M Gravel

S Kaschke

Jens Schneider

Places of Power in the Realm of Louis the Pious 1

ANR-DFG project "HLUDOWICUS. La productivité d'une crise : Le règne de Louis le Pieux (814-840) et la transformation de l'Empire carolingien -Produktivität einer Krise: Die Regierungszeit Ludwigs des Frommen (814-840) und die Transformation des karolingischen Imperium" (2008-2011) Nobody will deny the importance of space or the nearly constitutional function of several royal palaces in the ninth century, like Aachen during the reign of Charlemagne and Louis the Pious, like Compiègne for Charles the Bald or Regensburg for Louis the German. But today, in 2009, we have now had two thirds of a century of scholarship in which authority in the Early Middle Ages has generally been conceived as based on people and relations between them. Since the 1940s there has been a consensus among historians to focus on the relationships of the important noble families, a consensus which persisted through all epistemological fashions: the key word is network.

German historical tradition after the war established the idea of the 'Personenverbandsstaat', a conception of a medieval constitution which is not a state in the modern sense 2 . This

conception has been criticised and more or less abolished because, to point out just one reason, of the self-representation of the early medieval aristocracy as feeling a higher responsibility for the res publica3 . As Karl Ferdinand Werner demonstrated, this seems to have been an important marker of distinction for the noblesse4 . I do not pursue this idea here because it leads to a major theoretical argument, as part of which historians as Hans-Werner Goetz, Jörg Jarnut and Walter Pohl have defended the position that there were structures of 1 Paper presented at the International Medieval Congress, Leeds, 2009, July 14 th , as part of the session 'Louis the Pious and the Crisis of the Carolingian Empire' organized by the ANR-DFG project HLUDOWICUS. Among the important work published since 2010 on the problems this paper deals with I should like to mention GRAVEL, Distances, rencontres, communications, and MACLEAN, Palaces, Itineraries. to be published in M. Gravel & S. Kaschke (ed.), Politische Theologie und Geschichte unter Ludwig dem

Frommen -Histoire et théologie politiques sous Louis le Pieux, Ostfildern (Thorbecke) 2013 2 administration and government that we can already call a state 5 . I just want to stress the point that it is too simple to say that early medieval society was only people getting along with each other. So, 'Personenverbandsstaat' is history. Yet, historians continue to analyse society, both from an institutional or constitutional point of view -the rules and norms of a society -and from an anthropological perspective, analysing the rituals and taboos organizing life in groups or clans. Much important and admirable work has been done in the study of elites and réseaux de parenté 6 . Braudel's conception of space 7 as a determinant factor of history is far away in today's historical research. I do not intend to deny for my part the importance of studying these networks, nor do I claim to proceed without noticing political actors and their interactions. But I think that different approaches are possible and necessary.

Ethnological work has shown the importance of places for the representation of a society, but also for everyday life. This seems particularly true for the memory of the foundation of places. American Maya Indians of the classical period (4 th to 10 th century) lived in proper city-states showing an absolute respect for their foundation places; there were no signs of expansion although they were powerful enough to control dependant smaller cities. Instead, they proceeded to a permanent re-foundation within the limits of their city-states 8 . The Yanomami Indians in South America showed, on the other hand, dynamic spatial rituals 9 .

They used the same name for both their habitat and their community. Once they moved to the next place, they would use another name for themselves, the old name becoming a period in the past. Another well-documented example is that of the Australian Aborigines who do not mark the frontiers of their territory but who have their own rituals of memorizing and remembering it. These are the famous 'songlines': secret traditional songs that contain a very well-defined way to 'walk their land', as they put it 10 . At the same time the song is a way of paying respect to the land and the ancestors.

These three cases may show how earlier societies performed their given space. Joseph Morsel pointed out the meaning of the Aborigines' songlines as a technique that bound a community Christaller insists on one important point, which is that every central place is dynamic, in other words subject to change.

Then there is to mention the work of Eckhard Müller-Mertens who developed the method of cartographical visualization of the 'Reichsstruktur' especially for the tenth century but proceeded to its application on other periods afterwards 16 . Every historian working on places or itineraries knows his maps, which are, though, limited on the material aspects, that's to say the king's movements and property.

More recently, 'Topographies of Power' is a collection of some rather important case studies.

Janet Nelson's analysis of Aachen as the constructed Carolingian place of power par excellence goes much further than Müller-Mertens, Falkenstein or Binding17 . Aachen has nevertheless not always been the lieu de mémoire that we are used to18 , just as Metz seems to lose political weight in the later ninth and tenth century: the concrete importance of these two

Carolingian palaces has to be reviewed. Frans Theuws, taking the example of Maastricht in his contribution to the same volume, describes in anthropological terms the function of a relay, a centre of communicating spaces 19 . His point is to define the role of a centre by analysing both the more or less constructed symbolic past of a place and, at the same time, how it serves as a place of exchange, meaning both trade and social exchange.

This essay is focussed on Louis the Pious and the question of which places were the most important for royal administration and authority. Based on the cited studies I put forward five criteria for a place of power in the Early Middle Ages. I suggest the following categories:

(1) Episcopal sees (sedes episcoporum)

(2) mints

(3) royal palaces (palatia) (4) royal assemblies (placita)

(5) royal charters (diplomata) These categories will be examined in the following. Of course, this is no more than a rough sketch of a possible approach that needs to be refined; other categories might be added, the main routes for instance, as well as other vectors of communication.

Sedes episcoporum

Even if nowadays one does not believe anymore in the long term constancy of the territorial shape of the dioceses, the ecclesiastical civitas remains an eminent functional structure in the early middle ages 20 . In the Frankish kingdom of the ninth century Liège on the river Meuse had just replaced Maastricht as Episcopal see 21 . In the North of modern-day France, Arras and

Tournai cannot be taken into consideration because they were administrated by the bishops of Cambrai and Noyon until the eleventh century 22 .

Mints

Rosamond

Palatia

Far more than the mints, the royal palaces can be considered as symbols of royal or imperial authority 26 . The palatial system is older than the Carolingians: we know that the Merovingian kings used palaces in their cities, but also for their hunting expeditions, especially in the I now want to come back to my initial goal, that is to undertake a systematic approach to the presence of the king's body. In other words, I wish to ask how we might define his preferred and privileged places, the space covered by his immediate authority.

to be published in M. Gravel & S. Kaschke (ed.), Politische Theologie und Geschichte unter Ludwig dem Frommen -Histoire et théologie politiques sous Louis le Pieux, Ostfildern (Thorbecke) 2013

The five categories discussed above were the Episcopal sees, the mints, the royal palaces, the assemblies and the enactment of charters. There are only two places which answer to four of the five criteria. It will not come as a surprise that these are Aachen and Worms. Mainz would be a third but we do not have any certain evidence for a royal palace there at the time of Louis the Pious; the emperor might easily have stayed at the monastery of St Albans or rather at his nearby palace at Ingelheim 34 . Worms is one of the important palaces where assemblies took place and charters were given; it is an Episcopal see and it had a prestigious royal palace, rebuilt after it burned in 790/91 35 . Aachen was not a bishopric but it had a famous royal palace and a mint. Given that in Aachen by far the highest number of assemblies were held and highest number of charters given it seems to be justified to declare Aachen as the real sedes regni at the time of Louis the Pious, just as medieval authors did 36 . By a remarkable distance it is followed by Worms, then Mainz which had a mint, too, and finally, given the number of assemblies held there, Nijmegen and Compiègne.

