing the probabilities of the signal occupancy in terms of the number of bands N and the active cells set K as follows:

1. The good detecting probability

P d (N ) = Pr N = N ( 13
)
P d (K) = Pr K = K (14)
2. and the false alarm probability

P f a (N ) = Pr N > N (15)
P f a (K) = Pr | K| > |K| K ⊂ K ( 16
)
where |K| represents the cardinality of K. K and N represent, respectively, the 155 estimate of K and N given by the NUSSB.

In our proposed NUSSB, the threshold, η, is the only information assumed to be available on the input signal. It should be noted that these technique can be used only if the power dierence between each band is small.

As will be shown later in the paper through simulations, the results obtained 160 using Bartlett's method for spectral support detection are quite accurate.

Multi-Coset Reconstruction Block

After having found the set K, the vector s(f ) is reduced in dimension to z(f ), a vector of size q × 1 with r th element given by ( 17) :

z r (f ) = X(f + k r LT ). (17)
The matrix

A C becomes A C (K), a matrix of size p × q [A C (K)] ir = 1 LT exp(j2π c i L k r ) (18)
where 0 ≤ i ≤ p -1 and 0 ≤ r ≤ q -1. As a result ( 6) is reduced to

y(f ) = A C (K)z(f ) (19)
As our non uniform sampler (see.3.1) is recongurable, we always take p = q. So, the matrix A C (K) is a full column rank matrix, the unique solution of ( 6) is given by ( 20)

z(f ) = A -1 C y(f ), (20) where
A -1 C
is the inverse of A C (K) and f ∈ [0, 1 LT ]. Since x(t) ∈ M(B), its spectrum is Hermitian, the vector z(f ) dimension can be reduce to z + (f ). Then (20) become (21)

z + (f ) = A + C y(f ) (21
) where z + (f ) is a vector of size q 2 × 1 with r th element given by ( 22)

z + r (f ) = X(f + k r LT ) (22)
where 1 ≤ r ≤ q 2 and A + C , a matrix of size q 2 × q whose (i, l) th element is given by

A + C il = 2 A -1 C il (23) where 0 ≤ i ≤ q
2 -1 and 0 ≤ l ≤ q -1. Once z + (f ) is found, the inverse Fourier transform achieves the time domain representation of each cell, X(f

+ kr LT ) Fourier transform of x r (t) = x R r (t) + jx I r (t).
The reconstructed signal in time domain is written as :

x(t) = 2 q 2 -1 r=0 x R r cos(2π k r LT t) -x I r sin(2π k r LT t) (24)
The Multi-Coset reconstruction process is illustrated in (Fig. 6) for pM non uniform samples x(t n ), taken at times {t n } pM -1 n=0 , for LM , corresponding, Nyquist samples (see.2.2), with M > 0. The total number of real operations after each stage is given in the 5 (TABLE.1).

Complexity and power eciency

From (TABLE.1), the total number of real operations (addition and multiplication) to get a uniform Nyquist sample is given by (25) :

N total = 11p 2L log 2 (M ) + 37p 16 log 2 (M L) + 9p 2 2L + 6p L + p 2 4 + p 2 (25)
Since we have dened our NUS (see. 3.1) as an ADC which can be switch o at predened instances and have some circuitry which transforms the uniform clock to a non uniform one. Assuming insignicant the power consumption to transforms the uniform clock. We dene E dsb as the energy consumed by our DSB system to get a Nyquist sample by ( 26)

E dsb = τ E us + N total P cpu (26)
where τ = (p/L) is the compression ratio, E us represents the energy consumed by the ADC for a Nyquist sample and P cpu is the power eciency of the chosen processor, measured in Joule per Operations (J/OP). The proposed system is ecient only if E dsb < E us , then the processor power eciency must be chosen as (27)

P cpu < (1 -τ ) E us N total (27)
For example, if p = 10, L = 128, M = 512 then τ ≈ 0.078, we select a 14-bits 100 MSPS digitally calibrated pipeline ADC which consumes 250 mW per 100 MSPS [? ] then, E us = 2, 5 nJ and N total ≈ 408 operations. So from (27) , we deduce P cpu < 2.305 nJ/OP. Furthermore, to maintain a real-time processing, the execution time of the N total operations must be less than the sampling period. Then the chosen processor in this example must be performs more than 40 processes at 50 Giga operations per second with a power eciency around 57, 8 fJ/OP. Then our system could be more power eciency than regular ADC when the bands locations are unknown with a very sparse signal.

Optimal Average Sampling Rate Search Block (OASRSB)

The ideal sampling pattern is obtained when the condition number of A C (K) is as small as possible, [? ? ]. The condition number of a matrix U is dened as cond(U ) = U U -1 where * is a norm operation. The choice of C can be seen as the solution to the following minimization problem

C opt = arg min C:|C|=p cond(A C (K)) (28)
One way of solving (28) is to use greedy search algorithms. Authors in [? ] proposed to use the Sequential Forward Selection (SFS) to nd the set C. The SFS algorithm used in [? ] lacks the capability to remove elements from set C that become obsolete after the addition of other elements. For a given set L = 0, 1, ..., L -1, the SFS looks for a subset C = {c i } p i=1 which minimizes the function cond(A C (K)). L, T and K known, SFS algorithm searches for an optimal sampling pattern C which in turn minimizes the reconstruction error. Finally, C is used to compute the elements of the set T . Thus, for a given L, the non-uniform sampler operates at an optimal average rate which depends only on the number of active bands. As a result, the average sampling rate f of our proposed sampler is given by :

f = p LT = |K| LT (29) 4. Numerical Results
For simulations, we consider a multi band signal with a variable number of bands N ∈ {1, 2, ..., 6}, each with a bandwidth B=20MHz, we assume that the N bands have the same power. The wide band of interest is in the range of [0, 500] MHz i.e. f nyq = 1GHz. 16 QAM modulation symbols are used that are corrupted by the additive white Gaussian noise. To evaluate the performance of our Dynamic Non-uniform Sampler, we start by presenting in Sub-Section 4.1, the performance of our Non-Uniform Spectrum Sensing block versus to α = p/L ( the compression ratio), β (the threshold) and the SNR. In the Sub-Section 4.2, we illustrate the performance of the proposed DSB sampler by comparing it to regular MC scheme and the drawback of poor detection of the spectral support on its average frequency.

Performance of Bartlett Method

In this sub-section we will evaluate the performance of our NUSSB in terms of good detection of the number of bands N and the set of actives bands K (see.(3.2)). For simulation, we take L = 128, M = 16 and K = 32 as parameters in equation [START_REF] Palicot | Radio engineering: From software radio to cognitive radio[END_REF], and use SFS algorithm to choose the sampling instances.

In order to compute P d and P f a , we start by choosing an appropriate β. In Fig. 7, we have performed 1000 iterations at SNR=10dB and for dierent values of α and β. It can be seen here that whatever β, P d (K) increases with increasing α until reach a maximum at some β values.These values are reached around β = -6, -5.5 and -5 dB respectively for α = 0.7, 0.6 and 0.5. In the rest of the simulation, we take β = -6, -5.5 and -5 dB respectively for α = 0.7, 0.6 and 0.5. In Fig. 8, we plot the P f a(N ) and P f a(K) as a function of varying SNR. At low SNR i.e. -2 dB, the values for P f a(N ) and P f a(K) are quite high, especially for α ≥ 0.5. But as the SNR increases, the P f a(N ) and P f a(K) drop quickly, practically becoming zero at SNR=6dB for α=0.7. As expected, P f a(N ) and P f a(K) also depend on the number of non-uniform samples available to NUSSB for detection. As the average sampling rate for the DSB sampler is f avg = (p/L)f nyq , it can be suspected that a false estimation of N , K by NUSSB will directly eect 210 the f avg of the DSB sampler. This can happen because at a high P f a , the NUSSB shows that the estimated spectral support F of the signal is larger than the actual support, as a result, the DSB sampler is forced to sample at a higher, unoptimized sampling rate. This will be illustrated in the next sub-section.

