Claire Polo
email: claire.polo@ens-lyon.fr

Christian Plantin
email: christian.plantin@univ-lyon2.fr

Kristine Lund
email: kristine.lund@univ-lyon2.fr

Gerald Peter Niccolai
email: gerald.niccolai@ens-lyon.fr

Words to Reason and Argue about Drinking Water Management: Comparing Debates from Mexican, US and French Schools

Keywords: argumentative scenario, comparative analysis, education, discourse object, framing, schematization, socioscientific issues, textometrics

In ten videotaped socio-scientific debates related to water, students from Mexico, the USA and France tend to focus on few alternative positions. On the basis of Grize's definition of schematization, we followed their reasoning by studying how they put light on specific aspects of the discursive object 'water'. Through textometrical analysis of debate transcripts, we specified 6 characteristics of 'water' that are more or less emphasized depending on the prevailing national argumentative scenario.

INTRODUCTION

Argumentation is a great object of interest in education (e.g. [START_REF] Andriessen | Arguing to learn: Confronting cognitions in computer-supported collaborative learning environments[END_REF][START_REF] Driver | Establishing the norms of scientific argumentation in classrooms[END_REF][START_REF] Erduran | Argumentation in Science Education: Recent Developments and Future Directions[END_REF][START_REF] Von Aufschnaiter | Arguing to learn and learning to argue: Case studies of how students' argumentation relates to their scientific knowledge[END_REF][START_REF] Muller Mirza | Argumenter en classe de sciences[END_REF]. At the crossing of science, environmental and citizenship education, some topics challenge usual didactical perspective on argumentation: the socio-scientific issues (SSI). During a scientific café activity, students in Mexico, the USA and France proved capable of building complex arguments in response to SSI related to drinking water management, using knowledge, norms, values and emotions [START_REF] Polo | L'eau à la bouche: ressources et travail argumentatifs des élèves lors de débats socio-scientifiques sur l'eau potable. Etude comparée de 10 cafés scientifiques menés au Mexique, aux USA et en France[END_REF]. In this paper, we follow how they elaborate their reasoning, by analyzing the schematization processes affecting the discourse object 'water'. We base our work on Grize's perspective of schematization, as a discursive construct giving clues about the underlying cognitive process (1997, p. 65). A comparison among the 3 national corpuses allows us to identify the prevailing orientation focuses of the schematization of 'water' and characterize a typical argumentative framing for each country. While intercultural work in argumentation generally emphasizes matters of rhetorical style (e. g. [START_REF] Disson | D'une culture l'autre : argumentation et stratégies discursives au Japon[END_REF][START_REF] Oetzel | Face and facework in conflict: A cross-cultural comparison of China, Germany, Japan, and the United States[END_REF][START_REF] Taft | An empirical demonstration of contrastive rhetoric: Preference for rhetorical structure depends on one's first language[END_REF], our approach engages into a comparative study based on the substantial objects under discussion.

After detailing our theoretical orientations and research questions (2), we specify the context and data of our study (3). Then we detail our methodology (4), and present our main results (5), which significance is discussed in a final section (6).

THEORETICAL ORIENTATIONS AND CHALLENGES

2.1 'Schematization' [START_REF] Grize | Logique et langage[END_REF] Grize's 'natural logic' relies on the (re)construction of discourse objects through operations of schematization in dialogic communication. 'Schematization' refers both to the process and the result, a specific discursive representation of what the discussion is about (Grize, 1997, p. 29). Then, a schematization is inherently subjective and individual, even among argumentatively aligned participants. Nevertheless, people defending the same view tend to build similar schematization. This approach somehow extends the argumentative orientation of language described by [START_REF] Anscombre | L'argumentation dans la langue[END_REF]. Grize study both explicit and explicit argumentative moves in discourse:

As I understand it, argumentation considers the interlocutor not as an object to be manipulated, but rather as an alter ego who must be brought to share his vision. Acting on him, it's trying to modify the diverse representation which he can be thought to have, by emphasizing some aspects of things, hiding others, suggesting new ones, and all this thanks to an appropriate schematization. 1An appropriate schematization cats light on specific aspects of a discourse object to serve an argumentative purpose, by creating axiological inferences (Grize, 1997, p. 48). Last but not least, a schematization does not consist of an isolated utterance but corresponds to a whole 'system' (Id.,p. 73).

Grize considers the discourse objects from an extensive perspective, including their 'faisceau' (object 'ray cluster'), which is to say all the usual things generally associated to the object (intrinsic characteristics, typical relations to other objects, action schemes). A participant's contribution to the 'ray cluster' (faisceau) of a discourse object is argumentatively orientated to the defense of a preferred conclusion. Tracking and comparing such schematizations is key to do substantial argumentation analysis2 .

Research questions: comparing schematizations

Still, how can such tracking be operated? Generally, comparing schematization is done through qualitative discourse analysis, at a microscopic level. For instance, we proposed a monography of two rival schematizations in a ten-minute debate, with a special focus on the diverging emotional framing [START_REF] Polo | Quand construire une position émotionnelle, c'est choisir une conclusion argumentative[END_REF]. But when it comes to comparing schematizations at a larger scale, especially for intercultural analysis, other tools are needed. The use of textometrics can give us interesting clues, with a first automatic quantitative analysis. In this perspective, we defined three precise research questions: 1. Is the specific light cast on a discourse object embodied in the words used to argue? 2. If so, can we describe argumentative frames using textometrics? 3. How do such argumentative frames vary in different cultural and linguistic contexts?

To better understand the corresponding methodology (4), we now specify the context and the empirical data characterizing this study (3).

EMPIRICAL STUDY: CONTEXT AND DATA

Our data consists of video-taped debates run during a scientific-café-type activity in three countries, in 2011 and 2012. The two-hour-long pedagogical activity is extracurricular but organized at school, in usual class groups of students aged 12-14. The sequence is led by a duo of trained elder students, aged 15-18. It is based on a multiple-choice questionnaire, alternating quiz questions giving basic information and 'opinion questions'. The activity is oriented toward a final debate, on the 'main question' (MQ, figure 1), which is also presented at the beginning to introduce the topic. After following, three thematic steps, the students discuss it in group, choose a common answer, and then debate it with the whole-class. Finally, each student expresses an individual answer through an anonymous vote. Our corpus consists of 10 final debates, at the class level, on the MQ, 2 from each of the two Mexican schools, 3 from the US school, and 3 from the French school. It is essential to take into account the nature of the topic. In the literature, SSI are characterized upon 4 main properties (e.g [START_REF] Albe | Enseigner des controverses[END_REF][START_REF] Driver | Establishing the norms of scientific argumentation in classrooms[END_REF][START_REF] Gayford | Controversial environmental issues: a case study for the professional development of science teachers[END_REF]Kolsto, 2001;[START_REF] Legardez | L'école à l'épreuve de l'actualité[END_REF][START_REF] Simonneaux | Students' socio-scientific reasoning on controversies from the viewpoint of education for sustainable development[END_REF][START_REF] Zeidler | Beyond STS: A research-based framework for socioscientific issues education[END_REF]: interdisciplinarity, knowledge hybridizing, subjectivity [START_REF] Oulton | Reconceptualizing the teaching of controversial issues[END_REF] and controversy [START_REF] Albe | Enseigner des controverses[END_REF], Legardez, 2006, pp. 19-20). As a result, the debates studied here are very open compared to usual classroom discussions. This semi-formal activity lets the students use a great diversity of argumentative resources, and intend to make them feel comfortable with not all sharing the same view. The final debates on the MQ varied a lot among our data, with variations of the schematization of 'water'. Section 4 details our methodological to track these differences.

METHODOLOGY

Our research questions (see 2.2) can be operationalized into the three following problems:

1. Which words is the object 'water' associated with, in students' argumentative discourse? 2. Do such co-occurrences group into characteristic orientation focus or 'lights' ('éclairages')? 3. Can we observe regularities among national subcorpora? To answer these questions, we analyzed all the ten transcripts of the debates with TXM software3 , which automatically lists the co-occurrences between words. In order to get a complete picture of the use of the discourse object 'water', and not only the word itself, we proceeded into 3 analytical steps.

Researching contexts for the lemma 'agua', 'water', 'eau'

The first analytical step consists in researching co-occurrences for the lemmas corresponding to 'water', in each of the three subcorpora. We did not find any occurrence of 'water' using the plural. In the Mexican corpus, we found 117 occurrences of 'agua', which was the 5 th more frequent word. In the US corpus, 'water' was only the 9 th most frequent word, with a total of 91 occurrences. In the French transcripts, only 43 occurrences were found, 'eau' ranging as the 32 th most frequent word. Such global differences can be partly explained by the fact that the debates were not of equal duration: they lasted between 4 minutes and 04 seconds for the shortest, to 9 minutes and 47 seconds for the longest. Moreover, French debates were characterized by many topical digressions. These lemma-based requests in TXM gave us a the terms associated to each occurrence of the pivotal term 'water' (agua, eau), which can be alphabetically organized following the left context or the right context. Such visualization enables the analyst to start identifying, at a glance, repeated cooccurrences. Table 1 illustrates this type of data directly exported from the software, here twenty occurrences of 'water' in the US corpus, the first ones following the left context alphabetical order.