So far, so good: but where is the crisis? Did nothing change in Louis' whereabouts? It has been said, especially by German historians working on the charters, that his reign can be seen as the succession of three phases: 'the energetic beginning, the "stagnation and paralysing tension in the 820s", and finally the crisis of the last decade', as Mayke de Jong has put it in a deconstructivist perspective 37 . As a consequence of the crisis in the 820s, the centre of Louis

  and reinstallation on the throne in 83513 .If we focus on the importance of space, in particular distinct places for the exertion and the representation of power in his realm, we should ask what kind of criteria and what kind of geographical, or -better -spatial, categories may be appropriate in defining places of power 14 . To avoid the network idea of important bishoprics and monasteries to describe important places I tried to establish another, more systematic approach. One of the most important inspirations certainly is the theory of central places, published by Walter Christaller for the first time in 1933 15 . He defines, from a geographical point of view, nine criteria to describe a central place: institutions of (political) administration, cultural and ecclesiastical (!) institutions, institutions of public health and welfare, of social and economic live, of trade and monetary circulation, of production, of labour market, of traffic, and of market regulation. to be published in M. Gravel & S. Kaschke (ed.), Politische Theologie und Geschichte unter Ludwig dem Frommen -Histoire et théologie politiques sous Louis le Pieux, Ostfildern (Thorbecke) 2013 4

  McKitterick recently reminded us that mint distribution 'mirrors the concentration of the king's movements'23 . Thanks to the cooperation with a project on early medieval minting we have a revised list of the mint places under Louis the Pious 24 . On the first map we can observe a rather homogeneous net all over the empire but nothing east of the Rhine, except Regensburg on the river Danube. The mint places are essentially bishoprics and only a few trading places like Dorestad on the lower Rhine and Quentovic on the Channel coast. In Aquitaine there is a well-known mint at Melle (Metallum) near Poitiers, but not Limoges, though it had been known as a mint place at the end of the eighth century when Charlemagne struck coinage for his son Louis king of Aquitaine 25 . On the other hand we can see that coins were minted at Dax, which had lost its Episcopal see at the time of Louis the Pious.

  Ardennes and the Vosges, in Lotharingia 27 . Still, it was significant if a king built a new palace20 Cf MAZEL (ed.), L'espace du diocèse; PATZOLD, Raum der Diözese; ID., Episcopus. Charlemagne did in Paderborn and Nijmegen, or if he chose to enlarge an inherited palace as was done by Louis' sons, Louis the German with Frankfurt and Charles the Bald with Compiègne 28 . The difficulties in compiling a reliable catalogue of royal palaces are well known, as well as the problems of distinction between a completely equipped palatium, a smaller aula or just a villa 29 . For the present purpose, the extremely useful list of charters of Louis the Pious containing some 417 items, established by Jens Peter Clausen and Theo Kölzer, served as a basis, since the new edition prepared at Bonn is not yet published 30 .4. PlacitaAnother issue of Kölzer's editing project for the Monumenta Germaniae historica is the study of the assemblies held by Louis the Pious. From Daniel Eichler's table of 61 assemblies, the one held at Compiègne from October to November 833 has not been counted for the present purpose because Louis the Pious was absent 31 . It was then when Lothar and the Frankish bishops listed all the sins and faults of the dethroned emperor 32 . Eichler counts one each at Worms in the East and Compiègne in the West. Outside this frequently visited space there seem to have been limitrophic assemblies held on the Loire, the Saône and Rhone valley or at Augsburg and Paderborn. The later on in his reign, we deduce from the dark grey points, the more Louis the Pious was obliged to travel to the South-West of his empire. DiplomataWe get confirmation of this phenomenon in analysing the production of the chancery of Louis the Pious. The above-mentioned list by Kölzer contains 417 charters in total. Our third map shows the places where royal charters were given in four chronological phases, from yellow to red. At the same time, the map allows us to compare the principal places of enactment. If we relate the number of issued charters to the frequency and length of the visits Louis paid to these places, we can confirm that he preferred Aachen by far. Kölzer lists 191 charters given in this palace: Aachen is thus 'unbestrittene Hauptresidenz', followed by the middle Rhine region with Ingelheim (23), Frankfurt (23) and Worms (11)33 . At Thionville, Louis the Pious gave 12 charters and held five assemblies. These numbers confirm that the Thionville palace was an important element of the described triangular space where Louis spent most of his time: already more important than Metz that is outside of this central region.

	5 possible criteria of a place of power
	Frankish Empire under Louis the Pious (814-840)
		Aachen	Compiègne	Frankfurt	Ingelheim	Mainz	Thionville	Worms
	Episcopal see					x		x
	Mint	x				x
	Royal palace	x	x	x	x	?	x	x
	Assemblies	18	5	2	5	2	5	4
	Charters	191	9	23	23		12	11
	Criteria	4	3	3	3	3	3	4
	21 KUPPER, Liège, 78.
	22 MERIAUX, Gallia irradiata, 20.
	23 MCKITTERICK, Charlemagne, 168.
	24 DFG projekt 'Die Merowingischen Monetarmünzen als interdisziplinär-mediaevistische Herausforderung'
	(2007-2009). Cf JARNUT / STROTHMANN (ed.), Monetarmünzen. -Thanks to Jürgen Strothmann, University of 28 BAUTIER, Le poids de la Neustrie, 557; BINDING, Deutsche Königspfalzen, 118; ZOTZ, Le palais et les elites,
	Paderborn (now University of Siegen). 233; BARBIER, Le sacré dans le palais franc; MCKITTERICK, Charlemagne, 161.

25 COUPLAND, Trading Places, 221; cf ID., Money and coinage, 24 and Coinages of Pippin, 197. 26

Among the enormous bibliography I only mention BARBIER, Le système palatial; BRÜHL, Palatium und Civitas; EHLERS (ed.), Places of Power; ID. (ed.), Orte der Herrschaft; FENSKE / JARNUT / WEMHOFF (ed.), Splendor palatii; STAAB (ed.), Die Pfalz;

ZOTZ, Thomas, Pfalz und Pfalzen, in: RGA² 22 (2003)

, 640-645. 27 HENNEBICQUE [R. LE JAN], Espaces sauvages; WENSKUS, Reinhard, Forst, § 2: Historisches, in: RGA² 9 (1995), 348-350. to be published in M. Gravel & S. Kaschke (ed.), Politische Theologie und Geschichte unter Ludwig dem Frommen -Histoire et théologie politiques sous Louis le Pieux, Ostfildern (Thorbecke) 2013 as Paderborn, Augsburg, Vannes, Orléans, Tours, Le Palais-sur-Vienne, Langres, Tramoyes et Chalon-sur-Saône; two at Attigny, Frankfurt and Mainz; three at Quierzy; four at Worms; five at Nijmegen, Ingelheim, Compiègne and Thionville; and nearly four times as many again (18) held at Aachen. The second map thus shows, somewhat surprisingly, a concentration of Louis' 'parliamentary' activity, to use an anachronistic term, more or less along the middle and lower Rhine valley. After the nearly exclusive assembly place of Aachen in the first years we can observe a kind of triangle with the edge points at Nijmegen in the North, Frankfurt-Ingelheim-29

ZOTZ, Vorbemerkungen; BINDING, Deutsche Königspfalzen, 21-26. See for such a catalogue BARBIER, La fortune du prince (583 items, forthcoming). 30 KÖLZER, Kaiser Ludwig der Fromme, unpaginated appendix [40-65]. -Thanks to professor Theo Kölzer, University of Bonn. 31 EICHLER, Fränkische Reichsversammlungen, 111-113. 32 Relatio episcoporum a. 829 (see the present volume), 40-41. Cf DE JONG, penitential state, 228-234. to be published in M. Gravel & S. Kaschke (ed.), Politische Theologie und Geschichte unter Ludwig dem Frommen -Histoire et théologie politiques sous Louis le Pieux, Ostfildern (Thorbecke) 2013 9 5.