Flexibility and adaptivity of DSB 215

To start, we assume that the traditional MC sampler is designed not only to have an optimal reconstruction for the signals but also a perfect knowledge of (N, K) of the incoming signal while on the other hand, our proposed DSB sampler operates in blind mode and therefore has no information regarding the F and N of any kind of signals. In the following, we use the subscripts mc and dsb to dierentiate between simulation parameters for MC and DSB samplers, respectively. It should be noted that p mc allows to set the number of branches [? ] or number of uniform sampler placed in parallel [? ? ] on the MC sampler. For sampling process, we select L = 128 and let the samplers select their respective parameter p mc /p dsb depending on the signal's sparsity. We design the MC sampler so as to have an optimal reconstruction for wide band signals x(t) with frequencies carriers f i ∈ {15, 170, 250, 330, 410, 490}, each band of bandwidth B = 20MHz, the set of spectral indexes representing the set of actives cells K is given by ( 11) and ( 12), and the sampling pattern C is computed using SFS algorithm [? ]. Finally we have p mc = 50. We measure the performance of MC sampler and DSB sampler in terms of the Root Mean Square Error (RMSE) of the reconstructed time domain signal x(t). It is dened by (30) :

RMSE = x(t) -x(t) 2 x(t) 2 (30)
In Fig. 9, we have plotted the RMSE as a function of condition number of A C (K)) of signal x(t) without noise. The sampling pattern C is chosen randomly and by using the SFS algorithm (see. 3.5). It can be seen that SFS Algorithm make the reconstruction more ecient than a randomly choice of the set C. In the rest of the simulation the RMSE is calculated using (30) and is averaged over 1000 iterations. It should be noted that the parameters of the MC sampler remain unchanged due to the hardware rigidity regardless of the input signal characteristics. Since the spectral support is unknown to the DSB sampler, it always starts from the adaptation phase to nd F and N , and then goes ahead to operate in reconstruction phase until needed to recheck the spectrum of the input signal. The sampler can be programmed to periodically check the spectral changes. For simplicity we assume here that the instants of changes in spectrum are known. So, the sampler swhiches to the adaptation phase each time there is a change in the spectrum of the input signal to detect the new support. results obtained are very promising. We can conclude from Fig. 10 that the Bartlett method used for support detection makes our proposed sampler same as the non-blind one as. However, it is noted that such convincing results are obtained with a very sparse multiband signal.

In order to show the exibility and adaptivity of the proposed DSB sampler, we consider that the spectrum of the signal x(t) changes with time. In other word the number of bands N varies from 1 to 7. All the results provided in At the start when the time instant is equal to 1, the samplers start with N = 6. Based on the signal spectrum characteristics, both MC and DSB samplers start sampling at f mc =f dsb = f dsb with α = 0.7 390.62MHz with p mc = p dsb = p dsb with α=0.7 = 50, (see Figs. 1112). Note that f mc , f dsb and f dsb with α=0.7 represent the average sampling frequencies for non blind MC, non blind DSB samplers and blind DSB with α = 0.7 respectively. When N changes from 6→4 at time instant 3, our DSB sampler adapts accordingly by recalculating the signal support using Bartlett method in the adaptation phase. The updated signal support is then fed to the non-uniform sampler block. As shown in Fig. 11, at N = 4 the new updated average sampling frequency for our DSB sampler reduces to f dsb 260.31MHz with p dsb 34. Since the MC sampler lacks from an adaptive feedback mechanism, it continues to sample at f mc 390.62MHz with p mc = 50 depicted in Fig. 12. Fig. 11, shows that as N decreases to 2, the f dsb is further reduced to 130.47MHz which is 13 % of f nyq = 1GHz with p dsb 9 shown in Fig. 12. On the other hand, MC sampler continues to operate at f mc 390.62MHz with p mc = 50 even for N = 2 what, clearly depicts a sub-optimal performance compared to the proposed DSB sampler. This shows that our system adapts itself dynamically and reduces f dsb which results in reduced number of incoming samples p dsb that are to be treated later. At the opposite, when N increases from 2 → 3, the proposed DSB sampler adapts accordingly by re-evaluating the new support through the Bartlett In Fig. 13, we have plotted the variation in RMSE, for the two samplers, corresponding to Figs.11-12, as N changes over time. If we closely analyze Fig. 13, we observe that RM SE mc is slightly better compared to RM SE dsb when N decreases from 6 → 4 → 2. This happens because the MC sampler uses much more samples than the proposed one. The most important variation in RM SE mc to observe is when N changes from 5→7. Fig. 12 shows that as N changes from 5→7, our DSB sampler adjusts p dsb to 56. While for the MC sampler, p mc remains equal to 50 because of the lack of adaptability and lack of availability of 50 branches at the input of the MC sampler. This results in a drastic increase in RM SE mc i.e. of the order of -5 dB as shown in Fig. 13. Even if we suppose that there is a feedback mechanism in the MC sampler to set p mc =56, it cannot adjust p mc because of the hardware limitations, i.e. the MC sampler used in our simulation has only p=50 branches. On the other hand, adjust p dsb =56 and the RM SE dsb remains low i.e. of the order of -21dB. Therefore, our DSB sampler adapts quickly to avoid any loss of data. The proposed Adaptive non uniform sampler uses a single ADC which can be switched o at predened instances and can theoretically adapt p dsb to any value up to 128 depending on the signal.

In Fig. 14, F avg-opt , represents the optimal average sampling rate of DSB sampler when operating in non blind mode i.e. having full information of the spectral support. When the DSB sampler operates in blind mode, F avg = F nyq represents the worst case and the ideal case is when F avg = F avg-opt . It should be noted that the results in Figs. 14-15, show the impact of false detection (P f a (K)), performed in the adaptation phase, on the average sampling rate achieved in the reconstruction phase. In Fig. 14 it can be shown than the SNR impacts the averaging sampling rate F avg of the DSB sampler. This happens because at low SNR the false alarm probability is high. Then the NUSSB shows the estimated spectral's estimated support K as larger than the actual support, this make the DSB sampler to sample at a higher, un-optimized sampling rate. This becomes more clear Fig. 15, where F avg is plotted as a function of P f a (K) for dierent α. It can be seen that with the increase of P f a (K), F avg increases, whatever α, and approaches F nyq at P f a (K) = 1.

Conclusion

In this paper we proposed a new smart sampling system for cognitive radio, called the Dynamic Single Branch non-uniform sampler. To ensure optimal reconstruction with a small number of samples, the DSB adapts its parameters according to the input signal's sparsity. Its operates in two phases, the adaptation phase and the reconstruction phase. In the adaptation phase, the proposed scheme senses the spectrum and adapts its sampling rate by computing the proposed non-uniform spectrum sensing technique. We have shown that the proposed sensing model works eciently and shows high detection and low false alarm probabilities. The performance of the spectrum sensing model improves by increasing the number of available non-uniform samples to the sensing Figure 15: Estimation average sampling rate Favg plotted against P f a (K) for dierent α method. As the sampling instances of our Non-uniform sampler depends to the position of this one on the Nyquist grid, then the average sampling rate depends on the number of bands contained in the signal. Furthermore, we have shown the eect of false detection on it. The DSB sampler performance has been compared to those of a traditional, non exible Multi-Coset architecture which contains severals parallel uniform samplers. We have shown that our system is signicantly more ecient than the conventional MC sampler when the spectrum of signal changes. The system can be designed with existing ADC and falls in the category of software dened radio system.

1 .

 1 Multi-band signal Model Let M(B) be the class of continuous real-valued signals with nite energy and band-limited to a subset B.