Researching pronouns referring to water

To get a more complete idea of the way the object 'water' is constructed in students' discourse, we also took into account the occurrences of pronouns standing for water. We identified only a few ones actually used: 'la', 'lo', 'se', 'esa', 'le', 'que', 'qué' in the Spanish data; 'it', 'that' and 'ours' in the US corpus; 'ça', 'la', 'l'', 'en', and 'elle' for the French transcripts. This second analytical step enabled us to catch other occurrences of the discourse object water. Below are examples taken from the three corpuses, with the locutor name, the time when it is uttered in the video, the number of the debate, and own translation into English when necessary:

(1) LAU 1:48:09, 4 elle va devenir rare (it's gonna become scarce) (2) LIS 1:30:05, 1 a la vez la estamos contaminando (at the same time we're polluting it) (3) ABI 1:25:52, 2 they might not be able to get it and ours is very good\

Researching pronouns referring to water

In order to make sure that no occurrences of the object 'water' were missing, we followed a 3 rd analytical step, in TXM, making requests based on the results of steps 1 and 2. We researched any contextual term that has been identified as co-occurring with the lemma 'water' (3.1), or a pronoun standing for it (3.2). These requests were generally made using a radical form, since the automatic grammatical tagging tool used by TXM was not equally developed for the three languages. This last step, of course, mostly repeated results previously found, but it also enabled us to find new contextual terms, such as:

(4) OCE 1:49:28, 1 il y aura du manque (there will be a shortage) (5) MON 1:32:09, 1 y va a haber muy poquita (there will be very little) (6) ROS 1:29:23, 2 maybe for the future we will have enough

This three-step methodology enabled us to get a quite complete picture of the discourse construction of the object 'water' among students' transcripts. The last stage of our methodology, interpretation, is based on two sets of results: a) identifying and specifying the different orientation focuses or lights ('éclairages') conferred to 'water' by grouping co-oriented terms, and b) comparing their relative weight in each subcorpus. It is presented together with our main results, in section 5.

MAIN RESULTS

Our first result was to characterize precisely the alternative discursive constructions of the object 'water' in relation to the argumentative light being emphasized (5.1). Then, comparing the relative weight of each of these perspectives, we could identify coherent systems of predominant argumentative framing in each linguistico-cultural field (5.2).

Specification of 6 orientation focuses or lights ('éclairages')

On the basis of TXM results, we qualitatively grouped the contextual terms co-occurring with the discourse object 'water' into 6 main lights characterizing alternative schematizations. The full list of contextual terms associated to each group is reproduced in table 2. 1. A first group corresponds to rephrasing the general issue of accessing water. 2. A second group of associations made with water is characterized by a focus on natural water resources. 3. The third group of terms corresponds to apprehending water as related to the satisfaction of human needs, either from a qualitative or a qualitative perspective. In addition to the contextual terms listed in table 1, Two global expressions including left and right contexts are also part of this group: "as much water as we want, lower the water quality". 4. Another set of contextual terms cast light on the more or less environmentally friendly human uses of water. Two global expressions fall into group 4: "reduce the water that we search, leave the water on". 5. The fifth group of associated words considers water from the viewpoint of the water producer, concerned by technical processing problems or/and the matter of providing water to the consumers. This group, in addition to the terms listed in table 1, includes 3 global expressions: "change salt water into soft (convertir el agua salina en dulce), make water available, give water to other countries". 6. A sixth set of contextual words put emphasis on the monetary and commercial exchange related to access to water. It consists of the contextual terms listed in table 1, more the three global expressions: "use water as an economic thing (ocupar el agua como algo económico), see water as an economic thing (ver el agua como algo económico), income of the water to pay (ingreso del agua por pagar)". Whether the students' discourse cast light on one or the other of these aspects has different consequences in terms of argumentative framing of the debate. At first sight, group 1 associations do not seem to provide strong argumentative orientation toward a specific answer or argumentative conclusion. Still, the way the general issue is called gives a first insight about which orientation the speaker is trying to bring the debate toward. In combination with the analysis of the relative place of the 5 other groups of contextual terms, the analyst can specify the alternative argumentative framing being displayed. Since schematization doesn't rely on isolated utterances, but works as a meaning and orientating discursive network, a global picture is needed to actually describe this construct. We developed a synthetic word-cloud type visualization, for each subcorpus, of the textometrical analysis of co-occurrences, which makes it easier to combine these parameters and draw conclusions about the typical argumentative framing that emerges from the data (5.2).

Alternative schematizations of 'water': results for each country

Figures 2, 3 and 4 synthesize our results, respectively for the Mexican, the US and the French corpora. We designed this word-cloud visualization in order to present a global picture that articulates the analysis of cooccurrences made in TXM with the first step analysis consisting of qualitative grouping of occurrences into the 6 specific focuses described in 5.1. In the center of the visualization appears the pivotal lemma 'agua', 'water' or 'eau'. Then, left-context occurrences of the discourse object 'water' are reproduced on the left part of the figure, and the right-context occurrences in the right section of the plan, both in alphabetical order. When a contextual term co-occurred several times with the discourse object 'water', we reproduce the lemma (infinitive of verbs, adjectives and nouns in singular masculine), except if there was no variation. For instance, 3 occurrences of 'les riches' are reproduced as 'les riches', but the thirteen occurrences of the verb 'to get' used at different persons appear with the lemma 'get'. In order to get a meaningful representation of our results, contextual terms are bigger when they are more frequent, proportionally to the total number of co-occurrences. More frequent terms are also slightly more centered, appearing closer to the pivotal lemma. For the French corpus, it corresponds to terms used at least five times to characterize the discourse object 'water', while the threshold is three for the US corpus and four for the Mexican corpus. A number is associated to each contextual term, corresponding to its frequency in the corpus. Terms that share the same radical (ex: 'access', 'accessibility') or are semantically synonymous (ex: 'desalinize' and 'take the salt out') are reproduced close to each other, and in the police and place corresponding to the sum of their frequencies. Some occurrences cannot be classified as specifically left or right context terms but rather consist of global expression including left and right contexts for 'water'. These expressions were placed directly next to the central pivotal word. Last but not least, a color code was used to identify the different orientation focus (éclairages) characterizing the schematizations of the discourse object 'water': 1) in black appears the terms corresponding to the general problem of access to water; 2) blue is used for terms referring to water as a natural resource; 3) red is used for terms characterizing water in reference to human qualitative and quantitative needs; 4) in green are reproduced the terms dealing with human more or less environmentally friendly uses of water; 5) brown is the color corresponding to words describing water from the viewpoint of the water producer or provider; 6) purple is used for terms referring to water trade and the commercial transaction determining someone's access to water. These visualizations reveal a specific argumentative framing of the issue in each subcorpus.

Five key features characterize the Mexican corpus in terms of prevailing orientation focuses of schematization of 'water' (see figure 1). First, when water is considered as a natural resource (in blue), what is emphasize is how scarce (4 occurrences of 'escasez' or 'escasa' -scarcity or scarce), absent (2 occurrences of 'sin agua' -without water), or endangered (18 occurrences of 'acabarse' or 'agotarse' -extinguish or dry up) it is. Secondly, among human use of water (green), the idea that water must be saved (37 occurrences of 'ahorrar', 'guardar' or 'usar/utilizar menos' -save, keep, or use less) is very frequent. This focus is characterized by the extensive use of the verb 'cuidar' (take care), with 21 occurrences. Even if the light cast on human use of water here mostly consists of prescribing good practices, some behaviors that waste water are also mentioned (9 occurrences of 'desperdiciar', 'gastar' or 'desgastar'). Third characteristic of the Mexican corpus: when water is considered for its capacity to satisfy human needs (in red), two aspects are emphasized, the fact that water is vital (two occurrences of 'necesitar' -need-and one of 'recurso vital' -vital resource), and the capacity of people to adapt to the evolution of water resources (3 occurences of 'adaptar'). The expression 'recurso vital' is surrounded by a blue frame because the substantive also refers to water as a natural resource. The emotional tonality associated to the mentioned human needs is quite high, as they correspond to matters of life or death: 3 occurrences of 'tomar' or 'beber' (drink), and one occurrence of 'vivir' (live). On the contrary, this corpus presents very few contextual terms corresponding to the viewpoint of the water producer (in brown). A unique occurrence among the three national corpuses appears in Mexican data: the use of the verb 'compartir' (share), which is reproduced in green and surrounded by a brown line: here water provider is also presented as a couser. Fifth characteristic of the Mexican data: a large number of occurrences referring to water trade (in purple). Both selling and buying water are mentioned, but the students mostly present themselves from the viewpoint of the consumer (30 occurrences of 'comprar', 'pagar' or 'costar' -buy, pay, or cost, versus 6 occurrences of 'vender' or 'dar caro' -sell or give at a high price). Moreover, 13 terms deal with water price, 5 of them emphasizing how expensive it is, or mentioning that it is going up. Water is even considered as a potential currency twice ('pagar con' -pay with). Last but not least, the verb 'robar' (steal) appears twice. Drinking water, thought as a commodity, can be monopolized, traded, or stolen. Figure 3 synthesizes the results obtained for the US corpus. Here, the focus on water as a natural resource (in blue) is much more frequent than it is in the two other subcorpora. This perspective is embodied both in a general lexicon (10 occurrences of 'sources', 'resources' or 'supplies') and by reference to specific water sources: polar icecaps (1 occurrence), seawater (5 occurrences). If 3 occurrences mention that water is scarce or absent (2 occurrences of 'no water'; one of 'scarce'), there is also 2 expressions emphasizing its abundance ('a lot of', 'a ton of'). Two contextual expressions are ambivalent in this aspect: 'has been taken' and 'out to bigger countries', because they establish scarcity in a place corresponding to more water in other places, the USA falling into the second category (the 'bigger countries'). Such dichotomy is consistent with what appears in red, corresponding to water characteristics for the satisfaction of human needs. At a quantitative level, 5 occurrences mention the lack of water ('lack of' or 'run out of/low on'), 2 rather refer to an appropriate amount of water ('enough'; 'the needed amount'), and one occurrence describes a situation of abundance ("as much water as we want"). At the qualitative level, water is also framed as more or less good (6 occurrences of 'good', 'better', 'purified' or 'thin' versus 4 occurrences of 'bad', 'worst', or 'lower water quality'). When it comes to the uses of water (in green), 7 occurrences are quite neutral ('consume', 'consumption', 'usage' or 'use'), while 20 describe good practices to follow and 5 bad practices to be avoided. For good practices, the most frequent verb is 'save', referring to both preserving water and economizing it. The US corpus is also characterized by a very frequent schematization of the discourse object 'water' from the viewpoint of the water producer or provider (in brown). Water production is extensively discussed, with 4 general occurrences ('produce'; 'make water available'; 'make'; 'do'), and the mention of a large number of production techniques: the desalinization of seawater (4), the recycling of water ('reuse'; 'renew'), water purification (1), and even channeling a river (1). A lexicon referring to water distribution also embodies this orientation focus: 'lines'; 'plumbing'; 'give' (4); 'provide' (1). Water trade appears in the US corpus, as much from the viewpoint of the seller (2 occurrences of 'sell') as from the viewpoint of the buyer ('pay to get', 'pay for'). One occurrence of the neutral term 'interchange for' was also inventoried. About the water price, the emphasis is more on how cheap it is or might be (3 occurrences of 'cheap', one of 'affordable') than on how expensive it is or might be (only one occurrence of 'expensive'). Besides, in the US corpus, two words employed for the general problem of access to water deserve a specific attention. The first one is the massive use of the verb 'get', which embeds an idea of voluntary action that is absent in the verb 'have', for instance. Rather than considering access to water as determined by fatality, this framing is emotionally less intense than the French or Mexican ones. This tendency is confirmed by another specificity of the way the global problem is referred to, in the US corpus: the use of abstract words and a bigger emotional distance to the issue ('water problem', 'water issue', "it is wrong with their water"). In figure 4 are visualized the main results for the French corpus. Here, water is hardly considered as a natural resource (only 3 occurrences in blue). When such framing occurs, it is mainly to put emphasis on the fact that water is getting more and more rare. 'Economiser l'eau' is THE motto that stands out of the French data, when it comes to water uses (in green). In total, 16 occurrences refer to good practices to use less water (11 occurrences) or preserve it (5 occurrences). But, contrary to the other corpora, here no concrete situation is presented.