  activities would have been in the Eastern part of the Empire 38 . Is this so? Instead of a conclusion, a last map may bring some clarification. Louis the Pious has been officially and ritually dethroned in 833, a date that can be considered as a kind of institutional caesura. One might ask whether the royal charters given from 835 on indicate a change in Louis' government. Our map showing a correlation of the actum places with the chronological evolution of the charter production clearly confirms the observation Kölzer already made on the basis of his list, that Aachen becomes less important after 834 39 . If we compare the total number of 191 charters to only 17 in the last seven years, being 20 % of his reign from 814 to 840, we can ascertain that Aachen was losing its central status.The same observation is true for Frankfurt where he had built a new palace but did not have the occasion to spend much time: only four charters after 833. Worms, Ingelheim and Nijmegen each show two to three charters, rather a quantité négligeable compared to their former importance. Quierzy and Thionville are two palaces that already figured on the maps shown here; they maintain their status with five and six charters after 834.The fourth and last map we made finally resumes the six most frequent enactment places of the charters given from 834 until his death on the 20 th of June 840. The surprise here is Attigny, and especially Poitiers, both places where Louis did not give many charters before 833. With four and six charters in his last years they seem to have become more important for the Emperor. After the Attigny assembly of 822 when he sent out Lothar to Italy and Pippin to Aquitaine, Louis convoked another assembly there in November 834 40 . The Poitou region was familiar to the former King of Aquitaine who was born in the palace of Chasseneuil, some eight kilometres from Poitiers. Certainly, one should not overstrain this observation. Louis spent his last winter in Poitiers together with his wife Judith because he held a military campaign against Pippin II whose claim to his father's throne Louis would not accept 41 . Still, this short analysis of the charters proves that we cannot state that Louis the Pious moved his presence and political action to the East. The maps shown here, even if they represent a preliminary sketch, indicate a clear shift to the South-West after 833: more and more assemblies were held in what would become the Francia occidentalis of Louis' son Charles the Bald. The political equilibrium between the different geographical parts of his empire that has been seen as one big issue of the reign of Louis the Pious 42 appears so as the result of his last years' activities.The observation of Aquitaine as a region somehow apart during the process of supposed fragmentation shall have to be proved. There is the testimony of the assemblies, the charters and the mints, but the question is if, or why, Aquitaine was left out by Louis as long as Pippin I st was King there. It is remarkable that the only coins bearing a choronymic indication of the mint place during the reign of Louis the Pious are the AQVITANIA coins minted under Pippin I st in Bordeaux and perhaps Bourges, if we believe Simon Coupland 43 . The old Emperor trying to reintegrate the regnum of his beginnings recalls the case of Charles the Simple in Western Francia who, once he had acquired his 'greater heritage' 44 , took refuge in Lotharingia in the last years of his reign.All the maps shown here are part of the HLUDOWICUS project. I would like to thank Rémi Crouzevialle, University of Limoges, for the cartographical collaboration.

	34 I am obliged to professor Ludger Körntgen, Mainz, for confirming this opinion. Cf BRÜHL, Palatium und Civitas, 108-111, who suggested a 'Klosterpfalz' in Mainz. 35 BRÜHL, Palatium und Civitas, 128; EICHLER, Fränkische Reichsversammlungen, 60. 36 Regino Prumiensis, Chronicon, 98 (a. 869); cf Nithard, Histoire, 116 (lib. 4, c. 1), a scribe's marginal note: sedes prima Frantię. 37 Paper given at the International Medieval Congress, Leeds, 2009, session 706 organized by HLUDOWICUS on map 2: assemblies map 3: charter production and actum places July, 14 map 1: mint places map 4: frequent actum places after 834

th : 'Capitularies, Charters and the So-called Crisis of the Reign of Louis the Pious'. 38 KÖLZER, Kaiser Ludwig der Fromme, 32. 39 KÖLZER, Kaiser Ludwig der Fromme, 32. to be published in M. Gravel & S. Kaschke (ed.), Politische Theologie und Geschichte unter Ludwig dem Frommen -Histoire et théologie politiques sous Louis le Pieux, Ostfildern (Thorbecke) 2013 13 40 DEPREUX, Lieux de rencontre, 214; EICHLER, Fränkische Reichsversammlungen, 42. 41 Annales Bertiniani, 34-35 (a. 839). 42 BAUTIER, Le poids de la Neustrie, 555. 43 COUPLAND, Coinages of Pippin, 205-207. to be published in M. Gravel & S. Kaschke (ed.), Politische Theologie und Geschichte unter Ludwig dem Frommen -Histoire et théologie politiques sous Louis le Pieux, Ostfildern (Thorbecke) 2013 14 44 Actes de Charles III le Simple: largiore vero hereditate indepta (eschatocol, used from 911 on). I would like to thank Simon MacLean, St Andrews, who kindly accepted to read and correct this paper.

MAYER, Ausbildung der Grundlagen; TELLENBACH (ed.), Studien und Vorarbeiten.

Nithard, Histoire,[142][143][144] c. 7).

WERNER, Naissance de la noblesse,

MORSEL, Construire l'espace, 295.

For a survey see SCHNEIDER, Suche nach dem verlorenen Reich, 242-251; cf HEIMANN / SCHNEIDER, Kloster -Landschaft.

13 SCHIEFFER, Die Karolinger, 131; DE JONG, penitential state, 214-

259. 14 Cf DE JONG / THEUWS (ed.), Topographies of power; EHLERS (ed.), Places of

 Power. 15 CHRISTALLER, Die zentralen Orte; cf STEUER, Zentralorte.

MÜLLER-MERTENS, Die Reichsstruktur.

NELSON, Aachen; MÜLLER-MERTENS, Die Reichsstruktur; FALKENSTEIN, Charlemagne et Aix-la-Chapelle; BINDING, Deutsche Königspfalzen.

MCKITTERICK, Charlemagne, 158.


Vineeth S Varma
email: vineeth.varma@lss.supelec.fr

Salah Eddine Elayoubi
email: salaheddine.elayoubi@orange.com

Merouane Debbah
email: merouane.debbah@supelec.fr

Samson Lasaulce
email: samson.lasaulce@lss.supelec.fr

On the Energy Efficiency of Virtual MIMO Systems

The major motivation behind this work is to optimize the sleep mode and transmit power level strategies in a small cell cluster in order to maximize the proposed energy efficiency metric. We study the virtual multiple input multiple output (MIMO) established with each base station in the cluster equipped with one transmit antenna and every user equipped with one receive antennas each. The downlink energy efficiency is analyzed taking into account the transmit power level as well as the implementation of sleep mode schemes. In our extensive simulations, we analyze and evaluate the performance of the virtual MIMO through zero-forcing schemes and the benefits of sleep mode schemes in small cell clusters. Our results show that for certain configurations of the system, implementing a virtual MIMO with several transmit antennas can be less energy efficient than a system with sleep mode using OFDMA with a single transmitting antenna for serving multiple users.

I. INTRODUCTION

The energy consumed by the radio access network infrastructure is becoming a central issue for operators [START_REF] Saker | System selection and sleep mode for energy saving in cooperative 2G/3G networks[END_REF]. The goal of this work is to provide insights on how to design green radio access networks, especially in the framework of virtual MIMO systems. Indeed, classical network architectures are focused on integrated, macro base stations, where each cell covers a predetermined area, and inter-cell interference is reduced by the means of fixed frequency reuse patterns [START_REF] Elayoubi | Uplink intercell interference and capacity in 3G LTE systems[END_REF]. Heterogeneous Networks (HetNets) introduced a new notion of small cells where pico or femto base stations are deployed within the coverage area of the macro base stations [START_REF] Saker | Optimal control of wake up mechanisms of femtocells in heterogeneous networks, Selected Areas in Communications[END_REF]. Virtual MIMO is a step forward in this context that allows distributed systems of base stations/antennas that cover a common area and cooperate in order to increase the overall spectral efficiency [START_REF] Wang | Cooperative MIMO channel models: A survey[END_REF]. This paper focuses on these latter solutions and aims at addressing the problem from an energy efficiency point of view.