Figure 1 :

 1 Figure 1: Multi-Coset (MC) sampler implemented as a multi-channel system.

Figure 2 :

 2 Figure 2: Dynamic Single Branch Non-Uniform Sampler continues to gain popularity as it adapts intelligently to the radio environment,95

Figure 3 :

 3 Figure 3: L uniformly spaced Nyquist samples and corresponding p DSB samples.

130L

  like the regular MC sampler. But unlike the regular MC sampler, the DSB sampler uses a non uniform sampler witch samples only some portions of the received signal. It can be possible to switch o the ADC on a predened instances to reduce the power consumption. Most of the existing non uniform digitizer use a uniform clock as a reference clock, and have some circuitry which 135 transforms the uniform clock to a non uniform one [? ? ].

Figure 4 :Figure 5 :

 45 Figure 4: Non-uniform spectrum sensing model

Figure 6 :

 6 Figure 6: Multi-Coset Reconstruction block

Figure 7 :

 7 Figure 7: P d (K) plotted against varying β at SNR = 10 dB

Figure 8 :

 8 Figure 8: P f a(N ) and P f a(K) plotted against varying SNR for α = 0.5 to α = 0.7.

Figure 9 :

 9 Figure 9: RMSE as function of Condition number of A C (K) (no noise)

Fig. 10 ,

 10 Fig.10, shows that the performance of our proposed blind DSB sampler completely matches with the non-blind MC sampler in terms of RMSE. The

Figure 10 :

 10 Figure 10: Comparison between our proposed blind DSB sampler with non-blind DSB sampler in terms of RMSE.

Figs. 11 -

 11 [START_REF] Liang | A frequency-scalable 15-bit incremental adc for low power sensor applications[END_REF][START_REF] Rashidi | A wideband spectrum sensing method for cognitive radio using sub-nyquist sampling[END_REF], are for the scenario where the number of bands N changes from6 → 4 → 2 → 1 → 3 → 5 → 7 → 6.For simplicity we assume that the change in N occurs after every 2 time instants i.e. at time instants 1 and 2, N = 6, for time instants 3 and 4, N = 4 and N and so on. The evolution of N with time is indicated at the top of each of the Figs.11-12-13.

Figure 11 :

 11 Figure 11: Variation in f mc and f dsb as N changes over time at an SNR = 50dB.

Figure 12 :

 12 Figure 12: Variation in pmc and p dsb as N changes over time at an SNR = 50dB

Figure 13 :

 13 Figure 13: Variation in RM SEmc and RM SE dsb as N changes over time at an SNR = 50dB

Figure 14 :

 14 Figure 14: Average sampling estimation rate Favg of DSB sampler while operating in blind mode plotted for dierent α

Table 1 :

 1 .8 Giga operations per second. It is shown in [? ] a processor which Number of real operation in Multi-Coset Reconstruction block

	Multiplication	Addition


Guillaume Costeseque
email: guillaume.costeseque@inria.fr

Aurélien Duret
email: aurelien.duret@ifsttar.fr

Mesoscopic multiclass traffic flow modeling on multi-lane sections

Keywords: mesoscopic, multiclass, multilane, Hamilton-Jacobi, FIFO, diverge

1 Introduction

Motivation and quick literature review

Dynamic traffic flow models are an essential building block for many applications including traffic flow monitoring, estimation and control. Previous works have already shown that simulation of traffic flow on a large scale network thanks to such dynamic models can be cumbersome, despite a lot of research effort made in that direction in the past decades. Classical dynamic models embed microscopic and macroscopic models. Microscopic traffic models allow a detailed representation of traffic thanks to an individual tracking of vehicles but at a high computational cost and with obvious calibration difficulties, depending on the number of vehicles that are simulated. Macroscopic models and in particular the celebrated first order LWR model [START_REF] Lighthill | On kinematic waves II. A theory of traffic flow on long crowded roads[END_REF][START_REF] Richards | Shock waves on the highway[END_REF] provide a robust and easy way to compute traffic flow on networks but they generally fail to recapture some meaningful phenomenon like for instance the capacity drop, the bounded acceleration of vehicles or the stopand-go waves, without being specifically adapted. For a review of microscopic and macroscopic traffic models, the interested reader is referred to [START_REF] Bellomo | On the modeling of traffic and crowds: A survey of models, speculations, and perspectives[END_REF]. One interesting microscopic car-following model for us is Newell's one [START_REF] Newell | A simplified car-following theory: a lower order model[END_REF].

As underlined in [START_REF] Leclercq | A meso LWR model designed for network applications[END_REF], there exists a third class of traffic flow models that are well adapted for network applications. These models are called mesoscopic because while being consistent with macroscopic traffic flow rules, they also allow a vehicular description, say a microscopic description. The mesoscopic version of the LWR model can be also interpreted into the Hamilton-Jacobi (HJ) framework as it was first properly established in [START_REF] Laval | The Hamilton-Jacobi partial differential equation and the three representations of traffic flow[END_REF]. See also references therein about the three dimensions representation of traffic (Eulerian for t -x framework, Lagrangian for t -n one and Lagrangian-space for n -x one) and the variational techniques that can be used to solve such models.

To enrich the representation of macroscopic models, which basically use a one-lane aggregation, it is necessary, for real-world applications, to consider the variety of road users and of their behaviors.

Some efforts have been made in that direction since the beginning of the 2000's by introducing multiclass and multilane (MCML for short) features into classical models such as kinetic ones [START_REF] Hoogendoorn | Continuum modeling of multiclass traffic flow[END_REF] or as the seminal LWR model in Eulerian [START_REF] Wong | A multi-class traffic flow model-an extension of LWR model with heterogeneous drivers[END_REF] or Lagrangian coordinates [START_REF] Van Wageningen-Kessels | Multi-class continuum traffic flow models: Analysis and simulation methods[END_REF]. For an accurate overview of these MCML models, the reader is referred to [START_REF] Van Wageningen-Kessels | Multi-class continuum traffic flow models: Analysis and simulation methods[END_REF][START_REF] Van Wageningen-Kessels | Genealogy of traffic flow models[END_REF].

Another strategy to take into account the MCML features is to generalize the moving bottleneck (MB) problem for which the users are classically divided into a fast class, say the "rabbits", and a slower one, called the "slugs" (see [START_REF] Daganzo | A behavioral theory of multi-lane traffic flow. Part I: Long homogeneous freeway sections[END_REF] for the origin of these terms). The MB theory is now well structured since the pioneering works [START_REF] Gazis | The moving and phantom bottlenecks[END_REF][START_REF] Newell | A moving bottleneck[END_REF]. See for instance [START_REF] Laval | The Hamilton-Jacobi partial differential equation and the three representations of traffic flow[END_REF] for one description of the MB problem in both mesoscopic and HJ framework.

The aim of this paper is precisely to revisit the MCML problem on a network by gathering the main advantages of all the techniques that have been developed separately up to now. As an application, we are concerned with a diverge for which we introduced a relaxation parameter of the First-In-First-Out (FIFO) rule which is in some cases violated or not really realistic as pointed out in the literature [START_REF] Munoz | Experimental characterization of multi-lane freeway traffic upstream of an off-ramp bottleneck[END_REF][START_REF] Munoz | The bottleneck mechanism of a freeway diverge[END_REF] but which is still widely assumed in traffic engineering works for the sake of simplicity. Our guiding example is a diverge whenever at most one exit branch is congested.

Organization of the paper

The remaining of the paper is structured as follows: in Section 2, some basic theoretical elements which serve us as building blocks for our model, are recalled. Then in Section 3, the new MLMC model is introduced and the adapted numerical scheme is presented in Section 4. In Section 5, some numerical examples are provided. Finally, in Section 6, the main results are wrapped up and some future directions of research are pointed out.