In terms of schematization of water for its capacity to meet human needs (in red), both qualitative and quantitative aspects are mentioned in the French corpus. A unique interesting occurrence frames the need for access to water as a right to water. This is consistent with the fact that, among the 12 occurrences corresponding to the viewpoint of the water producer or provider (in brown), one specifically focuses on water distribution considered as a matter of public service. Some contextual terms also refer to the general action of producing water ('faire', 'fabriquer' -make), while the specific word 'refroidir' (cool down) refers to getting water from the air. The orientation focus on water trade (in purple) is quite well represented among the French data, with 9 occurrences referring to the price to be paid by the consumer to access water. Moreover, 10 occurrences embody a concern for the socioeconomic inequalities among people for access to water.

At the end of the day, the French corpus is strongly characterized by addressing the problem of access to water at a macro-social level, with a global motto ('économiser l'eau') rather than concrete, micro-level examples of good practices, the definition of a universal right to water, and the subsequent concerns of dealing with social inequalities and the matter of providing water to everyone.

CONCLUSION

The analysis of co-occurrences in TXM proved useful to specify and compare students' schematization of 'water' as a discourse object, between our three national corpora (Mexico, USA and France). Beyond effects of cultural rhetorical styles, substantial differences in terms of orientation focus (éclairages) distinguish the debates hold in the three countries. In the Mexican debates, water is mostly apprehended from the viewpoint of the consumer, worrying about daily practices for getting water at an affordable price and saving the vital endangered resource. On the contrary, in the US corpus, the viewpoint of the water producer prevails. Here what counts is finding technical solutions to provide water at a low price to the consumers, making the best out of the natural water resources of each territory. In France, social inequalities are mentioned from a macro-social viewpoint, and the addressed solutions mostly rely on a better global organization of the distribution of water, considered as a universal right and public service. These results about the specific argumentative framing converge with results obtained by two other methods, about the cognitive models of 'water' used by the students, the prevailing knowledge area in the debates in the three countries, and the analysis of students' votes during the activity (Polo, 2014, pp. 359-424). They contribute to the comparative characterization of the prevailing 'argumentative scenario' (Plantin, in press) in each setting. Innovative word-cloud visualization gives a global picture of the automatic textometrical results and the complementary qualitative analysis that was very helpful for the comparative dimension of our analysis.

More generally, we believe that the operationalization of the Grize's concept of schematization (1997), as a descriptive framing approach, can be applied in other argumentative situations and enrich the methodological toolbox of the community of argumentation studies. The textometrics provide promising ways of doing comparative analysis in argumentation that really focus on the objects under discussion. Moreover, the word-cloud-type visualization of mixt-method results that we developed in this study can be useful even in monographies, out of any comparative aim. It might be especially relevant to study the role of emotions in a specific argumentative corpus, enabling the analyst to see at a glance the prevailing orientations given to the debate, and the co-substantial emotional framing [START_REF] Polo | Quand construire une position émotionnelle, c'est choisir une conclusion argumentative[END_REF].

Figure 1 -

 1 Figure 1 -Main question of the café, debated in our data, analyzed with TXM (English version).

Figure 2 -

 2 Figure 2 -Schematization of 'water' in the Mexican data: visualization of co-occurring terms, their focus and frequency.

Figure 3 -

 3 Figure 3 -Schematization of 'water' in the US data: visualization of co-occurring terms, their focus and frequency.

Figure 4 -

 4 Figure 4 -Schematization of 'water' in the French data: visualization of co-occurring terms, their focus and frequency.

Table 1 -

 1 First occurrences of 'water' in left-context order, US corpus.

Table 2 -

 2 Grouping of contextual terms into 6 specific orientation focuses of 'lights' (éclairages).

Own translation from the French "Telle que je l'entends, l'argumentation considère l'interlocuteur, non comme un objet à manipuler, mais comme un alter ego auquel il s'agira de faire partager sa vision. Agir sur lui, c'est chercher à modifier les diverses représentations qu'on lui prête, en mettant en évidence certains aspects des choses, en en occultant d'autres, en en proposant de nouvelles et tout cela à l'aide d'une schématisation appropriée."(Grize, 1997, p. 40).

This approach is similar to the Anglo-Saxon frame analysis. As a discursive construct, 'frames', as schematizations, "induce us to filter our perceptions of the world in particular ways, essentially making some aspects of our multidimensional reality more noticeable than other aspects."(Kuypers, 2009, p.181). As a process, both 'framing' and 'schematization' "encourages the facts of a given situation to be interpreted by others in a particular manner."(Kuypers, 2006, p. 8). Nevertheless, frame analysis related to the critical tradition in discourse analysis, which does not fit with our radically descriptive epistemological approach.

This open-source tool is freely available and fully documented here: http://textometrie.ens-lyon.fr/.

ACKNOWLEDGEMENTS: This research is supported by LABEX ASLAN (ANR-10-LABX-0081) of Université de Lyon, within the program "Investissements d'Avenir" (ANR-11-IDEX-0007) of the French National Research Agency (ANR).


Andrzej Nałęcz
email: analecz@wz.uw.edu.pl

Internet Traffic Management under EU Regulation 2015/2120 -the Need for the Co-operation of Legal and Computer Scientists

Keywords: Basic Traffic Management Measures, Reasonable Traffic Management Measures, Exceptional Traffic Management Measures, Internet Congestion, Classes of Traffic

EU Regulation 2015/2120 introduces a network neutrality regime into European electronic communications law. The application and enforcement of the abstract rules established by the regulation influences all the stakeholders in the internet value chain. The regulation's provisions are ambiguous and inconclusive. Legal and computer scientists must co-operate to provide an interpretation of those provisions that would, first, respect the rights of the end-users of internet access services and provide a user experience in line with the regulation, and second, respect the technical realities of managing traffic on the internet. The paper presents the legal framework of traffic management under the regulation and exposes problems which may only be resolved by a synthesis of the legal and computer science approaches. These include: the classification of traffic management measures into basic, reasonable and exceptional categories, recognized by the regulation; the admissibility of traffic classification; the definition of congestion befitting the regulation; the role of traffic management measures in congestion control. If these issues are not resolved, the exercise of the rights of endusers of internet access services may be limited, and the providers of these services may be subjected by national regulatory authorities to unreasonable obligations.

The purpose of this article, written by a legal scientist, is to determine which problems related to internet traffic management have been solved by the law and which should be solved by computer scientists. While due to the principle of technology neutrality [START_REF] Maxwell | Technology Neutrality in Internet, Telecoms and Data Protection Regulation[END_REF] the Regulation 2015/2120 does not specify any concrete technical standards and relies solely on abstract rules, in the end it is applied in the real world, where suppliers of network devices and providers of internet access services face the problem of providing and utilizing solutions that are both effective and not illegal under the Regulation 2015/2120. Legal scientist and computer scientists should come together to make clear what these solutions should be.

2

The Terminology of the Regulation 2015/2120

Internet Access Services and Specialized Services

Article 2 of the Regulation 2015/2120 defines internet access services as publicly available electronic communications services that provide access to the internet, and thereby connectivity to virtually all end points of the internet, irrespective of the network technology and terminal equipment used. From the point of view of analyzing traffic management measures, the most important conclusion arising from the interpretation of the cited provision is the fact that the Regulation 2015/2120 pertains only to publicly available electronic communications servicesthat is those that are offered to the public, either on the consumer or business markets, or on both, rather than only to a select group of end-users. Thus, traffic management in private networks is not subject to the requirements of the Regulation 2015/2120 and is not mentioned or discussed further in the paper. Article 3 (5) of the Regulation 2015/2120 allows providers of internet access services and providers of content to offer services other than internet access services which are optimized for specific content, applications or services, or a combination thereof, where the optimization is necessary in order to meet the requirements for a specific level of quality. These are the so-called specialized services. Unlike internet access services, they come with Quality of Service guarantees. The distinction is important inasmuch that it shows that while traffic management of internet traffic may involve traffic classification and different treatment of the various classes of traffic (which is discussed below), it may only distinguish between classes of best-effort traffic types, since QoS optimization is limited to specialized services.