For classical macro networks, early works focused on designing energy-efficient power control mechanisms [START_REF] Goodman | Power Control for Wireless Data[END_REF]. Therein, the authors define the energy-efficiency of a communication as the ratio of the net data rate (called goodput) to the radiated power; the corresponding quantity is a measure of the average number of bits successfully received per joule consumed at the transmitter. This metric has been used in many works. Although fully relevant, the performance metric introduced in [START_REF] Goodman | Power Control for Wireless Data[END_REF] ignored the fact that transmitters consume a constant energy regardless of their output power level [START_REF] Desset | Flexible power modeling of LTE base stations[END_REF]. The impact of this constant energy has been studied for single user point-to-point MIMO systems in [START_REF] Varma | An Energy Efficient Framework for the Analysis of MIMO Slow Fading Channels[END_REF]. Sleep mode mechanisms have thus been regarded as a solution for this issue; they consist in deactivating network resources that have low traffic load, eliminating thus both the variable and constant parts of the energy consumption [START_REF] Saker | System selection and sleep mode for energy saving in cooperative 2G/3G networks[END_REF]. This mechanism has been applied to macro networks [START_REF] Saker | System selection and sleep mode for energy saving in cooperative 2G/3G networks[END_REF], as well as to heterogeneous networks with macro and small cells [START_REF] Saker | Optimal control of wake up mechanisms of femtocells in heterogeneous networks, Selected Areas in Communications[END_REF]. Our aim in this paper is to extend this concept to virtual MIMO networks, where an antenna that is not significantly contributing to the network capacity (for a given configuration of user positions and radio channels) is put into sleep mode.

The remainder of this paper is organized as follows. In section II, we present the system model and the resource allocation scheme. Section III presents our energy efficiency metric and optimizes it for a given system and channel configuration, using sleep mode mechanisms. Section IV presents some numerical examples and section V eventually concludes the paper.

II. SYSTEM MODEL

The wireless system under consideration is the downlink in a virtual MIMO system within a small cell cluster. To be precise, each of the small cell base stations are connected to a central processor and so they act as antennas for the virtual MIMO as shown in Fig 1 . We refer to the set of these base Fig. 1. An example illustration of a 2×2 virtual MIMO with g i,j representing the channel between BS antenna i and user j. stations as the "cluster". Each user is equipped with a single receive antenna. In order to eliminate interference zero-forcing is implemented. We consider a block-fading channel model where the channel fading stays is assumed to stay constant for the duration of the block and changes from block to block. The base stations require the channel state information available at the user end in order to implement the zero-forcing technique. Therefore, in each block channel a training and feedback mechanism happens, after which data is transmitted. We also assume that every base station is capable of entering into a "sleep-mode". In this mode, the base station does not send any pilot signals and therefore does not perform the training or feedback actions consuming a lesser quantity of power compared to the active base stations. Let there be M base stations in the cluster and K users. Define K = {1, 2, . . . , K} and M = {1, 2, . . . , M } the sets of users and base station antennas.

A. Power consumption model

As the transmit antennas are not co-located, each of them have an individual power budget. When a base station is active, it consumes a constant power of b due to the power amplifier design and training or feedback costs. Additionally, it consumes a power P m x m 2 proportional to the radiated power, where P m ≤ P max and x m ≤ 1 is the signal transmitted and P max is the power constraint [START_REF] Saker | System selection and sleep mode for energy saving in cooperative 2G/3G networks[END_REF] [START_REF] Desset | Flexible power modeling of LTE base stations[END_REF]. When it is placed on sleep mode, it is assumed that it only consumes power c where c < b. Denote by s the sleep mode state vector of the cluster with elements s m ∈ {0, 1}. The base station m is in sleep mode when s m = 1 and active when s m = 0. Thus the power consumption of the m-th base station is

cs i + (1 -s i )(b + P m x m
2 ). The total power consumption of the cluster is given by:

P tot = M m=1 cs m + (1 -s m )(b + P m x m 2 ) (1)
For any given state of the cluster, we define ω(s) as the total number of base stations that are active. This value can be calculated as ω(s) = Mm s m . If M < K, zero-forcing can not be used. However, if M > K, and ω(s) ≥ K, then the zero-forcing technique can be implemented by choosing K base stations to transmit the data signals after all ω(s) active base stations train and obtain feedback on their channels.

B. The zero-forcing scheme

As there are K users connected to the ω(s) active base stations, there are a total of ω(s) × K number of channels. Let ζ = {1, 2, . . . , ω(s)} be the set of active base stations. We denote by G with elements g l,k ∈ R the path fading between base station l ∈ ζ and user k ∈ K, while H with elements hl,k ∈ C denotes the fast fading component, resulting in a signal at k given by:

y k = ω(s) l=1 g l,k P l σ 2 hl,k x l + z (2)
where x is the signal transmitted with x l as its elements; z is the normalized noise and σ 2 the noise strength. Note that g l,k can be determined based on the user location while hl,k are i.i.d. zero-mean unit-variance complex Gaussian random variables. We define H(G, H) as the combined channel matrix with elements hl,k = √ g l,k hl,k .

In our work, as we focus on the small-cell scenario where the antennas are distributed over the cell in a dense manner, we assume that every user can have a similar level of signal strength. Define N = {1, 2, . . . , N } as the set of transmitting antennas that perform zero forcing beam-forming. We define β ∈ N → ζ as the function that describes which base stations in ζ will be picked to transmit the data signals. Given BS j ∈ N , the corresponding label for the BS in ζ is given by β(j). Given ω(s) active base stations that perform training and receive feedback on H, we define the effective channel matrix as H( H, β), where its elements h j,k = hβ(j),k . For zeroforcing, we require that the number of transmitting antennas is equal to the number of receiving antennas or K = N . With this constraint, if H is an invertible matrix, we define:

x = (H( H, β)) -1 u (3)
where u is a vector of length K, where u k which is the signal received by the k-th user. In this work, we take u k 2 = 1 so that each user obtains identical signal strengths. This constraint has several benefits:

1) This results in a very "fair" beam forming scheme as all users experience equal signal strength and thus similar data rates. 2) As the base station antennas are spread over the cell, there is no user on the "cell edge" or "cell center". In this situation, equal SINRs for all users can result in less congestion when user traffic patterns are taken into account 3) Finally, the resulting system is far less complex and easier to analyze than one with arbitary values for u k 2 .

With these definitions, we can now precode the transmitted signal as:

x = x α( H, β) (4)
where α( H, β) = max(x m ). This pre-coding works only if all the P j are equal, and so we chose P j = P 0 . From this point onwards, P 0 represents the transmit power level of the system with this pre-coding. The signal received by each of the K users is given by:

y k = P 0 σ 2 u k α( H, β) + z (5)
Thus the SINR at each user is now given by

γ k = P 0 α( H, β) 2 σ 2 (6)

III. ENERGY EFFICIENCY OPTIMIZATION

This work aims at minimizing the energy consumption by base stations. If each user in the network is connected to download some data, then the total energy consumed by the network is the total power consumed multiplied by the total duration for which the user stays connected. Energy efficiency (EE) is a metric that is often used to measure this, and maximizing the energy efficiency leads to minimizing the total energy consumed.