2 Recap of some useful concepts

Mesoscopic modeling

The mesoscopic resolution of the well-known LWR model proposed in [START_REF] Leclercq | A meso LWR model designed for network applications[END_REF] allows to compute the passing times of vehicles at a certain number of specific points of the network (the intersections for instance): it is thus called an event-based model keeping track of vehicles. The name "mesoscopic" came from the fact that the approach combines vehicular description with macroscopic behavioral rules. It is noteworthy that this is not related to gas-kinetic models. This n -x representation is fully consistent with the macroscopic (Eulerian) framework, say the original LWR model, and with the microscopic (Lagrangian) framework, say microscopic car-following models. The mesoscopic resolution is able to track vehicles on the network, which is an advantage compared to previous Eulerian approach resolution (for instance for network application). The mesoscopic resolution presents a calculation time that only depends on the number of events during the simulation which is again an advantage compared to the previous microscopic resolution for which the calculation time increases when the network is congested.

Following [START_REF] Laval | The Hamilton-Jacobi partial differential equation and the three representations of traffic flow[END_REF], let us introduce the headway h (time gap between two successive vehicles), the pace r (travel time per space unit) and the spacing s (spatial gap between two successive vehicles). One has the following relations

q = 1 h , v = 1 r , k = 1 s
, q = kv, and h = rs where q, v and k denote respectively the flow, the speed and the density.

We now define the Lagrangian variable as

n = N(t, x) := ∞ x k(t, ξ)dξ
where N(t, x) gives us the label of the vehicle located at position x at time t. We are now ready to introduce T (n, x) the passing time of vehicle n at location x as follows

T (n, x) := ∞ x r(n, ξ)dξ.
One can verify that

∂ t N = q, (flow) ∂ x N = -k, (density)
, and

∂ n T = h, (headway) ∂ x T = r, (pace)
We recall that the first order LWR model [START_REF] Lighthill | On kinematic waves II. A theory of traffic flow on long crowded roads[END_REF][START_REF] Richards | Shock waves on the highway[END_REF] is given by the following scalar conservation law

∂ t k + ∂ x Q(k) = 0 (1)
with k(t, x) the density and Q : R → R the flow-density Fundamental Diagram (FD). One has

Q (k(t, x)) := k(t, x)V (k(t, x))
where V : R → R denotes the speed-spacing FD.

1

Applying the above change of coordinates, (1) becomes in Lagrangian-space framework

∂ n r -∂ x h = 0 (2)
where we assume that the headway is a function of the pace h = H(r).

2

The variational theory, say the Hamilton-Jacobi (HJ) formulation of the conservation law (2), teaches us that if r = ∂ x T solves (2), then T satisfies the following Hamilton-Jacobi equation

∂ n T -H (∂ x T ) = 0, for (n, x) ∈ [0, +∞) × [0, +∞). (3)
Consider the following boundary conditions

T (n, 0) = ǧ(n), for any n ∈ [0, +∞), T (0, x) = G(x), for any x ∈ [0, +∞). (4)
For sake of simplicity, assume here that we consider a one-lane section (N = 1). If moreover we consider a triangular Fundamental Diagram (also called Hamiltonian) H,

H(r) =    1 κ r + 1 wκ , if r ≥ 1 u , +∞, otherwise, (5)
then the representation formula known as the Lax-Hopf formula, for the associated HJ equation ( 3)-(4) for which T (n, 0) for any n (passing times of all vehicles at upstream position) and T (0, x) for any x (passing times of the first vehicle) are known and piecewise constant,

T (n, x) = sup (n 0 ,x 0 ,X(•)) s.t. X(n 0 )=x 0 , X(n)=x T (n 0 , x 0 ) + n n 0 H ⊠ -Ẋ(ξ) dξ ,
simply boils down to a maximization problem between two terms (see [START_REF] Laval | The Hamilton-Jacobi partial differential equation and the three representations of traffic flow[END_REF]):

T (n, x) = max T (n, 0) + x u , T (0, x + nσ) + nτ , (6)
with u denotes the free flow speed, σ := 1 κ the jam spacing (or minimal spacing) and τ := 1 wκ the wave trip time between two consecutive vehicles in congestion (recall that w > 0). Indeed, the concave Fenchel transform H ⊠ of H is given by

H ⊠ (s) := inf p∈Dom(H)

{H(p) -ps}

and if H is given by [START_REF] Leclercq | A meso LWR model designed for network applications[END_REF], then it follows

H ⊠ (s) =    1 κ 1 u - 1 w - s u , if s ≥ 1 κ , -∞, otherwise.
See also [START_REF] Jin | Some Theoretical and Practical Perspectives of the Travel Time Kinematic Wave Model: Generalized Solution, Applications, and Limitations[END_REF] for explicit (generalized) Lax-Hopf formula to compute the solution of (3) with the assumption of a piecewise affine Hamiltonian and under piecewise affine internal boundary conditions.

Moving bottleneck theory

Following [START_REF] Laval | The Hamilton-Jacobi partial differential equation and the three representations of traffic flow[END_REF][START_REF] Gazis | The moving and phantom bottlenecks[END_REF][START_REF] Newell | A moving bottleneck[END_REF], a moving bottleneck is defined as a vehicle (or a platoon of vehicles) defining by itself a class of vehicles, for instance trucks or buses, which has a maximal free flow speed which is lower than the free flow speed of its immediate following vehicle (or immediate following platoon of vehicles).

A moving bottleneck is said to be active if and only if it generates queues at upstream positions with respect to the moving bottleneck and if the upstream state is different from the downstream state.

We just recall the notations and the main steps for the computation of the passing rate (say the flow that can overtake the moving bottleneck) in the Eulerian framework (see Figure 1). Let us denote by:

• ξ N (t) the trajectory of the moving bottleneck in time-space plane t -x,

• ξ T (n) the location where the vehicle n crosses the bottleneck in Lagrangian-space plane n -x,

• v B (t) := ξN (t) the moving bottleneck speed which is upper bounded by v b ,

• k D := k (t, ξ N (t) + ) the density downstream of the bottleneck,

• q D := Q(k D ) the flow downstream of the bottleneck,

• R(v B , q D ) the maximum bottleneck passing rate

R(v B , q D ) := q D -k D v B . Notice that ξT (n) = v B R(v B , q D ) . v B -w -w v B k Q (N -1) lanes N lanes R(v B , q D ) (D) (U ) ξ N (t) x (U ) (D) v B k D (N -1)κ N κ q D N C Q * (v B ) u FIGURE 1
       ∂ t k + ∂ x (Q(k)) = 0, for any t > 0, x ∈ R, Q(k(t, ξ N (t))) -ξN (t)k(t, ξ N (t)) ≤ αQ * ξN (t) , for any t > 0, ξN (t) = min {v b , V (k(t, ξ N (t) + ))} , for any t > 0, (7)
together with the following initial conditions

k(0, x) = k 0 (x), on R, ξ N (0) = ξ 0 . (8)
In [START_REF] Hoogendoorn | Continuum modeling of multiclass traffic flow[END_REF], the PDE leads the evolution of the density of vehicles while the ODE gives us the trajectory of the moving bottleneck. The coupling condition is the constraint imposed by the moving bottleneck on the traffic flow. For a complete presentation of the model in Eulerian coordinates, the reader is referred to [START_REF] Delle Monache | Scalar conservation laws with moving constraints arising in traffic flow modeling: an existence result[END_REF] and references therein.