2.2

The Lack of a Legal Definition of "Traffic Management" in the Regulation 2015/2120

The Regulation 2015/2120 does not define the term "traffic management". However, traffic management was discussed in the debate leading up to the regulation being legislated. The Body of European Regulators for Electronic Communications (BEREC), which was tasked by the European Commission with investigating issues related to network neutrality and coordinated the discussion and presently coordinates the regulation's enforcement, leans towards a very broad understanding of traffic management, describing it as "the way the traffic is forwarded in the networks (…), which may include both regular first-come-first-serve management of traffic and more advanced ways of shuffling traffic through the networks" [2]. Such a broad understanding of traffic management differs from the one presented in computer science literature. In [START_REF] Constantine | Traffic Management Benchmarking[END_REF], Constantine and Krishnan describe the traffic management capabilities of network devices as those pertaining to: traffic classification, traffic policing, traffic scheduling, traffic shaping and active queue management. This approach, then, excludes the simplest technical solutions, such as passive queue management, from the scope of the term "traffic management".

The broad approach to traffic management in interpreting the Regulation 2015/2120 is necessitated by the requirements of network neutrality and by the need to safeguard the rights of end-usersas shaped by the regulation. Only a broad approach may prevent the providers of internet access services from circumventing the network neutrality rules by claiming a practice they are engaging in does not in fact constitute traffic management subject to the Regulation 2015/2120. Thus, traffic management under the Regulation 2015/2120 should be understood to include all the technical solutions, be it simple or complex, used to measure, analyze, classify and forward traffic in the network.

3

Traffic Management Measures under the Regulation 2015/2120

Primal and Dual Congestion Control Under the Regulation 2015/2120

One of the objectives of utilizing traffic management measures under the Regulation 2015/2120 is congestion control. In the networking literature [START_REF] Papadimitrou | Open Research Issues in Internet Congestion Control[END_REF], "primal congestion control refers to the algorithms executed by the traffic sources for controlling their sending rates or window sizes. (…) TCP algorithms fall in this category. Dual congestion control is implemented by the routers through gathering information about the traffic traversing them. A dual congestion algorithm updates, implicitly or explicitly, a congestion measure or congestion rate and sends it back, implicitly or explicitly, to the traffic sources that use that link." While the Regulation 2015/2120 itself does not mention these two types of congestion control, they are referenced by BEREC and in the legal literature as "end-point based" and "network-internal" congestion control [START_REF]BEREC Guidelines on the Implementation by National Regulators of European Net Neutrality Rules[END_REF][START_REF] Piątek | Rozporządzenie UE Nr 2015/2120 w zakresie dostępu do otwartego internetu. 1st edn[END_REF]. The former does not contravene the provisions of the Regulation 2015/2120 on traffic management [START_REF]BEREC Guidelines on the Implementation by National Regulators of European Net Neutrality Rules[END_REF] and is in fact outside their scope [START_REF] Piątek | Rozporządzenie UE Nr 2015/2120 w zakresie dostępu do otwartego internetu. 1st edn[END_REF]. Thus, the regulation applies only to the latter, network-internal measures.

Classification of Traffic Management Measures under the Regulation 2015/2120

The Regulation 2015/2120 implies the existence of basic traffic management measures, and explicitly mentions two other types of measuresreasonable traffic management measures, regulated by Article 3 (3) second subparagraph of the regulation, and those going beyond the reasonable, regulated by Article 3 (3) third subparagraph. The latter are called exceptional traffic management measures by BEREC and in the legal literature [START_REF]BEREC Guidelines on the Implementation by National Regulators of European Net Neutrality Rules[END_REF][START_REF] Piątek | Rozporządzenie UE Nr 2015/2120 w zakresie dostępu do otwartego internetu. 1st edn[END_REF]. Basic measures are the simplest ones and do not involve any traffic classification. Reasonable measures may rely on traffic classification and may involve different treatment of specific categories of best-effort traffic. Exceptional measures may be employed only under exact circumstances, enumerated in the Regulation 2015/2120, and they may go as far as traffic blocking. All types of measures are described in further detail below.

Basic Traffic Management measures

Article 3 (3) first subparagraph of the Regulation states that Providers of internet access services shall treat all traffic equally, when providing internet access services, without discrimination, restriction or interference, and irrespective of the sender and receiver, the content accessed or distributed, the applications or services used or provided, or the terminal equipment used. The cited provision implies the recognition of a type of traffic management measures, unnamed in the regulation, and known in the legal literature as basic traffic management measures [START_REF] Piątek | Rozporządzenie UE Nr 2015/2120 w zakresie dostępu do otwartego internetu. 1st edn[END_REF]. The key phrase in the provision is that all traffic should be treated equally. Since only reasonable and exceptional traffic management measures allow traffic classification and the differentiated treatment of specific classes of traffic, the equal treatment of traffic in this context must be understood in its most narrow, linguistic meaning. Thus, basic traffic management measures must entail the identical treatment of every data packet traversing the network. As has been explained in the legal literature, they involve a transmission of data packets without any interference and completely independent of the class of traffic they belong to, and are therefore completely neutral from the point of view of the end-users [START_REF] Piątek | Rozporządzenie UE Nr 2015/2120 w zakresie dostępu do otwartego internetu. 1st edn[END_REF].

Handling traffic using Passive Queue Management (PQM), involving first-in-firstout and tail drop algorithms, and giving all data packets the same priority undoubtedly constitutes basic traffic management measures. However, the status of classless Active Queue Management (AQM) as either a basic or reasonable traffic management measure is unclear under the Regulation 2015/2120. BEREC seems to indicate that all AQM (be it class-based or classless) should be assessed under the requirements for reasonable rather than basic measures [START_REF]BEREC Guidelines on the Implementation by National Regulators of European Net Neutrality Rules[END_REF]. This is unconvincing. As will be explained below, reasonable traffic management measures rely on traffic classification. Since classless AQM, such as Random Early Detection (RED) do not distinguish between traffic classes, they should be considered basic traffic management measures. RED with In and Out (RIO) and Weighted RED (WRED) algorithms allow a differentiated treatment of packets and therefore should be considered reasonable traffic management measures under the Regulation 2015/2120 (described in the following section), rather than basic ones. This is one of the areas where the application and enforcement of the Regulation 2015/2120 would greatly benefit from the input of computer scientists.

Reasonable Traffic Management Measures

Requirements for Reasonable Traffic Management -Introduction. Article 3 (3) second subparagraph of the Regulation 2015/2120 introduces the concept of reasonable traffic management measures and sets out the requirements which must be met by a measure for it to be considered reasonable. Under the cited provision, such measures must be transparent, non-discriminatory and proportionate, and must not be based on commercial considerations but on objectively different technical quality of service parameters of the specific categories of traffic. They must not monitor the specific content and must not be maintained for longer than necessary. Article 3 (3) third subparagraph, which generally pertains to exceptional traffic management measures, also exemplifies what kinds of the treatment of traffic may not be considered reasonable. All these requirements are discussed in detail below.

Transparency. The transparency of a traffic management measure pertains to its disclosure to both the end-users and the national regulatory authorities. While the endusers must only very broadly be informed of traffic management measures by means of the contract including internet access services, the information provided to the national regulatory authority must be detailed enough for the regulator to be able to determine whether a given measure satisfies the requirement of reasonableness. Non-discrimination. Traffic classes. Reasonable traffic management measures must be non-discriminatory. Non-discrimination is a well-known concept in EU law. It "is not equivalent to the distinction as such but refers to the disadvantageous treatment of an individual based on a ground prohibited by law [and] it occurs when comparable situations are treated differently and different situations are treated in the same way", when it is not objectively justified [START_REF] Maliszewska-Nienartowicz | Direct and Indirect Discrimination in European Union Law -How to Draw a Dividing Line[END_REF]. In the case of internet traffic management, the requirement of non-discrimination pertains not to individuals but rather to data packets transferred over the internet. All packets must be transmitted according to the same rules, e.g. using the first-in-first-out algorithm, unless there is an objective justification for differentiation. The Regulation 2015/2120 unequivocally states in Article 3 (3) second subparagraph that specific categories of traffic may be treated differently based on their objectively different quality of service requirements (the regulation uses the term "traffic categories" rather than "traffic classes", used in computer science). This harmonizes with the objective of reasonable traffic management, described in recital 9 of the Regulation 2015/2120 as contributing to an efficient use of network resources and to an optimization of overall transmission quality. Treating all data packets in the same way regardless of the circumstances, especially when network congestion occurs, would lead to an inefficient allocation of network resources, such as capacity, and it would also adversely affect the user experience. Thus, the key issues of interest to both legal and computer scientists are: establishing classes of traffic with objectively different quality of service requirements, identifying data flows as belonging to a specific class of traffic (traffic classification), and specifying the circumstances under which the treatment of the various classes of traffic should be differentiated. The Regulation 2015/2120 does not specify which network devices may be involved in classifying traffic flows as belonging to specific categories of trafficit may therefore occur in any device, e.g. in routers or in specialized middleboxes. Proportionality. Reasonable traffic management measures must also be proportionate.

Proportionality is a well established general principle of EU law. It "requires that there be a reasonable relationship between a particular objective to be achieved and the means used to achieve that objective" [START_REF] Klatt | The Constitutional Structure of Proportionality. 1 st edn[END_REF]. When an action is undertaken that is subject to a proportionality review, a test may be carried out to evaluate whether the action satisfies several specific requirements: the pursuit by that action of legitimate aims; the suitability of the means, referring to its ability to further the achievement of the objective; the necessity of the means, that is whether it affects the rights of individuals as little as possible; and finally, proportionality in the narrow sense, that is the weighing of the reduction of the enjoyment of rights against the level of realization of the objective [START_REF] Klatt | The Constitutional Structure of Proportionality. 1 st edn[END_REF].