A. Defining the EE metric

Before defining the EE, we first calculate the total power consumption of the network. From ( 1) and (4), the total power consumed is given by:

P tot (P 0 , H, β) = M m=1 cs m + (1 -s m ) ×   b + P 0 (H -1 ( H, β)u) β -1 (m) α( H, β) 2   (7)
Here we define

∀m ∈ M; β -1 (m) = j if j ∈ N exists s.t β(j) = m 0 otherwise.
(8) and () j is the j -th element if j = 0 and is 0 if j = 0. In this scenario, we define the instantaneous energy efficiency as:

η(P 0 , H, β) = k f (γ k (P 0 , H, β)) P tot (P 0 , H, β) (9)
where f () gives the effective throughput as a function of the SINR. f (γ k ) = log(1 + γ k ) for example. However when we study the base station energy efficiency for a longer duration, the effects of fast fading in H gets averaged and in this case a more reasonable definition for the EE is:

η(P 0 , G, β) = E H[ k f (γ k (P 0 , H(G, H), β))] E H[P tot (P 0 , H(G, H), β)] (10)

B. Optimization w.r.t the transmit power

In this section, we show some properties of our energy efficiency metric w.r.t P 0 . If the goal of a system is to be energy efficient using power control, then one important question arises: Is there a unique power for which the energy efficiency is maximized ? We answer this question with the following proposition:

Proposition 1: Given a certain path loss matrix G and a selection of transmitting base stations β in the virtual MIMO system, the average EE η(P 0 , G, β) is maximized for a unique P * 0 and is quasi-concave in P 0 . Proof: Consider the SINR for each user γ k . It can be observed that

∂γ k (P0, H,β)) ∂P0 is a constant. So if f () is concave in γ k , it is also concave in P 0 . Now consider the numerator of (10), E H[ k f (γ k (P 0 , H(G, H), β))]
. This is a weighted sum of several concave functions and is hence also concave itself. Similarly, ∂Ptot(P0, H,β) ∂P0 can also be verified to be a constant.

Hence, ∂ E H[Ptot (P0, H,β)]
∂P0 is a constant. Thus η(P 0 , G, β) is the ratio of a concave function of P 0 to a linear function of P 0 . This is known to be quasi-concave and has a unique maximum P * 0 satisfying:

∂ η(P * 0 , G, β) ∂P 0 = 0 (11)
This proposition guarantees that given a certain choice of transmit antennas and a path fading matrix, the energy efficiency can always be optimized w.r.t the transmit power level P 0 .

C. Optimizing the selection of transmitting base stations

Given a certain sleep mode state s, there are ω(s) base stations active that train and obtain feedback. From this set ζ, K base stations have to be picked for zero-forcing. This choice is mathematically expressed by β. The β that optimizes the energy efficiency depends on the channel state H. The following proposition details the method of choosing the β that optimizes EE.

Proposition 2: When P0 b → 0, the β * that maximizes η(P 0 , G, β) is obtained by:

β * = arg min[α( H, β); β ∈ {N → K}] (12)
Proof: By observing ( 6) it can be seen that γ k (P 0 , H, β) is maximized by picking β * . And so k f (γ k (P 0 , H, β)) is maximized when β = β * . When P0 b → 0, for β * and any β 1 we have:

lim P 0 b →0 η(P 0 , G, β * ) -η(P 0 , G, β 1 ) = (13) E H[ k f (γ k (P 0 , H, β * )) -k f (γ k (P 0 , H, β 1 ))] M m=1 cs m + (1 -s m )b ≥ 0 (14)
This implies that we pick β such that α( H, β) is minimized for optimizing EE when b >> P 0 . From a practical point of view, the above result is useful as it proposes an algorithm to pick the right base stations based on the CSI obtained from all the base stations that are active. The condition b >> P 0 is most applicable when the users are close to the base stations resulting in a low P 0 being used for maximizing EE.

IV. NUMERICAL RESULTS

In this section we use simulations and numerical calculations to study the effectiveness of our proposal as well as the advantages offered. For the purpose of a thorough numerical study, we will analyze two kinds of system settings A and B. For both the configurations the common parameters are:

1) c = b 10 W 2) P max = 2 W 3) f (γ) = B log(1 + γ) 4) σ 2 = 1 mW
Where B = 10 6 hz is the bandwidth.

The fast fading co-efficient we consider is hi,j = o(π m,k )Ω + 0.1ξ. Where ξ ∈ CN (0, 1), a is the direct line of sight factor which plays a dominant role in most small cell networks, o π m,k ∈ 0, 1 is the shadow factor and o(pi m,k ) = 1 with probability π m,k . Here π m,k is the probability that the receiver k has line of sight with the BS antenna m. We take π m,k = 0.5 ∀(k, m) for our simulations.

The presented results study the case of two users K = 2 served by a small cell cluster of three base stations, i.e M = 3. In addition to zero-forcing, when there are two users a single base station could also alternately use Orthogonal Frequency-Division Multiple Access (OFDMA) to serve the two users and keep the other two BS in sleep mode (i.e . Our numerical simulations study all these possible scenarios and compare their respective EE performances.

In both of the settings presented below, we study two main regimes of interest:

1) b = 1W : This regime represents the futuristic case where power amplifier efficiencies are quite high and the constant power consumed is lower than the maximum RF output power. 2) b = 10W : This regime represents the more current state of the art w.r.t power amplifier efficiency where in small cell antennas, a large portion of the power is lost as a fixed cost. We also consider two possible values of Ω, the line of sight factor. The case Ω = 10 is representative of pico-cells that are deployed externally, whereas the case Ω = 0 represents femtocells deployed internally and no line of sight communication is possible.

A. Setting A

The deployment of antennas and the user locations are shown in Fig 2 . In this setting we take g m,k = 1 ∀m, k. In Fig 3 we study the EE of a VMIMO system with a very efficient power amplifier. In this figure, we notice that using all available base station antennas is more efficient when line of sight communications are possible. In this case, no BS is in sleep mode and all the antennas train and obtain feedback on their channels. The choice of β is very much relevant in this scenario. However in the regime where there is no direct line of sight (users are inside buildings), it becomes more efficient to just use 2 BS antennas and put one on sleep mode. As the configuration is symmetric, the choice of the base station in sleep mode is not relevant.

In Fig 4 we study the EE of a VMIMO system with an inefficient power amplifier. In this figure, we notice that using all available base station antennas is not efficient even when line of sight communications are possible. In this case, having one BS in sleep mode and obtaining a 2×2 virtual MIMO with the other remaining antennas is the most efficient. Suprisingly, in the case of no line of sight, i.e Ω = 0, we observe that using OFDMA with one BS active is the most efficient solution. This is explained by the relative inefficiency of zero-forcing in the low SNR regime, causing less energy to be spent by having two BS antennas in sleep mode and just one antenna transmitting for the two users in orthogonal frequencies.

B. Setting B

The deployement of antennas and the user locations are shown in Fig 5 . In this setting we take g 1,1 = g 2,1 = 4, g 3,1 = g 1,2 = g 2,2 = 0.1 and g 3,2 = 10.

In Fig 6 , similarly to what was done in the previous setting, we study the EE of a VMIMO system with a very efficient power amplifier. In this figure, for both Ω = 0 and Ω = 1 we see that having to use 2 BS antennas and put one on sleep mode is the most efficient. In this setting, the configuration of BS and users are asymmetric and the BS to be put in sleep mode has to be chosen carefully. BS 1 and 2 are symmetric and are close to user 1, but 3 is closer to user 2. In this case In this setting, we see from Fig 7 that unlike in Setting A, using OFDMA to divide resources between the two users is not as efficient as ZF due to the higher SNR when served by nearby BS antennas. Like in Fig 6, choosing s 1 = 1 or s 2 = 1 and zero-forcing is always the most efficient solution.