The coefficient α ∈]0, 1[ in [START_REF] Hoogendoorn | Continuum modeling of multiclass traffic flow[END_REF] gives the reduction rate of the road capacity due to the presence of the moving bottleneck. It is indeed the ratio of available lanes for the overtaking maneuvers. If N denotes the number of lanes on the road section and if the moving bottleneck is located on the rightmost lane, then

α = N -1 N .
The upper bound on the passing rate in [START_REF] Hoogendoorn | Continuum modeling of multiclass traffic flow[END_REF] is defined as a function of the moving bottleneck speed

v B (t) → Q * (v B (t))
and it matches the Legendre-Fenchel transform of the flow-density FD Q (for the N lanes)

Q * (v) := sup ρ∈Dom(Q) {Q(ρ) -vρ} .
The Eulerian model ( 7) can be easily extended to the Lagrangian-space framework. It will be presented in the next section for our MCML application. The full solution details for the mesoscopic LWR model are given in Section 5.2 of [START_REF] Laval | The Hamilton-Jacobi partial differential equation and the three representations of traffic flow[END_REF]. However, it is noteworthy that in any case (say if the vehicle is constrained by the downstream traffic conditions -the moving bottleneck is inactiveor if the vehicle is constrained by the moving bottleneck -the moving bottleneck is active-), the passing time of a vehicle n at position x of the network is determined as the maximum between the passing time of vehicle n at upstream position (x -∆x) plus the travel time obtained at free flow speed on the section [x -∆x, x] (or eventually reduced due to the active moving bottleneck)

and the passing time of the leader vehicle (n -1) at downstream position (x + ∆x) translated by the jam speed.

It is worth noticing that all the material of this Section 2 is extracted from existing works. These results will served as the fundamental background for our building blocks that are presented in next Section 3.

3 Mesoscopic formulation of multiclass multilane models

Choice of the modeling

For tackling the multi-class and multi-lane modeling (whatever the network is), we can consider:

• either generic multi-class models (see for instance [START_REF] Van Wageningen-Kessels | Genealogy of traffic flow models[END_REF]) that classically fall into the Generic Second Order Model (GSOM) family [START_REF] Lebacque | Generic second order traffic flow modelling[END_REF] with a class-dependent fundamental diagram. In this case, on can define the vehicular attribute as the class of the considered vehicle.

• or the moving bottleneck theory thanks through a coupled system of conservation laws (or equivalently of Hamilton-Jacobi equations) extending what was previously done in the literature (see for instance [START_REF] Delle Monache | Scalar conservation laws with moving constraints arising in traffic flow modeling: an existence result[END_REF]).

In the first case, we cannot use the strength of the mesoscopic formulation since we have to define the indices of the vehicles, once and for all, and it is not straightforward to deal with the re-labeling of the vehicles once we consider overtaking maneuvers between each class.

In the second case, each class has its own labeling and by assuming that inside a class the vehicles respect the FIFO rule, they stay ordered at any time. Apart from this fundamental difference, both approaches are quite close since we consider a fundamental diagram (or Hamiltonian) which depends on an "attribute" denoted by I and defined as the vehicle class and the lane (which is the case in the remaining of the paper) or the Origin-Destination pattern or a combination of both.

Modeling assumptions

Consider a stretch of road [x 0 , x 1 ] composed by N separate lanes with traffic stream composed of fast and slow vehicles defining two classes of users called respectively "rabbits" (I = I 1 = 1) and "slugs" (I = I 2 = 2). In all that follows, notice that we will use indifferently the terms rabbits or fast vehicles and slugs or slow vehicles.

It is assumed that one vehicle cannot belong simultaneously to more than one class but its class can evolve with respect to time (according to the traffic state or to the mental stress of the driver or to its path on the network for instance). In the remaining, the class attribute is assumed to not depend on time.

Slugs cannot go to the passing lane(s) and are limited to the shoulder lane, while the fastest vehicles can use the passing lane(s) and the shoulder lane.

Let us introduce the class-dependent headway-pace FD (or Hamiltonian)

H : (r, I) → H(r, I)
for a given class attribute I ∈ {1, 2}.

(A0) We assume that r → H(r, •) is convex.

Capacity drop parameter

One of the main drawback of the LWR model (whatever the chosen resolution is) is that the capacity drop phenomenon is not represented. Capacity drop can be characterized as a reduction in discharge flow after queue formation that is observed at the downstream of an active (moving) bottleneck location. Basically this difference of flow is interpreted as a recovery flow from low speeds (vehicles stuck in a congestion) to free flow (vehicles accelerating). Below, we describe a strategy to take into account this well-known feature of traffic flows.

Capacity drop and moving bottleneck

Consider a parameter δ ∈ [0, 1] that tells us how much the traffic flow complies to the First-In-First-Out (FIFO) rule, with the two extreme cases:

• If δ = 0, the traffic is strictly non-FIFO, say all the vehicles can overtake each other without any restriction, say they can use the entire residual headway h B of the passing lane(s).

• If δ = 1, the traffic is strictly FIFO, say the vehicles keep ordered at all time, independently of their origin and/or destination and even if there are some available passing lane(s). The vehicles behave as if the residual headway is equal to zero.

An intermediate value δ ∈]0, 1[ can be interpreted as a penalty for overtaking maneuvers by reducing the residual headway of the passing lane(s) (say the available headway for vehicles arriving on the closest moving bottleneck and that want to overtake it) to a modified residual headway (say the real available headway for vehicles that are going to overtake)

hB := 1 1 -δ h B . (U) (D) (U) Case FIFO Case non-FIFO (U) (D) Flow (MB) v B R(v B , q D ) δ = 1 δ = 0 r H 1 κ 1 u r B = 1 v B h B = 1 q D 1 R(v B , q D ) v B 1 C 1 wκ FIGURE 2
Moving bottleneck graphical solution in Lagrangian-space coordinates, for a triangular Hamiltonian or headway-pace FD H(r) defined in (5) with r the pace, and for different values of the capacity drop parameter δ. (U) and (D) denote respectively the upstream and downstream traffic state with respect to the moving bottleneck position.

In Figure 2, one can notice that with a strict non-FIFO behavior, whenever the moving bottleneck is active, the upstream traffic state is different from the downstream traffic state. Conversely, if one assumes a strict FIFO behavior, all the vehicles upstream have to adopt the moving bottleneck

speed v B .
Interestingly, our approach finds a different application in a work by Laval [19] that deals with the modeling of two-lane rural roads. In this framework, the opposite traffic flow acts as a reduction of the ability of vehicles to overtake a moving bottleneck. This is very similar to our capacity drop parameter δ. Still in [START_REF] Laval | A macroscopic theory of two-lane rural roads[END_REF], Laval gives the analytical expression of the travel time as a function of the proportion of slow vehicles.

Relaxing the FIFO assumption at a diverge

The parameter δ introduced in the previous subsection for road sections, can be extended to spatial discontinuities of the network. In this paper, we are particularly concerned with diverges. In such a case, introducing the parameter δ is equivalent to relax the FIFO assumption.

Let us consider a diverge composed of one incoming road that divides into J outgoing branches (with J ≥ 2). The original model of Daganzo [START_REF] Daganzo | The cell transmission model, part II: network traffic[END_REF] (equivalent to the first diverge model used in [START_REF] Newell | A simplified theory of kinematic waves in highway traffic, part I-III[END_REF] and also to the one in [START_REF] Lebacque | The Godunov scheme and what it means for first order traffic flow models[END_REF], see [START_REF] Ni | A simplified kinematic wave model at a merge bottleneck[END_REF] for a detailed review of the literature on diverge models) relies on the FIFO assumption that is reasonable for the following traffic cases:

• the traffic state is in free flow condition on all downstream branches,

• the traffic state is congested on all downstream branches. Nonetheless, as soon as at least one downstream branch is in free flow condition while another downstream branch is congested, the FIFO assumption is no longer realistic.

The idea is then to introduce a relaxation parameter δ ∈ [0, 1] that represents the percentage of vehicles that can go through the diverge in direction of the exiting branch, even if the node is congested. This parameter is exactly the same than the capacity drop parameter introduced in the previous section. In the case of a diverge, this parameter allows us to relax the FIFO assumption upstream to the diverge.

Let us denote by

• ∆ 0 , the upstream demand,

• ∆ j , the proportion of the demand that wants to exit on downstream branch j, ∆ j := γ j ∆ 0 , for any j,

• Σ j , the downstream supply on branch j,

• C j , the capacity of branch j, say the maximal flux that can be passed on the road,

• γ j , the proportion of the flow that goes in direction of branch j. Note that one has j γ j = 1 and without loss of generality, we assume that γ j > 0 for any j.