A traffic management measure may be considered reasonable only when it passes the testthat is when it cumulatively satisfies all its four criteria, described below in further detail.

Legitimate aim. A traffic management measure pursues a legitimate aim when it is not based on commercial considerations (which is discussed below) but seeks to achieve an efficient use of network resources and contributes to the user experience, while not infringing on the rights of the end-users enshrined in the Regulation 2015/2120. E.g. a measure giving precedence to traffic generated by a specific application (rather than to a class of traffic generated by all the applications sharing similar quality of service requirements) would not be considered as pursuing a legitimate aim, even if it contributed to the efficient use of network resources, since it would discriminate between specific applications, which the regulation forbids to protect the end-users' rights. Suitability. A measure is suitable when it indeed effectively serves the purpose for which it was introduced. E.g. if the purpose of employing a specific measure is the control of congestion, it will be considered suitable only if it can be proved that it in fact contributes to limiting congestion. Necessity. A measure is necessary when the aim it pursues is not achievable by using another measure, which would influence the right of the end-user to access any internet content to a lesser extent. E.g. if the efficient use of network resources and a satisfying user experience may not be achieved by using classless traffic management measures, solutions distinguishing between several classes of best-effort traffic types may be utilized.

Proportionality in the narrow sense. Finally, a traffic management measure is proportionate in the narrow sense when its introduction does not adversely affect the enduser's rights to an excessive extent, e.g. a measure introduced to provide efficient access to one category of internet content (translating into a specific class of traffic) may not lead to another category of content becoming effectively unusable by the end-user. Commercial Considerations. Reasonable traffic management measures may not be based on commercial considerations. This means that a provider of internet access services may not make using or refraining from the use of specific traffic management measures conditional upon receiving monetary or non-monetary remuneration from any party, be it an internet content provider, an end-user, or anybody else. Monitoring specific content. Under Article 3 (3) second subparagraph of the Regulation 2015/2120, reasonable traffic management measures may "not monitor the specific content". This is understood by BEREC as a prohibition on monitoring the specific content of a data packet, i.e. the transport layer protocol payload [START_REF]BEREC Guidelines on the Implementation by National Regulators of European Net Neutrality Rules[END_REF]. On the other hand, the information contained in the IP packet header or transport layer protocol header is considered by BEREC as generic rather than specific content and therefore reasonable traffic management measures may rely on it [START_REF]BEREC Guidelines on the Implementation by National Regulators of European Net Neutrality Rules[END_REF]. This includes such common identification methods as those based on the connection's port numbers [START_REF] Dainotti | Issues and Future Directions in Traffic Classification[END_REF]. In practice, the prohibition on monitoring specific content means that reasonable traffic management may not under any circumstances involve deep packet inspection techniques (DPI). This may prove problematic in practice, as such solutions are often used in middleboxes [START_REF] Bremler-Barr | Deep Packet Inspection as a Service[END_REF]. However, the interpretation of the Regulation 2015/2120 does not preclude the use of machine learning solutions, if they do not rely on DPI. Thus, the utilization of pattern recognition algorithms described in [START_REF] Dainotti | Issues and Future Directions in Traffic Classification[END_REF] is allowed.

How Long Reasonable Traffic Management Measures May be Maintained? The Congestion Control Controversy.

Under the Regulation 2015/2120, reasonable traffic management measures may not be maintained for longer than necessary, which relates to the proportionality principle. Both BEREC [START_REF]BEREC Guidelines on the Implementation by National Regulators of European Net Neutrality Rules[END_REF] and legal scientists [START_REF] Piątek | Rozporządzenie UE Nr 2015/2120 w zakresie dostępu do otwartego internetu. 1st edn[END_REF] distinguish between trigger functions and employing traffic management measures as such. A trigger function may be "implemented and in place (…) on an ongoing basis inasmuch as the traffic management measure only becomes effective in times of necessity" [START_REF]BEREC Guidelines on the Implementation by National Regulators of European Net Neutrality Rules[END_REF]. Thus, employing a measure which continuously or regularly affects traffic is questionable from the point of view of the requirement of necessity [START_REF]BEREC Guidelines on the Implementation by National Regulators of European Net Neutrality Rules[END_REF][START_REF] Piątek | Rozporządzenie UE Nr 2015/2120 w zakresie dostępu do otwartego internetu. 1st edn[END_REF]. Technical solutions involving class-based AQM algorithms that trigger specific treatment of data packets to control congestion, but which allow for first-in-first-out, class-less functionality when there is no congestion seem to be in line with the Regulation 2015/2120 and fall within the category of reasonable traffic management measures. However, this sensible conclusion is weakened to the extent of being muddled by BEREC's stance and the imprecise wording of the regulation. BEREC indicates that class-based measures preventing "impending congestion" constitute exceptional traffic management measures and are subject to the limitations of Article 3 (3) subparagraph three rather than subparagraph two [START_REF]BEREC Guidelines on the Implementation by National Regulators of European Net Neutrality Rules[END_REF]. If that interpretation were to be accepted, then there would appear to be no circumstances under which any reasonable traffic management measures could ever be used. Why else reasonably manage traffic than to prevent and mitigate congestion, if class-based differentiation based on commercial considerations is banned altogether? Computer scientists' input on the issue would be invaluable. What treatment of traffic may not be considered reasonable. Article 3 (3) third subparagraph of the Regulation 2015/2120 exemplifies what may not be involved in reasonable traffic management. The actions mentioned include, but are not limited to: blocking, slowing down, altering, restricting, interfering with, degrading or discriminating between specific content, applications or services, or specific categories thereof. Such actions are limited to exceptional traffic management measures, as described by BEREC [START_REF]BEREC Guidelines on the Implementation by National Regulators of European Net Neutrality Rules[END_REF] and in the legal literature [START_REF] Piątek | Rozporządzenie UE Nr 2015/2120 w zakresie dostępu do otwartego internetu. 1st edn[END_REF]. The interpretation of the cited provision is problematic, especially with respect to the slowing-down of traffic. Best-effort, classbased AQM involves priority schemes for data packets, effectively slowing some packets down in relation to others. This should be considered as belonging to reasonable traffic management measures (as long as traffic classes are based on objective and genuine QoS requirements), rather than exceptional ones, as the Regulation implies. Computer scientists' elaboration on the issue would be greatly appreciated.

Exceptional Traffic Management Measures

Exceptional Traffic Management Measures -Introduction. Under Article 3 (3) third subparagraph of the Regulation 2015/2120, in certain, very specific circumstances, enumerated in the regulation and discussed below, traffic management measures going beyond those deemed reasonable may be used. In the literature and BEREC documents, these measures are called exceptional traffic management measures [START_REF]BEREC Guidelines on the Implementation by National Regulators of European Net Neutrality Rules[END_REF][START_REF] Piątek | Rozporządzenie UE Nr 2015/2120 w zakresie dostępu do otwartego internetu. 1st edn[END_REF]. Even though it is not clearly stated in the Regulation 2015/2120, a legal interpretation of the provisions leads to the conclusion that these measuresjust like the reasonable onesmust be transparent and proportional, may not be based on commercial considerations and may not be maintained for longer than necessary. Unlike reasonable measures, however, the exceptional ones may under some circumstances monitor the specific content. They may also involve the actions mentioned in the previous paragraph above, including the blocking of traffic. EU and national legislators would be advised to cooperate with computer scientists in choosing the most appropriate, specific measures to be used. Preservation of network integrity and security. Under Article 3 (3) (b) of the Regulation 2015/2120, exceptional traffic management measures may be employed to preserve the integrity and security of the network, of the services provided via that network, and of the terminal equipment of end-users. This is rather uncontroversial and has been convincingly explained by BEREC in [START_REF]BEREC Guidelines on the Implementation by National Regulators of European Net Neutrality Rules[END_REF].

Circumstances under Which Exceptional

Preventing and Mitigating Congestion. Under Article 3 (3) (c) of the Regulation 2015/2120, exceptional traffic management measures may be employed to "prevent impending network congestion and to mitigate the effects of exceptional or temporary network congestion, provided that equivalent categories of traffic are treated equally". The cited provision is problematic for two reasons: first, the Regulation 2015/2120 does not define network congestion, and second, it does not contain provisions that would sufficiently differentiate between cases when reasonable or exceptional traffic management measures should be employed to control congestion. Understanding what the term "congestion" used in the Regulation 2015/2120 is supposed to mean is not as straightforward as it may seem. In networking literature congestion is "defined as a state or condition that occurs when network resources are overloaded, resulting in impairments for network users as objectively measured by the probability of loss and/or delay" [START_REF] Papadimitrou | Open Research Issues in Internet Congestion Control[END_REF]. However, other definitions of congestion are also presented in the literature [START_REF] Bauer | The Evolution of Internet Congestion[END_REF]. Legal and computer scientists should work together to provide a definition of congestion best suited to the application and enforcement of the Regulation 2015/2120.

Taken literally, Article 3 (3) (c) of the Regulation 2015/2120 implies that any "impending network congestion" justifies the use of exceptional traffic management measures. Such an interpretation however is unacceptable. As has been mentioned above, under the Regulation 2015/2120 solutions implemented in the network to aid in congestion control, such as Active Queue Management, generally fall into the category of reasonable traffic management. The use of exceptional rather than reasonable traffic management measures in order to control impending congestion is certainly justified in cases when a congestion collapse of the network is predicted based on the analysis of the network traffic. Other cases should be elaborated upon by computer scientists, considering both the need to safeguard the end-users' rights enshrined in the Regulation 2015/2120, and the technical realities of network traffic management.