V. CONCLUSION

This paper studies the performance of virtual MIMO systems from an energy efficiency perspective. It defines an energy efficiency metric that takes into account the capacity as well as the energy consumption, and considers both fixed and variable parts of this latter. We optimize the power allocations of the different antennas and show that sleep mode can bring a significant energy efficiency gain. This involves deactivating antennas that do not have a significant contribution to the system capacity, for a given number of users and radio channel conditions.

This work is applicable only for the specific case of a small cell cluster with a centralized network and CSIT. Thus, many extensions of the proposed work are possible. The most relevant extension is to apply the proposed framework

Fig. 2 .

 2 Fig. 2. Setting A schematic

Fig. 3 .Fig. 4 .

 34 Fig. 3. Setting A: EE v.s P 0 for b = 1 W

Fig. 5 .

 5 Fig. 5. Setting B schematic

Fig. 7 .

 7 Fig. 7. Setting B: EE v.s P 0 for b = 10 W

VI. ACKNOWLEDGMENTS

This work is a joint collaboration between Orange Labs, Laboratoire des signaux et systémes (L2S) of Supélec and the Alcatel Lucent Chair of Supélec. This work is part of the European Celtic project "Operanet2".


Mariem Mhiri
email: mariem.mhiri@gmail.com

Vineeth S Varma
email: vineeth.varma@lss.supelec.fr

Maël Le Treust
email: mael.le.treust@emt.inrs.ca

Samson Lasaulce
email: samson.lasaulce@lss.supelec.fr

Abdelaziz Samet
email: abdelaziz.samet@ept.rnu.tn

On the benefits of repeated game models for green cross-layer power control in small cells (Invited Paper)

Keywords: distributed power control, energy efficiency, repeated game, channel state information

In this paper, we consider the problem of distributed power control for multiple access channels when energy-efficiency has to be optimized. In contrast with related works, the presence of a queue at each transmitter is accounted for and globally efficient solutions are sought. To this end, a repeated game model is exploited and shown to lead to solutions which are distributed in the sense of the decision, perform well globally, and may rely on limited channel state information at the transmitter.

I. INTRODUCTION

Designing energy-efficient communication systems has become a critical issue in modern day wireless networks. The problem treated in this work deals with power control when energy efficiency (EE) has to be optimized. This metric (EE) has been defined in [START_REF] Goodman | Power control for wireless data[END_REF] as a ratio of the net data rate (goodput) to the transmit power level. The problem was formulated as a non-cooperative game where each transmitter aims at selfishly maximizing its individual energy-efficiency. The considered solution is the Nash equilibrium (NE) which is shown to be unique but generally Pareto inefficient. To deal with this inefficiency, an operating point (OP) was proposed in [START_REF] Treust | A repeated game formulation of energyefficient decentralized power control[END_REF] where repeated game was exploited. Authors in [START_REF] Treust | A repeated game formulation of energyefficient decentralized power control[END_REF] showed that when playing with the developed OP according to a cooperation plan, only channel state information (CSI) is needed and transmitters can improve the social welfare (sum of utilities). Recently, a generalized EE metric has been proposed in [START_REF] Varma | A cross-layer approach for energy-efficient distributed power control[END_REF] for two important transport layer protocols (Transmission Control Protocol (TCP) and User Datagram Protocol (UDP)). The new EE metric is based on a cross-layer approach and takes into account the effects of the presence of a queue with a finite size at the transmitter. An interference channel system was studied and it was shown that a unique NE exists for a noncooperative game. In this paper, we consider the problem of distributed power control with the new EE metric according to UDP protocol developed in [START_REF] Varma | A cross-layer approach for energy-efficient distributed power control[END_REF] and for multiple access channels (MAC) system. Our goal is to find another unique solution concept which is efficient and may rely on limited CSI at the transmitter. We refer to a repeated game model (RG) developed in [START_REF] Treust | A repeated game formulation of energyefficient decentralized power control[END_REF] and try to apply the results on the crosslayer power control game. One of the major mathematical distinction between the two metrics used is the presence of a constant power term in the denominator of the EE metric. Although it appears to be a small change, the structure of the equilibrium solution is quite different. The optimal SINR when using the [START_REF] Goodman | Power control for wireless data[END_REF] metric is independent of the channel state. This property is lost when accounting for the constant power consumption, and motivates us to propose a new OP for the cross-layer metric. The main contributions of this work are:

1) Study the RG when using the cross layer EE as the utility of the game; 2) Establish the threshold on the game length beyond which the equilibrium policy can be pareto-optimal; 3) Propose a new OP that is efficient and can be reached in a distributed manner.

This paper is structured as follows. In section II-A, we introduce the system model under study. Then, we define (in section II-B) the static power control game. This is followed (section II-C) by a review of the non-cooperative one-shot game. In section III, we give the formulation of the RG model.

In section IV, we introduce the new OP and an equilibrium for the finite RG is proposed. Numerical results are presented and discussed in section V. Finally, concluding remarks are proposed in section VI.

II. PROBLEM STATEMENT

A. System model

The communication network under study is that of a MAC system, where N small transmitters are communicating with a receiver and are operating in the same frequency band. Transmitter i ∈ {1, . . . , N } sends a signal √ p i x i with power

p i ∈ [0, P max i
] where P max i > 0 is the maximum transmit power. The channel gain of the link between transmitter i and the destination is denoted as g i . Thus, the baseband signal received is written:

y i = g i √ p i x i + N j=1 j =i g j √ p j x j + n i , (1)
with n i is additive white Gaussian noise (AWGN) with mean 0 and variance σ 2 i . We assume that σ 2 i is identical for all the transmitters such that: σ 2 i = σ 2 . Therefore, the resulting SINR γ i at the receiver is given by:

γ i (p) = p i |g i | 2 σ 2 + 1 L N j=1 j =i p j |g j | 2 , (2)
where p = (p 1 , p 2 , . . . , p N ) is the power vector which will describe later the power actions of the N transmitters and L refers to the spreading factor [START_REF] Varma | A cross-layer approach for energy-efficient distributed power control[END_REF]. We assume that the described system is based on the IP (Internet Protocol) stack where packets arrive from an upper layer into a finite memory buffer of size K (in packets). Here, the considered protocol is UDP for which the packet arrival process follows a Bernoulli process with a constant probability q, independent from the SINR. This results in an effective packet loss denoted by Φ(γ i ) and an energy efficiency η i given by:

η i (p i , p -i ) = Rq(1 -Φ(γ i (p))) b + qp i (1 -Φ(γ i (p))) f (γ i (p)) , (3)
where p -i = (p 1 , .., p i-1 , p i+1 , .., p N ), R is the used throughput (in bit/s) and b represents the fixed consumed power when the radiated power is zero [START_REF] Varma | A cross-layer approach for energy-efficient distributed power control[END_REF].

B. Static power control game

The major motivation behind this work is in order to establish an efficient equilibrium point to which a completely distributed system can converge to. A non-cooperative game has been introduced in [START_REF] Varma | A cross-layer approach for energy-efficient distributed power control[END_REF] where the existence of a unique Nash equilibrium was proved. Here, we are looking for more efficient solutions which are distributed in the sense of the decision making, but may rely on limited channel state information at the transmitter. As motivated in [START_REF] Varma | A cross-layer approach for energy-efficient distributed power control[END_REF], the power control can be modeled by a strategic form game (see e.g., [START_REF] Lasaulce | Game Theory and Learning for Wireless Networks: Fundamentals and Applications[END_REF]).