• q 0 , the flow on the unique upstream branch and q j the flow on downstream branch j. We have that for any j      q j = γ j q 0 , (Flow distribution)

q j ≤ Σ j ≤ C j , ( Supply constraint)
There are several cases to distinguish:

1. If no upstream demand exceeds its downstream supply, say ∆ j ≤ Σ j for any j or equivalently

∆ 0 ≤ min j Σ j γ j
, then the solution is provided by the classical Daganzo model q 0 = ∆ 0 , q j = ∆ j = γ j ∆ 0 = γ j q 0 , for any j.

2. If all upstream demands exceed the downstream supplies, say ∆ j > Σ j for any j or equiv-

alently ∆ 0 > max j Σ j γ j
, then the solution is (also) provided by the classical Daganzo

model    q 0 = j Σ j < j ∆ j = ∆ 0 , q j = Σ j < ∆ j ,
for any j.

3. If at least one upstream demand exceeds its downstream supply, but one of the downstream branch is in free flow condition, say min

j Σ j γ j < ∆ 0 < max j Σ j γ j
, we denote by J * the set of all downstream branches that are in free flow condition and we set

q := min k Σ k γ k .
Then the solution is provided by the modified model (with the parameter δ ∈ [0, 1])

             q 0 = δ q + (1 -δ)   j / ∈J * Σ j + j∈J * ∆ j   ,
q j = δγ j q + (1 -δ)Σ j , for any j / ∈ J * , q j = δγ j q + (1 -δ)∆ j , for any j ∈ J * .

For a full-non FIFO model (δ = 0), we get

         q 0 = j / ∈J * Σ j + j∈J * ∆ j < j ∆ j = ∆ 0 , q j = Σ j ,
for any j / ∈ J * , q j = ∆ j , for any j ∈ J * , while taking δ = 1 gives us the classical FIFO model    q 0 = q < j ∆ j = ∆ 0 , q j = γ j q < ∆ j , for any j.

All the details are given for flows for the reader convenience because that the Eulerian framework is the most well-known in the community up to now. However, notice that all the approach can be conversed to headway without any difficulty.

Expression of the MCML model

In this subsection, we finally summarize the previous contributions (old -Section 2-and new -Section 3-) into a unique multiclass and multilane mesoscopic model, expressed in Lagrangianspace coordinates.

Assume that each class satisfies to a LWR model with a specific fundamental diagram that depends on the attribute I. Consider two headway-pace fundamental diagrams H(•, I 1 ) and H(•, I 2 ). We assume that the vehicles belonging to class 1 are faster than the ones of class 2. Thus, one has u 1 > u 2 (where we recall that u i is the top speed of class i). In the discrete application presented in next Section, we consider that H are piecewise linear. See [START_REF] Daganzo | A behavioral theory of multi-lane traffic flow. Part I: Long homogeneous freeway sections[END_REF] for an example of the chosen Hamiltonians. In that setting, the maximal speeds of each class are different but the congestion wave and the jam density are conserved.

Moreover, we define the concave Legendre-Fenchel transform of the class-specific Hamiltonian H(•, •) with respect to its first variable, as follows

H ⊠ (s, I) := inf r∈Dom(H(•,I))
{H(r, I) -sr} , for any (s, I) ∈ R × R.

Let (n 0 , x 0 , x 1 ) ∈ R 3 be fixed such that x 0 < x 1 . The complete model in HJ framework writes as a system of coupled PDEs

                 ∂ n T 1 -H (∂ x T 1 , I 1 ) = 0, for (n, x) ∈ [n 0 , +∞) × [x 0 , x 1 ], ∂ n T 2 -H (∂ x T 2 , I 2 ) = 0, for (n, x) ∈ [n 0 , +∞) × [x 0 , x 1 ], H (∂ x T 1 (n, ξ(n)), I 1 ) -(1 -δ) ξ (n * 2 ) ∂ x T 1 (n, ξ(n)) ≥ 1 α H ⊠ (1 -δ) ξ (n * 2 ) , I 1 for n ∈ [n 0 , +∞), T 2 (n, ξ(n)) ≥ T 1 (n * 1 , ξ(n)) + H (∂ x T 1 (n * 1 , ξ(n)), I 2 ) , for n ∈ [n 0 , +∞), (9) where n
* 2 := argmax T 2 (•, ξ(n)) and ξ(n) = ξ(n * 2 ), for n ∈ [n 0 , +∞), n * 1 := argmax T 1 (•, ξ(n)) for n ∈ [n 0 , +∞),
and with the following mixed Neumann-Dirichlet boundary conditions

     ∂ n T i (n, x 0 ) = ǧi (n), on [n 0 , +∞), ∂ n T i (n, x 1 ) = ĝi (n), on [n 0 , +∞), T i (n 0 , x) = G i (x), on [x 0 , x 1 ], for i ∈ {1, 2} . ( 10
) (A1) Assume that ǧi , ĝi ∈ C 0 (R) and G i ∈ C 1 (R).
In [START_REF] Van Wageningen-Kessels | Multi-class continuum traffic flow models: Analysis and simulation methods[END_REF], the first two equations express the conservation of vehicles in Lagrangian-space coordinates and inside each vehicle class (independently of the other one).

Assume that n * i (i ∈ {1, 2}) denotes the nearest leader from class i for vehicle n of class j = i (j ∈ {1, 2}). It is computed as the last vehicle that has passed through the current location of vehicle n.

The third equation is the coupling condition from the point of view of class 1, say the moving bottleneck condition whenever a vehicle of class 1 catches a slower vehicle of class 2, whose label is assumed to be n * 2 . This inequality gives a lower bound on the modified passing headway. Due to the capacity drop parameter δ ∈ [0, 1[ (which also stands for the relaxation of the FIFO assumption on a diverge), the modified passing headway H is determined as follows

H = 1 1 -δ h B - ξT (n * 2 ) u 1 where h B = 1 (N -1)C
is the residual headway available up to the moving bottleneck since there remains only (N -1) passing lanes (see Figure 3). This bound is defined as the Legendre-Fenchel transform H ⊠ of the Hamiltonian H(•, I 1 ) applied to ξT (n * 2 ) the spacing imposed by the current moving bottleneck.

1 u r 1 (N -1)κ ξ(n * 2 ) ξ(n * 2 )
N lanes

(N -1) lanes 1 N κ h B = 1 (N -1)C r B 1 N C 1 N wκ H * ξ(n * 2 ) h B - ξ(n * 2 ) u H FIGURE 3
Moving bottleneck treatment in Lagrangian-space coordinates for a triangular Hamiltonian or headway-pace FD H(r) defined in [START_REF] Daganzo | A behavioral theory of multi-lane traffic flow. Part I: Long homogeneous freeway sections[END_REF] with r the pace. (U) and (D) denote respectively the upstream and downstream traffic state with respect to the moving bottleneck position. For the sake of readability, the capacity drop parameter δ is not represented here.

The fourth equation stands for the coupling condition from the point of view of class 2. We assume that if a vehicle of class 2 catches a vehicle of class 1, then it behaves according to a "follow-theleader" rule. Indeed, class 2 is restricted to the shoulder lane and cannot go on the passing lane(s).

Moreover, such a situation may happen if the nearest leader of class 1 is stuck in a congestion and it makes no sense that vehicle of class 2 can overtake. The coupling conditions are thus asymmetric between classes 1 and 2.

Remark 3.1 (Multilanes modeling)

The multilane behaviour of the model ( 9) is not straightforward at this (continuous) stage. In reality, in the discrete approach, we will use the attribute as a vector in R 2 containing the information about the class of the vehicle and about the lane on which the vehicle is circulating. The lane is updated for each vehicle that can overtake. In a continuous model, it is actually not clear how to express this lane-changing behavior apart from saying that İ(n, x) = ϕ(x) for a given ϕ.