Conclusions

Since the Regulation 2015/2120 came into force, all traffic management measures used by providers of internet access services in the EU must meet the requirements introduced by its provisions. Thus, the law has become one of the factors that have to be considered when designing and applying internet traffic management measures employed in networks used to provide publicly available internet access services in Europe. The proper interpretation of the Regulation 2015/2120, serving its sensible application and enforcement, necessitates the co-operation of legal and computer scientists. The former may not disregard the technical realities of managing internet traffic, which should be elaborated on by the latter, in turn taking legal considerations into account.

Computer scientists' input would be invaluable in clearing up the many ambiguities created by the Regulation 2015/2120 with respect to the various types of traffic management measures categorized by it, namely the basic, reasonable and exceptional ones. The distinction between these three categories of measures is of the utmost importance to the application of the Regulation 2015/2120, and yet it was not made clear enough by the legislators or BEREC. Computer scientists undoubtedly may help in determining which concrete, technical solutions or types thereof fall within each of the three categories. Specifically, even though the need to control congestion in the network is apparently one of the main reasons of introducing any traffic management measures, the Regulation 2015/2120 is unclear on when reasonable or exceptional measures may be used to control congestion. Computer scientists' input in resolving this conundrum, as well as interpreting how the term "congestion" should be understood for the purpose of applying and enforcing the regulation would be appreciated.

Computer scientists may also greatly assist in resolving the issues related to the treatment of various classes of traffic. The Regulation 2015/2120 allows the objectively justified differentiation in the treatment of traffic classes, based on their different technical quality of service requirements. Computer scientists may delineate classes of traffic with objectively different quality of service requirements, determine the means of identifying data flows as belonging to a specific class of traffic (traffic classification), and specify the circumstances under which the treatment of the various classes of traffic should be differentiated.

1 Introduction

 1 Internet traffic management was not traditionally regulated by EU law. However, that changed with the entering into force of the Regulation 2015/2120 of the European Parliament and of the Council of 25 November 2015 laying down measures concerning open internet access and amending Directive 2002/22/EC on universal service and users' rights relating to electronic communications networks and services and Regulation (EU) No 531/2012 on roaming on public mobile communications networks within the Union (further referred to as Regulation 2015/2120). The regulation introduced a network neutrality regime into EU electronic communications law, establishing the rights of end-users of internet access services and requiring the providers of those services to manage internet traffic respecting the rules set by the law.

  Traffic Management Measures May be Used. The circumstances under which exceptional traffic management measures may be used have been enumerated in Article 3 (3) (a)-(c) of the Regulation 2015/2120 and are discussed below. Compliance with Legislation. Under Article 3 (3) (a) of the Regulation 2015/2120, exceptional traffic management measures may be employed to ensure compliance with EU or national legislation, and with acts that comply with that legislation, e.g. decisions by courts or other public authorities. This involves inter alia the blocking of illegal content, such as child pornography under national laws resulting from the transposition of Article 25 of the Directive 2011/92/EU of the European Parliament and of the Council of 13 December 2011 on combating the sexual abuse and sexual exploitation of children and child pornography, and replacing Council Framework Decision 2004/68/JHA. The relevant legislation usually determines what specific traffic management measures should be employed to enact it, e.g. Domain Name Service blocking.


Florian Schanzenbächer
email: author.florian.schanzenbacher@ratp.fr

Nadir Farhi

Fabien Leurent

Gérard Gabriel

A discrete event traffic model explaining the traffic phases of the train dynamics on a linear metro line with demand-dependent control

In this paper we present a mathematical model of the train dynamics in a linear metro line system with demanddependent run and dwell times. On every segment of the line, we consider two main constraints. The first constraint is on the travel time, which is the sum of run and dwell time. The second one is on the safe separation time, modeling the signaling system, so that only one train can occupy a segment at a time. The dwell and the run times are modeled dynamically, with two control laws. The one on the dwell time makes sure that all the passengers can debark from and embark into the train. The one on the run time ensures train time-headway regularity in the case where perturbations do not exceed a run time margin.

We use a Max-plus algebra approach which allows to derive analytic formulas for the train time-headway and frequency depending on the number of trains and on the passenger demand. The analytic formulas, illustrated by 3D figures, permit to understand the phases of the train dynamics of a linear metro line being operated as a transport on demand system.

I. INTRODUCTION AND LITERATURE REVIEW

Innovative public transportation systems are nowadays more and more turned towards transport on demand systems. Transport on demand signifies that the transportation offer is adjusted in real-time to satisfy the passenger demand. This features trains stopping at stations just as long as needed to let passengers debark and embark. Irregularities on the train time-headway shall be recovered quickly to avoid a cascade effect, that is a longer headway causes a longer dwell time because of the accumulation of passengers, which causes an even longer headway. Therefore, transport on demand systems necessitate an efficient control. The latter detects and controls perturbations to offer a reliable service to costumers with a constant train frequency that equalizes the passenger charge over all trains.

Several approaches for control of mass transit metro systems have been proposed. The authors of [START_REF] Fernandez | Predictive traffic regulation for metro loop lines based on quadratic programming[END_REF] have applied a quadratic programming algorithm for optimal traffic control to a circular metro line, taking into account a constant passenger arrival rate at the platforms. In [START_REF] Schanzenbächer | Fluidification du trafic Transilien : approche prédictive et optimisation quadratique[END_REF] an optimal control strategy to gain on headway regularity for the traffic on a mass transit railway line using a quadratic programming algorithm has been proposed. This method has been extended by [START_REF] Li | Robust model predictive control for train regulation in underground railway transportation[END_REF] and [START_REF] Li | Joint optimal train regulation and passenger flow control strategy for high-frequency metro lines[END_REF], considering uncertain time-variant passenger demand. Moreover, the authors of [START_REF] Yin | Energy-efficient metro train rescheduling with uncertain time-variant passenger demands: An approximate dynamic programming approach[END_REF] have developed a real-time control approach for a metro system which deals with stochastic passenger demand on the one side, and optimizes energy consumption, on the other side. However, the models [START_REF] Fernandez | Predictive traffic regulation for metro loop lines based on quadratic programming[END_REF], [START_REF] Schanzenbächer | Fluidification du trafic Transilien : approche prédictive et optimisation quadratique[END_REF], [START_REF] Li | Robust model predictive control for train regulation in underground railway transportation[END_REF], [START_REF] Li | Joint optimal train regulation and passenger flow control strategy for high-frequency metro lines[END_REF], [START_REF] Yin | Energy-efficient metro train rescheduling with uncertain time-variant passenger demands: An approximate dynamic programming approach[END_REF] allow to represent only one train per inter-station [START_REF] Breusegem Van | Traffic Modeling and State Feedback Control for Metro Lines[END_REF], which limits their practical relevance, especially in networks with longer inter-station distances. The Max-plus approach features many advantages for the control of discrete event systems, like metro lines, which are discretized into segments. For more details on the Max-plus algebra refer to [START_REF] Baccelli | Synchronization and linearity : an algebra for discrete event systems[END_REF].

Demand-dependent control of transportation systems modeled in Max-plus algebra, has only recently begun to be treated in scientific research. The authours of [START_REF] Farhi | Traffic modeling and real-time control for metro lines. Part I -A Max-plus algebra model explaining the traffic phases of the train dynamics[END_REF] have presented a Max-plus model for a linear metro line with constant dwell and run times. The authors of [START_REF] Farhi | Traffic modeling and real-time control for metro lines. Part II -The effect of passengers demand on the traffic phases[END_REF] have adopted this model for a linear metro line with dynamic dwell times depending on the passenger demand. Their model includes an efficient control strategy which stabilizes the system in case of perturbations on the train time-headway. In their approach, longer headways are recovered by shortening dwell times of retarded trains. This control strategy is efficient since, in case of perturbations, trains running on schedule are not retarded, but trains running behind schedule are accelerated via shorter dwell times to recover delays and to harmonize headways. However, retarded trains generally face higher demand at the upcoming stops, because passengers have been accumulating over a longer time interval. Consequently, recovering perturbations by cutting down dwell times is counterintuitive.

Therefore, in this paper we propose a mathematical Maxplus model for a linear metro line with demand-dependent dwell and run times. The dwell time takes into account passenger upload and download rate as given by the vehicle characteristics, as well as passenger demand emissions and attractions of the platforms. Perturbations on the train timeheadway are recovered by shorter run times on the following inter-stations to avoid the amplification of the perturbation and a cascade effect. The here presented mathematical model features a derivation of analytic formulas of the average train time-headway and frequency depending on the number of trains on the line and on the passenger demand.

In our future work, we will extend this model to cope with the stochastic nature of the passenger demand. The authors of [START_REF] Loenhout Van | Model predictive control for stochastic switching max-plus-linear systems[END_REF] have published on model predictive control for stochastic switching Max-plus linear systems.

II. REVIEW ON MAX-PLUS ALGEBRA

As having been said above, Max-plus algebra has several advantages for modeling discrete event systems.

Max-plus algebra [START_REF] Baccelli | Synchronization and linearity : an algebra for discrete event systems[END_REF] is the idempotent commutative semiring (R ∪ {-∞}, ⊕, ⊗), where the addition and the multiplication of a, b is defined by: a ⊕ b = max{a, b} and a ⊗ b = a + b. This allows to model the dynamics of a metro system in a matrix form which is highly advantageous for deriving analytic formulas of the train dynamics, as the average train time-headway and frequency, as well as for real-time traffic control.

For more details on how to model the train dynamics on a metro line in Max-plus algebra and how to technically derive analytic formulas of the average train time-headway and the frequency refer to [START_REF] Farhi | Traffic modeling and real-time control for metro lines. Part I -A Max-plus algebra model explaining the traffic phases of the train dynamics[END_REF]. The main Theorem we use here to derive the analytic formulas for the average asymptotic train time-headway and frequency is from [START_REF] Cochet-Terrasson | Numerical computation of spectral elements in maxplus algebra[END_REF] and has already been applied in [START_REF] Farhi | Traffic modeling and real-time control for metro lines. Part I -A Max-plus algebra model explaining the traffic phases of the train dynamics[END_REF].