Definition 2.1: The game is defined by the ordered triplet G = N , (A i ) i∈N , (u i ) i∈N where

• N is the set of players. Here, the players of the game are the sources/transmitters, N = {1, . . . , N }; • A i is the set of actions. Here, the action of source/transmitter i consists in choosing p i in its action set A i = [0, P max i ]; • u i is the utility function of each user according to UDP given by:

u i (p i , p -i ) = η i (p i , p -i ) (4)
The function f : [0, +∞) → [0, 1] is a sigmoidal efficiency function which corresponds to the packet success rate verifying f (0) = 0 and lim x→+∞ f (x) = 1. The function Φ identifies the packet loss due to both bad channel conditions and the finiteness of the packet buffer. This can be calculated as:

Φ(γ i ) = (1 -f (γ i ))Π K (γ i ) (5)
where Π K (γ i ) is the stationary probability that the buffer is full and is given by:

Π K (γ i ) = ω K (γ i ) 1 + ω(γ i ) + . . . + ω K (γ i ) (6)
with:

ω(γ i ) = q(1 -f (γ i )) (1 -q)f (γ i ) (7)
In [START_REF] Varma | A cross-layer approach for energy-efficient distributed power control[END_REF], the authors prove that the non-cooperative game with rational players, G, allows for a unique pure Nash equilibrium (NE). This NE is the set of powers from which no player has anything to gain by changing only his own strategy unilaterally. This is explained in the following section.

C. Review of the non-cooperative game

The non-cooperative power control game has been investigated in [START_REF] Varma | A cross-layer approach for energy-efficient distributed power control[END_REF] where the quasi-concavity of the utility function given in (4) was proved. Accordingly, as the NE represents the fundamental solution for a non-cooperative game, existence and uniqueness of such a solution have been studied and demonstrated as well. Thus, the optimal power denoted as p * i is obtained by setting ∂u i /∂p i to zero, which leads to solve the following equation:

bγ i Φ (γ i ) + q 1 -Φ(γ i ) f (γ i ) 2 [f (γ i ) -p i γ i f (γ i )] = 0, ( 8
)
where

γ i = dγ i dp i = γ i p i , f = df dγ i and Φ = dΦ dγ i .
However, the NE solution is not always Pareto efficient for many scenarios. An example is presented in Fig. 1 where we stress that the NE is far from the Pareto frontier. Motivated by the need to design an efficient solution relying on limited CSI at the transmitter, we move to the repeated game framework.

III. REPEATED POWER CONTROL GAME

In repeated games (RG), as the name suggests, the same game is played several times. The long-term interactions between the players in such a situation is studied under the RG framework. The players react to past experience by taking into account what happened in all previous stages and make decisions about their future choices [START_REF] Hart | Robert aumann's game and economic theory[END_REF], [START_REF] Sorin | Repeated games with complete information[END_REF]. The resulting utility of each player is an average of the utility of each stage. A game stage t corresponds to the instant in which all players choose their actions simultaneously and independently and thus a profile of actions can be defined by p(t) = (p 1 (t), p 2 (t), . . . , p N (t)). When assuming full monitoring, this profile choice is observed by all the players and the game proceeds to the next stage [START_REF] Sorin | Repeated games with complete information[END_REF]. The sequence of actions p i (t) of a transmitter i at time t defines his history denoted as h(t) = p i (t) = (p i (1), p i (2), . . . , p i (t -1)) and which lies in the set H t = P t-1 i . Before playing stage t, all histories are known by all the players [START_REF] Treust | A repeated game formulation of energyefficient decentralized power control[END_REF]. According to the above descriptions, a pure strategy δ i,t of player i ∈ N is a mapping from H t to the action set A i = [0, P max i ] specifying the action to choose after each history [START_REF] Treust | A repeated game formulation of energyefficient decentralized power control[END_REF], [START_REF] Sorin | Repeated games with complete information[END_REF]:

δ i,t : H t → [0, P max i ] h(t) → p i (t) (9)
We define the joint strategy δ = (δ 1 , δ 2 , . . . , δ N ) as the vector of all the players strategies.

In this paper, we are interested in the finite repeated game, i.e the game is played for a finite number of steps (T steps). The utility function of each player results from averaging over the instantaneous utilities over all the game stages. At each stage t, the instantaneous utility of player i is a function of the profile of actions of all the players p(t).

Definition 3.1: The utility function of the i th player for the finite RG is the arithmetic average of the sum of the utilities for the initial T first stages [START_REF] Sorin | Repeated games with complete information[END_REF], [START_REF] Aumann | Long-term competition-a gametheoretic[END_REF]. We have [START_REF] Treust | A repeated game formulation of energyefficient decentralized power control[END_REF]:

v T i (δ) = 1 T T t=1 u i (p(t))
for the finite RG (10)

where T ≥ 1 defines the number of game stages in the finite RG.

An equilibrium solution of the RG is defined in the following manner: Definition 3.2: A joint strategy δ satisfies the equilibrium condition for the finite repeated game if for all players i ∈ N , for all other strategies δ i , we have

v T i (δ) ≥ v T i (δ i , δ -i ).
It means that no deviating strategy δ i can increase the utility v T i (δ) of any one player. This equilibrium solution is exactly what we are interested in, as a strategy δ satisfying the above condition would be precisely what rational players in a RG would play. In a RG with complete information and full monitoring, the Folk theorem characterizes the set of possible equilibrium utilities [START_REF] Treust | A repeated game formulation of energyefficient decentralized power control[END_REF], [START_REF] Sorin | Repeated games with complete information[END_REF]. It states that the set of Nash equilibrium in a RG is precisely the set of feasible and individually rational outcomes of the one-shot game (non-cooperative game) [START_REF] Hart | Robert aumann's game and economic theory[END_REF], [START_REF] Sorin | Repeated games with complete information[END_REF]. In a RG, interesting patterns of behavior between players can be seen and studied. This includes: rewarding and punishing, cooperation and threats, transmitting information and concealing [START_REF] Hart | Robert aumann's game and economic theory[END_REF].

IV. AN OPERATING POINT AND REPEATED GAME CHARACTERIZATION

A. New OP for the game G Consider the operating point (OP) described in [START_REF] Treust | A repeated game formulation of energyefficient decentralized power control[END_REF]. It is identified by a subset of points of the achievable utility region such that p i |g i | 2 = p j |g j | 2 for all (i, j) ∈ N . The optimal subset of powers consists of the solutions of the following system of equations:

∀(i, j) ∈ N , ∂u i ∂p i (p) = 0 with p i |g i | 2 = p j |g j | 2 (11)
with u i is the utility function defined in (4). Due to the presence of the parameter b which we consider different from 0, it can be observed that there will be N different solutions corresponding to equation (11) in terms of p i and thus the operating point from [START_REF] Treust | A repeated game formulation of energyefficient decentralized power control[END_REF] is not well defined when using the utility defined in [START_REF] Varma | A cross-layer approach for energy-efficient distributed power control[END_REF]. To deal with this problem, a new OP is proposed. The new OP consists in setting p i |g i | 2 to a constant denoted as α that can be optimized. We propose to determine a unique optimal α by maximizing the expected sum utility over all the channel states as follows:

α = arg max E g N i=1 u i (α, g) (12)
When playing at the OP, the power played by the i th player, denoted as pi , is given by:

pi = α |g i | 2 (13)
In the following section, we focus on the characterization of the finite RG.