Numerical scheme 4.1 Setting of the IBV Problem

Let us introduce the finite steps ∆n, ∆x > 0.

Assume that we consider a set of n max vehicles which are divided into k max platoons of ∆n vehicles (obviously n max = n 0 + k max ∆n). It is noteworthy that for the sake of clarity, we consider that the label step ∆n is the same whatever the vehicle class is. These vehicles are moving on a link [x 0 , x 1 ] divided into l max sections which length is denoted by ∆x (obviously

x 1 = x 0 + l max ∆x).
We adopt the following notation

n k := n 0 + k∆n, for any k ∈ {1, . . . , k max } , x l := x 0 + l∆x,
for any l ∈ {1, . . . , l max } .

Consider the IBV problem ( 9)-( 10) on the computational domain

[n 0 , n max ] × [x 0 , L].
Assume that the class-specific Hamiltonians H are triangular, say for any i ∈ {1, 2},

H(r, I i ) =    1 κ N(I i ) r + 1 w , if r ≥ 1 u i , +∞ , otherwise, (11)
where u i := u(I i ) denotes the free-flow speed of the class i and N(I i ) stands for the number of lanes that are accessible for the class i

N(I i ) := N, for i = 1, 1, for i = 2.
Below, we make precise the definitions of the initial and boundary values ǧi , ĝi , G i for i ∈ {1, 2}. We define the piecewise affine (Eulerian) upstream boundary condition for the Hamilton-Jacobi equation ( 9) as follows

T i (n K , x 0 ) = T i (n 0 , x 0 ) + K k=1 h i,k ∆n, for any i ∈ {1, 2}
with the headway values

h i,k = 1 q i,k for any k ∈ [1, k max ] and i ∈ {1, 2} .
We assume moreover that for any k ∈ [1, k ′ max ] and i ∈ {1, 2}

h i,k = 1 ∆n n k n k-1 ǧi (η) dη holds true.
i,k ′ } k ′ ∈[1,k ′ max ]
for each group of ∆n vehicles of class i ∈ {1, 2}, at a fixed (and given) location x 1 (thanks to a loop detector for instance).

We define the piecewise affine (Eulerian) downstream boundary condition for the Hamilton-Jacobi equation ( 9) as follows

T i (n K ′ , x 1 ) = T i (n 0 , x 1 ) + K ′ k=1 ĥi,k ∆n, for any i ∈ {1, 2}
with the headway values

ĥi,k = 1 qi,k , for any k ∈ [1, k ′ max ] and i ∈ {1, 2} .
We assume moreover that for any We define the piecewise affine (Lagrangian) boundary condition for the Hamilton-Jacobi equation ( 9) as follows

k ∈ [1, k ′ max ] and i ∈ {1, 2} ĥi,k = 1 ∆n n k n k-1 ĝi (η) dη holds true.
T i (n 0 , x L ) = T i (n 0 , x 0 ) + L l=1 r i,l ∆x, for any i ∈ {1, 2}
with the pace values

r i,l = 1 v i,l
for any l ∈ [1, l max ] and i ∈ {1, 2} .

We assume moreover that for any l ∈ [1, l max ] and i ∈ {1, 2}

r i,l = 1 ∆x x l x l-1 G ′ i (ξ) dξ holds true.

Lax-Hopf formulae for the MCML model

Representation formulae for PDEs [START_REF] Evans | Partial differential equations[END_REF] such as the Lax-Hopf formula for the Hamilton-Jacobi equation [START_REF] Hopf | Generalized solutions of non-linear equations of first order[END_REF][START_REF] Lax | Hyperbolic systems of conservation laws II[END_REF], enable a fast and exact computation of solutions for the associated PDE.

For instance, Lax-Hopf formula for the LWR model in traffic flow has been used in the Eulerian framework in [START_REF] Claudel | Lax-Hopf based incorporation of internal boundary conditions into Hamilton-Jacobi equation[END_REF][START_REF] Claudel | Lax-Hopf based incorporation of internal boundary conditions into Hamilton-Jacobi equation[END_REF][START_REF] Mazaré | Analytical and grid-free solutions to the Lighthill-Whitham-Richards traffic flow model[END_REF], in the Lagrangian framework in [START_REF] Han | Lagrangian-based hydrodynamic model: Freeway traffic estimation[END_REF] and in the Lagrangian-space framework in [START_REF] Jin | Some Theoretical and Practical Perspectives of the Travel Time Kinematic Wave Model: Generalized Solution, Applications, and Limitations[END_REF]. For a synthetic review of these formulae, the interested reader is referred to [START_REF] Laval | The Hamilton-Jacobi partial differential equation and the three representations of traffic flow[END_REF].

Then for any (n, x) ∈ [n 0 , n max ] × [x 0 , x 1 ], consider the following Lax-Hopf like formulae (so to speak, here we use a Bellman dynamic programming principle)

T 1 (n, x) = max T 1 (n, x -∆x) + ∆x u 1 , T 1 (n -∆n, x + ∆x) + ∆x w , T 2 (n * 2 , x) + 1 1 -δ h B with n * 2 := argmax T 2 (•, x) (12)
and

T 2 (n, x) = max T 2 (n, x -∆x) + ∆x u 2 , T 2 (n -∆n, x + ∆x) + ∆x w , T 1 (n * 1 , x) + H T 1 (n * 1 , x) -T 1 (n * 1 , x -∆x) ∆x , I 2 with n * 1 := argmax T 1 (•, x) (13)
with the condition ∆x = ∆n κ .

We recall that w > 0 denotes the wave speed, n * i is the label of the last vehicle of type I = I i which has passed through position x and h B is the residual headway available on the remaining lanes, when crossing the moving bottleneck.

Some mathematical results

By generalizing the methodology proposed in [START_REF] Delle Monache | Scalar conservation laws with moving constraints arising in traffic flow modeling: an existence result[END_REF] for the moving bottleneck problem [START_REF] Hoogendoorn | Continuum modeling of multiclass traffic flow[END_REF] in Eulerian coordinates, one can show that Proposition 4.4 (Existence of a continuous solution to (9)-( 10)) Assume that r → H (r, I i ) is convex for any i ∈ {1, 2}. Then, the IBV problem ( 9)-( 10) admits a solution

(T 1 , T 2 ) continuous on [n 0 , +∞] × [x 0 , x 1 ].
The proof is out of the scope here. We only assume that this result holds true.

We are now interested in showing the following result: Proposition 4.5 (Convergence) Assume that r → H (r, I i ) is convex for any i ∈ {1, 2}. Then the numerical solution (T ε 1 , T ε 2 ) of the scheme ( 12) and ( 13) converges towards a continuous solution (T 1 , T 2 ) of the IBV problem ( 9)- [START_REF] Van Wageningen-Kessels | Genealogy of traffic flow models[END_REF] when ε := (∆n, ∆x) goes toward zero.

Proof For the sake of space, we only give the sketch of the proof.

It is worth noticing that the first two terms in the maximum in [START_REF] Gazis | The moving and phantom bottlenecks[END_REF] and ( 13) are strictly similar to the classical case, say the Lax-Hopf formula (6) for solving the Hamilton-Jacobi equation (3). Indeed, these two terms stand for the usual mesoscopic resolution of the LWR model. The convergence of these two terms is of no concern for us, as the Lax-Hopf formula ( 6) has already been proved

to give an explicit solution to the Hamilton-Jacobi equation [START_REF] Bellomo | On the modeling of traffic and crowds: A survey of models, speculations, and perspectives[END_REF]. The interested reader is referred to [START_REF] Laval | The Hamilton-Jacobi partial differential equation and the three representations of traffic flow[END_REF].