III. TRAIN DYNAMICS

This work is based on [START_REF] Farhi | Traffic modeling and real-time control for metro lines. Part I -A Max-plus algebra model explaining the traffic phases of the train dynamics[END_REF] where a discrete event traffic model explaining the traffic phases of the train dynamics on a linear metro line has been proposed. In [START_REF] Farhi | Traffic modeling and real-time control for metro lines. Part I -A Max-plus algebra model explaining the traffic phases of the train dynamics[END_REF], lower bounds are imposed to the run and dwell times, independent of the passenger travel demand. We extend here the model presented by the authors of [START_REF] Farhi | Traffic modeling and real-time control for metro lines. Part I -A Max-plus algebra model explaining the traffic phases of the train dynamics[END_REF], accounting for the passenger travel demand in the run and dwell time dynamics. As the authors of [START_REF] Farhi | Traffic modeling and real-time control for metro lines. Part I -A Max-plus algebra model explaining the traffic phases of the train dynamics[END_REF], we consider a space discretization of the metro line into n segments, as shown in Fig. 1. Let us use similar notations to those used in [START_REF] Farhi | Traffic modeling and real-time control for metro lines. Part I -A Max-plus algebra model explaining the traffic phases of the train dynamics[END_REF].

d k j
the k th departure time of trains on segment j. a k j the k th arrival time of trains on segment j. r j the average running time of trains on segment j (between nodes j -1 and j).

w k j = d k j -a k j the k th dwell time on node j. t k j = r j + w k j the k th travel time from node j -1 to node j. g k j = a k j -d k-1 j the k th safe separation time (or close-in time) at node j. h k j = d k j -d k-1 j = g k j + w k j the k th departure time-headway at node j. s k j = g k+bj j -r j . b j ∈ {0, 1}. It is 0 (resp. 1) if there is no train (resp. one train) at segment j. bj = 1 -b j .
Lower and upper bounds are written r j , w j , t j , g j , h j and wj , tj , ḡj , hj where w j is the minimum dwell time, regarding the demand, and r j the minimum run time, given by the fastest speed profile. The average on k and j of the quantities above are denoted r, w, t, g, h and s. We have the following relationships [START_REF] Farhi | Traffic modeling and real-time control for metro lines. Part I -A Max-plus algebra model explaining the traffic phases of the train dynamics[END_REF].

g = r + s, (1) t
= r + w, (2) h = g + w = r + w + s = t + s. (3)
We replace now the static run and dwell times by a model giving these two variables as a function of the demand and the vehicle characteristics. Moreover, we suppose that the demand for the next train is completely satisfied by this train, that means all the passengers willing to debark will get off and all the ones on the platform will board. This assumption is necessary since, at this point, we do not take into account the capacity of platforms and trains. It will be done in a further step. Furthermore, the following passenger demand model supposes an average demand for each platform. We use the following additional notations. λ ij passenger travel demand from platform i to platform j, when i and j denote platforms, and We define x j as a passenger demand parameter

λ ij = 0 if i or j is not a platform node. λ in i = j λ ij the
x j = λ out j α out j + λ in j α in j , (4)
such that x j h j represents the time needed for passenger down-and upload at platform j. The model presented in [START_REF] Farhi | Traffic modeling and real-time control for metro lines. Part I -A Max-plus algebra model explaining the traffic phases of the train dynamics[END_REF] considers two time constraints:

• A constraint on the travel time on every segment j.

d k j ≥ d k-bj j-1 + t j . (5)
• A constraint on the safe separation time at every segment j.

d k j ≥ d k-bj+1 j+1 + s j+1 . (6)
In [START_REF] Farhi | Traffic modeling and real-time control for metro lines. Part I -A Max-plus algebra model explaining the traffic phases of the train dynamics[END_REF], t j in constraint ( 5) is written t j = r j + w j since running times r j are assumed to be constant over time, and dwell times w j are lower bounded without any dependence on the passenger travel demand. In the model we present here, we modify constraint (5), whereas we keep constraint (6) unchanged. We replace constraint (5) with the following.

d k j ≥ d k-bj j-1 + t k j (h k j , x j ) = d k-bj j-1 + r k j (h k j , x j ) + w k j (h k j , x j ), (7)
In constraint [START_REF] Li | Robust model predictive control for train regulation in underground railway transportation[END_REF], r k j , w k j and thus t k j are functions of the train time-headway h k j , and of the passenger demand. The function t k j (h k j , x j ) is the control law which makes the train dynamics adaptive with respect to train delays and to passenger demand.

We pass here from static travel times, which are supposed to respect lower bounds, to a travel time model which is a function of the passenger demand.

Moreover, the dwell time, the first component of the travel time, is a linear function of the train time-headway. The run time, the second component of the travel time, is as well a function of the train time-headway and will be used to recover perturbations on the dwell time. We detail below the two functions r k j (h k j ) and w k j (h k j ).

A. The dwell time control law

Let us first recall that in [START_REF] Farhi | Traffic modeling and real-time control for metro lines. Part II -The effect of passengers demand on the traffic phases[END_REF], two dwell time models depending on the passenger travel demand have been considered. The first model is of the form

w k j = λ j α j h k j . (8)
We can see that the dwell time takes into account the needed upload time without taking into account the needed download time. This model permits to adjust the dwell times on platforms in function of the arrival demand. However, it has been shown in [START_REF] Farhi | Traffic modeling and real-time control for metro lines. Part II -The effect of passengers demand on the traffic phases[END_REF] that the train dynamics is not stable in this case. That means train delays are amplified over time and propagate backwards through the metro line.

To deal with the latter problem, the authors of [START_REF] Farhi | Traffic modeling and real-time control for metro lines. Part II -The effect of passengers demand on the traffic phases[END_REF] have proposed a second model for the train dwell times at platforms. The model is of the form

w k j = wj -θ j λ j α j h k j , (9)
with θ j being a control parameter to be fixed. The authors have shown that the model ( 9) guarantees the stability of the train dynamics, and that the dynamics admit an asymptotic regime with an asymptotic average train time-headway. The latter is derived by simulation in function of the number of trains and of the level of the passenger demand. However, with the control law of model ( 9), a delayed train at platform j, which induces an accumulation of passengers at platform j, will reduce the train dwell time at that platform, what is absurd for passengers.

We propose here a model which resolves all those problems. The dwell time model is similar to the model ( 8), but it also takes into account the attraction term of the travel demand. In order to deal with the instability issue, we complete the dwell time model with a running time model which cancels the terms that causes instability in the dwell time model. More precisely, for a train delay at a platform j, the dwell time model extends the dwell time at that platform in order to satisfy the needed download and upload times. The running time model will reduce the running time from platform j to platform j + 1 in order to compensate in such a way that the whole travel time (t = w + r) is stable.

The dwell time model is the following.

w k j (h k j , x j ) = min(x j h k j , wj ), (10)
where the k th dwell time on platform j is the minimum time for passenger up-and download on the platform, and a maximum dwell time wj to avoid train congestion behind a train stopping too long. The lower bounds on w j are fixed as follows (upper bounds are fixed accordingly).

w j = x j h j , (11)
where h j are derived from given g j , from the formula h = g + w of (3), and from the dwell time law [START_REF] Schanzenbächer | A discrete event traffic model explaining the traffic phases of the train dynamics in a metro line system with a junction[END_REF], as follows.

We have

h j = g j + w j = g j + x j h j . (12)
Then we get

h j = 1/(1 -x j ) g j , (13)
and then

w j = X j g j , (14)
where X j := x j /(1 -x j ). Let us notice, that (14) corresponds to the minimum dwell time satisfying the demand. The upper bounds are fixed similarly. From the point on we observe a small perturbation on the train time-headway, the following longer dwell time, modeled in equation ( 10), will make this perturbation growing, since longer headways result in longer dwell times which results in even longer headways.

Our aim is to recover perturbations on the dwell time by a dynamic running time model in order to stabilize the system.

B. The running time control law

We propose the following running time law.

r k j (h k j , x j ) = max r j , rj -x j h k j -h j , ( 15
)
where rj is the average (nominal) running time of trains on segment j. The model (15) gives the running time as the maximum between a given minimum running time r j and a term that subtracts x j h k j -h j from the nominal running time. The term x j h k j -h j expresses a deviation of the upload and download time, due to a deviation of the train time-headway. We notice here that the term x j h k j , appearing in the dwell time law (10) with a sign "+", appears in the running time law (15) with a sign "-".

Combining the dwell time law [START_REF] Schanzenbächer | A discrete event traffic model explaining the traffic phases of the train dynamics in a metro line system with a junction[END_REF] with the running time law (15), we obtain the following train travel time law.

t k j (x j ) = r k j (h k j , x j ) + w k j (h k j , x j ). ( 16
)
Let us use the notations.

∆h j := hj -h j , ∆g j := ḡj -g j , ∆w j := wj -w j , ∆r j := rj -r j , with hj being the longest headway observed. It is then easy to check the following.

∆w j = x j ∆h j = X j ∆g j , ∀j.

Then we have the following result. Theorem 1: If h 1 j ≤ hj = 1/(1 -x j ) ḡj , ∀j and ∆r j ≥ ∆w j = X j ∆g j , ∀j, then the dynamic system ( 6)-( 7) is a Max-plus linear system, and is equivalent to

d k j ≥ d k-bj j-1 + rj + X j g j . ( 17
)
d k j ≥ d k-bj+1 j+1 + s j+1 . (18)
Proof: By induction, let us first show that ( 6)-( 7) is equivalent to (17)-( 18) for k = 1.

• On the one side, we have x j h 1 j ≤ x j hj = wj . Therefore, the first term realizes the minimum in [START_REF] Schanzenbächer | A discrete event traffic model explaining the traffic phases of the train dynamics in a metro line system with a junction[END_REF]. That is, ( 10) is equivalent to

w 1 j (h 1 j , x j ) = x j h 1 j . ( 19
)
• On the other side, we have rj -

x j (h 1 j -h j ) ≥ rj -x j hj -h j = rj -∆w j ≥ rj -∆r j = r j .
Therefore, the second term realizes the maximum in (15). That is, ( 15) is equivalent to

r 1 j (h 1 j , x j ) = rj -x j h 1 j -h j . ( 20
)
Consequently, (16) gives

t 1 j (x j ) = rj + x j h j = rj + X j g j . (21)
Then ( 7) can be written

d 1 j ≥ d 1-bj j-1 + rj + X j g j .
Let us now show, that if ( 6)-( 7) is equivalent to (17)-( 18) for a given k, then it holds also for k + 1. Since it holds for k, then a Max-plus linear dynamics (17)-(18) will be applied for k. We notice by f the Max-plus map of the Maxplus dynamics. We know that Max-plus linear maps are 1-Lipschitz for the sup. norm. The assertion holds for k means that h k j ≤ hj , ∀j.

Then ||d k -d k-1 || ∞ ≤ hj . Hence ||d k+1 -d k || ∞ = ||f (d k )-f (d k-1 )|| ∞ ≤ ||d k -d k-1 || ∞ ≤ hj .
Therefore h k+1 j ≤ hj , ∀j. Then, we can easily show (as done for k = 1) that ( 6)-( 7) is equivalent to (17)-( 18) for k + 1.

Let us interpret the two conditions of Theorem 1.

• Condition h 1 j ≤ hj = 1/(1 -x j ) ḡj , ∀j, k limits the initial headway h 1 j (i.e. the initial condition) to its upper bound hj , which is given by the level of the passenger travel demand x j at platform j, and by the upper bound ḡj on the safe separation time at the same platform. We notice here that a big value of h k j corresponds to a delay of the k th train passing by platform j. Therefore, this condition, tells that if all the delays expressed by h k j , ∀j are limited to hj , ∀j, then the dynamic system is Maxplus linear, and is then stable, and admits a stationary regime. In other words, the train dynamics is stable under small disturbances.

• Condition ∆r j ≥ ∆w j = X j ∆g j , ∀j limits the margin on the train dwell times to the margin on the train running times. Indeed, as explained above, the model consists in responding to disturbances and train delays by first extending the train dwell times so that the passengers accumulated on the platforms have time to embark, and second, by recovering the dwell time extension by reducing the train run times on the downstream inter-stations. In order to ensure this recovering to be possible, and then the dynamic system to be stable, we need the margin on the running times to be at least equal to the margin on the train dwell times at the platforms. It is important to notice that fixing bounds on r and on g will impose bounds on s, according to [START_REF] Baccelli | Synchronization and linearity : an algebra for discrete event systems[END_REF].

g j = r j + s j .
In (17), the travel time t k j (x j ) depends on the demand X j , weighted by the minimum safe separation time g j . This dependence is linear on X j . We notice also that, in general we have λ out j , λ in j << α out j , α in j . Therefore x j can be assumed to be close to 0. By consequent, a linearity with respect to X j is a kind of exponential relationship with respect to x j , since X j = x j /(1 -x j ). Condition ∆r j ≥ ∆w j = X j ∆g j , ∀j of Theorem 1 makes a link between two important parameters which are the margin on the running time control ∆r j , and the level of the travel demand X j . Finally, the dynamic modeling of the dwell and run time allows to adjust the run time margin ∆r j depending on the demand. The system can be optimized with regard to train frequency or stability.

In the following section, we give directly the analytic formulas for the average asymptotic train time-headway h and the average asymptotic frequency f for the three traffic phases, which can be derived as in [START_REF] Farhi | Traffic modeling and real-time control for metro lines. Part I -A Max-plus algebra model explaining the traffic phases of the train dynamics[END_REF].

IV. THE TRAFFIC PHASES

The Max-plus theorem derived in ([4], Theorem 1) gives the average asymptotic train time-headway on a linear metro line as a function of the number of trains circulating on the line. Let us consider the following notations. m the number of trains. n the number of segments on the line. X the vector of demand parameters X j , ∀j.

Since the here presented model can, as the model with static run and dwell times in [START_REF] Farhi | Traffic modeling and real-time control for metro lines. Part I -A Max-plus algebra model explaining the traffic phases of the train dynamics[END_REF], be written in Max-plus algebra, we can directly replace t j in ([4], Theorem 2) by t j (x j ). h(m) = max j t j (x j ) m , max j (t j (x j ) + s j ), j s j n -m .

(22) By replacing t k j (x j ) using ( 16), we obtain the following result.

Theorem 2: The average asymptotic train time-headway of the linear Max-plus system with dynamic dwell and run times depending on the demand is given by the average asymptotic growth rate of the system. The average headway depends on the number of trains m on the line and the passenger travel demand for every station X j .

h(m, X) = max              j (g j Xj +rj )
m , max j ((g j X j + rj ) + s j ), j s j n-m . Proof: Under the conditions of Theorem 1, the system can be written in Max-plus algebra. It has been shown in [START_REF] Farhi | Traffic modeling and real-time control for metro lines. Part I -A Max-plus algebra model explaining the traffic phases of the train dynamics[END_REF] that in this case, the average asymptotic growth rate of the system can be analytically derived and corresponds to the average headway. We replace t j in the Max-plus theorem of [START_REF] Farhi | Traffic modeling and real-time control for metro lines. Part I -A Max-plus algebra model explaining the traffic phases of the train dynamics[END_REF] and obtain demand-dependent headway equation. We notice that h depends not only on the number of trains m, but furthermore on a weighted mean of the demand j (g j X j + rj ), and on the maximum of all the station demands max j ((g j X j + rj ) + s j ).

We obtain the analytic formulas for the average train frequency on the line directly from Theorem 2.

Corollary 1: The average frequency of the linear Maxplus system with demand-dependent dwell and run times is a function of the number of trains and the travel demand. Table I depicts the three headway phases derived in Theorem 2, as well as the three frequency phases of Corollary 1 for a linear metro line with 18 stations from the RATP network in Paris. The demand level has been assumed to be equal for all the platforms. The graphs in the left column are given in function of the average demand parameter x = (λ out /α out + λ in /α in ) , whereas the ones in the right column depict the main results over X = x/(1 -x).

f (m, X) = min                m j (g j Xj +rj ) ,
Let us interpret the first two graphs with the results of Theorem 2. The headway in the free flow traffic phase depends on X and on m. For a fixed m, the headway is linear in X. In the maximum frequency phase, the headway is given by the maximum max j ((g j X j + rj ) + s j ). The headway in the maximum frequency phase is here linear in X and moreover independent of m. In the congested traffic phase, the analytic formula shows that the headway is independent of the demand. Furthermore, the headway increases with m.

On the left side, the headway is depicted over x. In the free flow traffic phase, the headway is non linear, even for a fixed m. Since X = x/(1 -x), the headway is non linear in the maximum frequency phase. Finally, in the congested traffic phase, the headway is independent of the demand and increases with m.

Let us now look at the frequency, as given by Corollary 1. In the free flow traffic phase, for a fixed m, the frequency is non linear in X, since the headway is linear here and f = 1/h. For a fixed X, the frequency is linear in m. The same can be observed for the maximum frequency phase, which is non linear in X, because the headway is linear in X here. In the congested traffic phase, the frequency is linear in m.

For the frequency given over x, the free flow traffic phase in non linear for fixed m, but it is linear in m for a fixed x. In the maximum frequency phase, the frequency is independent of the number of trains m, and it is non linear in the demand parameter x. Finally, as above, the frequency is linear in m in the congested traffic phase, whereas it degrades linearly.

V. CONCLUSIONS

We have presented here a mathematical model which describes the train dynamics of a linear metro line, and proposes a traffic control strategy. The model considers two control laws, one for the train dwell and one for the run times, taking into account the passenger demand. It is a discrete event traffic model. The approach is based on the Max-plus algebra which has allowed us to derive analytic formulas for the average asymptotic train timeheadway and frequency, in function of the number of trains and of the passenger demand, with the characteristics of the infrastructure and the vehicles being taken into account, too.

Our approach solves the problem that delayed trains facing a higher demand on the upcoming stations, are more and more delayed, which leads to a serious destabilization of the system. The model detects such a possible cascade effect. In case of a delayed train to a given platform, where passengers have been accumulating, the train dwell time is extended to respond to the passenger demand. To avoid the amplification of the train delay, the running time is reduced in the following inter-station to recover the dwell time extension. Consequently, the train dynamics responds to the passenger demand, and remains stable.

Theorem 1 has shown that dwell times can only be extended up to the margin on the running time in the following inter station. However, as given in Theorem 2, the average asymptotic headway increases with the margin on the run time. That means, a compromise has to be found to satisfy both a high frequency service on the one side, and to guarantee system stability, on the other side.

In our future work, we will apply this approach to a metro line with a junction. Moreover, we will propose a model for the passenger stock at the platforms and in the trains. In an ultimate version, we will finally handle the uncertainties in the system, due to the stochastic nature of the passenger demand.

Fig. 1 .

 1 Fig. 1. A linear metro line representation.

  average passenger arrival rate on origin platform i to any destination platform. λ out j = i λ ij the average passenger arrival rate on any origin platform to destination platform j. α in j average passenger upload rate on platform j. α out j average passenger download rate on platform j.

  i λ ij h i /α out j time for passenger download at platform j.

  for passenger upload at platform j.

  Directly from Theorem 2 with f = 1/h.