The novelty is the third term that appears in the max in ( 12) and ( 13). In each case, we have to ensure that these terms satisfy the coupling conditions between both mesoscopic LWR models. If the coupling conditions are fulfilled, then the inf-morphism property (see for instance [START_REF] Claudel | Lax-Hopf based incorporation of internal boundary conditions into Hamilton-Jacobi equation[END_REF][START_REF] Jin | Some Theoretical and Practical Perspectives of the Travel Time Kinematic Wave Model: Generalized Solution, Applications, and Limitations[END_REF] and references therein) allows us to conclude. Indeed, for each class point of view, the coupling can be seen as an external constraint. The global solution is thus given by the maximum between all the partial solutions obtained for each condition (initial and boundary ones).

The result is straightforward for [START_REF] Newell | A moving bottleneck[END_REF].

For [START_REF] Gazis | The moving and phantom bottlenecks[END_REF], let us fix (n, x) and assume that

T 1 (n, x) = T 2 (n * 2 , x) + 1 1 -δ h B .
Moreover, we assume that at a given location x there is no vehicle between the vehicle of class 1 denoted n 1 and its nearest leader of class 2, say n * 2 . This means in particular that

T 1 (n 1 -∆n, x) ≤ T 2 (n * 2 , x).
Thus, we have that for any s < 0

H T 1 (n 1 , x) -T 1 (n 1 -∆n, x) ∆n , I 1 -(1 -δ)s T 1 (n 1 , x) -T 1 (n 1 -∆n, x) ∆n ≥ H T 1 (n 1 , x) -T 2 (n * 2 , x) ∆n , I 1 -(1 -δ)s T 1 (n 1 , x) -T 2 (n * 2 , x) ∆n ≥ h B -(1 -δ) s u 1 ≥ N -1 N h B -(1 -δ) s u 1 ≥ H ⊠ ((1 -δ)s, I 1 ) .
Taking the limit ∆n going to zero and setting s = ξ(n *

2 ), we get the result. This ends the proof.

5 Numerical examples 5.1 Impact of a single "slow vehicle"

The numerical scheme ( 12)-( 13) has been applied to a link composed of two lanes located immediately upstream a diverge (at x = x 1 ) with two output branches. The incoming link length is set to L = x 1 -x 0 = 1000m. The demand at x = x 0 is set to q = 1 vps (vehicle per second) for In the remaining of the section, "fast vehicles" (resp. "slow vehicles") designate vehicles from class 1 (resp. 2). Here it has been assumed that no capacity drop occurs, δ = 0. Then four levels of capacity drop

FIGURE 6 Simulation results

Here again, slow vehicles act as active moving bottlenecks. The flow downstream slow vehicles is bounded and fast vehicles accumulate behind.

FIGURE 7 Simulation results

It can also be seen from figure 7 that travel times experienced by fast vehicles oscillate around 60s.

This mean travel time is a trade-off between 40s (when no interaction occurs between classes) and 100s (when class 1 and 2 strictly respect the FIFO rule). It is directly related to the simulation parameters, and first of all δ which drives the level of relaxation of the FIFO rule. We conclude from this scenario that the proposed multiclass model:

• provides a more realistic framework to simulate traffic condition upstream diverge. It allows for relaxing the FIFO rule while considering the interactions between vehicle classes.

• the calibration of δ is of paramount importance for practical application. Further research is needed to propose accurate methodology to calibrate δ accurately and to identify factors that impact δ (lane ratio, class distribution, speed difference, etc.).

Conclusion and perspectives

We have introduced a new event-based mesoscopic model for multi-class traffic flow modeling on multi-lane sections. The basis model was first introduced by Leclercq and Bécarie [START_REF] Leclercq | A meso LWR model designed for network applications[END_REF] and deals with traffic flow on mono-pipe links, even if it can take into account simple junction models and two-flows representation thanks to the theory of moving bottlenecks. The model has been proven to be very attractive to simulate traffic flow on wide networks since it only keeps track of individual passing times at the boundaries of the network.

Among the perspectives, we would be interested in dealing with real traffic data to validate the continuous model, even if we are aware that these data need to be precise enough such that one can make the distinction between vehicle classes for example cars and trucks. A step further would be data assimilation for real-time applications such as traffic state estimation and forecast. We are also looking forward coupling our approach with lane flow distribution as proposed for instance in [START_REF] Duret | Lane flow distribution on a three-lane freeway: General features and the effects of traffic controls[END_REF] where the authors proposed a mechanism to show how variable speed limits help to delay the congestion onset.

In a quite different fashion, another extension of this work would be to consider the mesoscopic formulation of the GSOM family (as discussed in Section 3.1) and to assume that the moving bottlenecks are (known) internal boundary conditions. A numerical solution could be obtained by extending the algorithm described in [START_REF] Costeseque | A variational formulation for higher order macroscopic traffic flow models: numerical investigation[END_REF], when assuming a triangular fundamental diagram H(•, I).

  FIGURE 1 Moving bottleneck graphical solution in Eulerian coordinates, for a triangular Hamiltonian or flow-density FD defined as Q(k) = min {uk, w(κ -k)}, with k the density. (U) and (D) denote respectively the upstream and downstream traffic state with respect to the moving bottleneck position.

Definition 4 . 1 (

 41 PWA Eulerian upstream boundary condition) Assume that one can measure the flow values {q i,k } k∈[1,kmax] for each group of ∆n vehicles of class i ∈ {1, 2}, at a fixed (and given) location x 0 (thanks to a loop detector for instance).

Definition 4 . 2 (

 42 PWA Eulerian downstream boundary condition) Assume that one can measure the flow values {q

Definition 4 . 3 (

 43 PWA Lagrangian boundary condition) Assume that one can measure the speed values {v i,l } l∈[1,lmax] for each discrete position distant of ∆x, for a given leader vehicle n 0 of the class i ∈ {1, 2}.

class 1 .

 1 A single vehicle from class 2 drives through the link with a maximum speed u 2 = 10m/s, while vehicles from class 1 have a maximum speed u 1 = 25m/s. The congested parameters are set to w = 5m/s and k x = 0.14veh/m for both classes. The simulation duration is set to 200s with δ = 0.

FIGURE 4 Figure 4 1 (

 441 FIGURE 4Impact of single slow vehicle (class 2) on fast vehicles (class 1)

5

 5

6

  have been tested: δ = {0.2, 0.4, 0.6, 1}. The results are presented in Figure5.

7 Time 2 FIGURE 5 10 δ = 1 , 13 "

 72510113 FIGURE 5 Impact of δ on traffic states surrounding an active MBObviously the slow vehicle surrounding conditions highly depend on the capacity drop parameters, 8

14 5. 2 Simulation with a mixed traffic 15 Finally

 14215 , an advanced simulation scenario has been implemented with the same link characteristics,16where the vehicle classes have the same characteristics as above. The demand at x = x 0 is set to 17 q = 1 vps with the following distribution per class: class 1 = 95% and class 2 = 5%. The simulation 18 results are illustrated in Figure6.

Table of symbols

 of

	1
	Symbol	Meaning
	x	spatial location
	t	time
	n	vehicle label, Lagrangian coordinate
	q	flow
	k	density
	v	speed
	κ	maximal density for one lane
	C	maximal flow for one lane
	-w	wave speed (with w > 0)
	u	free flow velocity
	k → Q(k)	flow-density fundamental diagram
	r → H(r)	headway-pace fundamental diagram
	h = 1 q	headway
	r = 1 v s = 1 k v B	pace spacing speed of the moving bottleneck
	h B	residual capacity up to the moving bottleneck
	δ	capacity drop parameter
	N	number of lanes
	I	vehicle class

TABLE 1 Table

 1

of symbols used along the paper.

Acknowledgements

The authors are sincerely grateful to Prof. Ludovic Leclercq for fruitful discussions during the preparation of this paper and to the three anonymous referees whose comments helped to improve the presentation of this work.