B. Repeated power control game characterization

As a first step, we determine the minimum number of stages (T min ) corresponding to the finite RG. When the number of stages in the game is less than T min , the equilibrium of the RG is to simply play at the NE. However, if T > T min , a more efficient equilibrium point can be reached. Assuming that channel gains

|g i | 2 lie in a compact set [ν min i , ν max i ] [2],
we have the following proposition: Proposition 4.1 (Equilibrium solution for the finite RG): For a finite RG, if T > T min , then the corresponding equilibrium solution is given by [START_REF] Treust | A repeated game formulation of energyefficient decentralized power control[END_REF]:

δ i,t : pi for t ∈ {1, 2, . . . , T -T min } p * i for t ∈ {T -T min + 1, . . . , T } P max i for any deviation detection (14)
where T min is:

Tmin=        Aν max i bν min i +γ i σ 2 B - Gν max i bν min i + αH Eν min i bν max i +γ * i ( σ 2 + 1 L j =i p * j ν max i ) F - Cν min i bν max i + γ i( σ 2 + 1 L j =i p max j ν max i ) D        (15)
The proof for this proposition is given in Appendix A, as well as the quantities A, B, C, D, E, F , G and H. γ * i is the SINR at the NE while γi and γ i are the SINRs related to the utility max and the utility minmax respectively (see Appendix A).

V. NUMERICAL RESULTS

We consider a scenario with a MAC where N transmitters are communicating with their corresponding receiver (e.g. base station). The efficiency function is assumed to be f (x) = e -c/x where c = 2 R R 0 -1 with R and R 0 are the throughput and the used bandwidth and supposed to be 1 Mbps and 1 MHz respectively. The other parameters are set as follows:

•

σ 2 = 10 -3 W • b = 10 -2 W • K = 10 • q = 0.5 • P max i = P max = 10 -1 W
The channel gains are assumed to be |g i | 2 = 1 and |g j | 2 = 0.5. Our simulations consist in showing firstly the advantage of the OP regarding the NE of the one shot game. Thus, we plot the achievable utility region, the NE and the proposed OP when considering a system of two transmitters and a spreading factor L = 2. In Fig. 2, the region delimited by the Pareto frontier and the minmax level defines, according to the Folk theorem, the possible set of equilibrium utilities of the RG. In addition, we highlight that the new OP dominates in terms of Pareto the NE and it is Pareto efficient. Fig. 3 represents the ratio w F RG w N E for the finite RG as a function of the number of stages. We have:

w F RG w N E = N i=1 ( T -Tmin t=1 ũi (p(t)) + T t=T -Tmin+1 u * i (p(t))) N i=1 T t=1 u * i (p(t)) (16)
We consider the same system as in Fig. 2 (2 transmitters and a spreading factor L = 2). We proceed to an averaging over channel gains lying in a compact set such that 10 log 10 νmax νmin = 20. According to equation (15), the minimum number of stages T min is equal to 4800. According to this figure, we deduce that the social welfare can be improved when playing a RG. 3. Improvement of the social welfare in finite repeated game vs the Nash equilibrium. While the efficiency of the RG while using the traditional metric defined in [START_REF] Goodman | Power control for wireless data[END_REF] seems to be higher, it requires a longer game than in the cross layer model. Fig. 3 plots the improvement of the social welfare as defined in ( 16). This improvement obtained is compared for case when using the metric defined in [START_REF] Goodman | Power control for wireless data[END_REF] to the cross-layer metric used. The required time for profiting from the RG scenario is much lower in the cross-layer case, but the improvement seems to be relatively smaller. However, note that the NE in the crosslayer game itself is more efficient than the NE in [START_REF] Goodman | Power control for wireless data[END_REF] and so in absolute terms, the proposed OP is still quite efficient and can be utilized for shorter games. This validates our approach and shows that the RG formulation is a useful technique for efficient distributed power control.

VI. CONCLUSION

In this paper, we study an efficient solution for a relevant game with a new EE metric considering a cross-layer approach and taking into account the effects of the presence of a queue with a finite size at the transmitter. As the NE is generally inefficient in terms of Pareto, we design a new OP and exploit a repeated game model to improve the performance of a MAC system. We contribute to express the analytic form for the minimum number of stages in a finite RG. Moreover, our approach provides an efficient solution relying on limited CSI at the transmitter when comparing to the NE and contributes to considerable gains in terms of social welfare for the finite RG. As a first step, we determine the power p i maximising u i and which we denote as ṗi . This amounts to reduce ∂u i /∂p i to 0. We recall that we consider the following notations: γ i = dγi dpi = γi pi , f = df dγi and Φ = dΦ dγi . The power ṗi maximising u i is then the solution of the following equation:

b γ i p i Φ (γi)+q 1-Φ(γ i ) f (γ i ) 2 [f(γi)-γif (γi)]=0 (17)
Therefore, the expression of the maximum utility function writes as:

ui( ṗi,p-i)= Rq(1-φ( γi )) b+ ṗi q(1-φ( γi )) f ( γi )
, with:

γi= ṗi |g i | 2 σ 2 + 1 L j =i p j |g j | 2
In a second step, we are interested in studying the behavior of ui ( ṗi , p -i ) as a function of p j for j = i; which amounts to calculating the sign of ∂ ui( ṗi,p-i) ∂pj ,which is shown to be negative in [START_REF] Varma | A cross-layer approach for energy-efficient distributed power control[END_REF]. Therefore ∂ ui( ṗi,p-i) ∂pj < 0. As ui is a decreasing function of p j , it reaches its maximum when p j = 0 and it is minimum when p j = p max j (for all j = i).

A. Expression of ūi

The utility ui reaches its maximum when p j = 0. When substituting p j = 0 in the SINR expression γi , this allows the determination of the optimal power ṗi :

b |g i | 2 σ 2 Φ (γi)+q 1-Φ(γ i (p i )) f (γ i (p i )) 2 [f(γi(pi))-γif (γi(pi))]=0 (18)
As the latter equation is a function of the SINR, the solution will be in terms of SINR and will be denoted as γi . The corresponding optimal power is pi = γiσ 2 |gi| 2 . Then, we have:

ūi= Rq(1-φ(γ i )) b+ γi σ 2 |g i | 2 q(1-φ(γ i )) f (γ i )

B. Expression of u i

We proceed as described previously and determine the optimal SINR denoted as γ i which is the solution of the following equation:

b|g i | 2 σ 2 + 1 L j =i p max j |g j | 2 Φ (γi)+q 1-Φ(γ i ) f (γ i ) 2 [f(γi)-γif (γi)]=0 (19)
Then, we have:

ui= Rq(1-φ( γ i )) b+ γ i |g i | 2 ( σ 2 + 1 L j =i p max j |g j | 2 ) q(1-φ( γ i )) f ( γ i )
C. Existence proof of γ i and γi

Both equations ( 18) and ( 19) are resulting from the same equation (17) for two different forms of the SINR (γ i for p j = 0 and γ i for p j = p max j ). Showing the existence of these two solutions amounts to prove the existence of the solution of equation (17). However, according to the study established in [START_REF] Varma | A cross-layer approach for energy-efficient distributed power control[END_REF], it was proved that u i is quasi-concave in (p i , p -i ) and then it exists γ + such that the first derivative of u i regarding to p i is strictly positive on [0, γ + ] and strictly negative on [γ + , +∞] for all p j ∈ [0, p max j ] : the first derivative is continous and is equal to zero in γ + . According to the utility which we are studying (max or minmax), γ + is either γi (eq. ( 18)) or γ i (eq. ( 19)).

D. Proof

The SINR γi refers to the SINR when playing the new OP. In order to simplify expressions, we use the following notations:

A = Rq(1 -φ(γ i )) B = q(1-φ(γi)) f (γi) C = Rq(1 -φ( γ i )) D = q(1-φ( γi)) f ( γi) E = Rq(1 -φ(γ * i )) F = q(1-φ(γ * i )) f (γ * i ) G = Rq(1 -φ(γ i )) H = q(1-φ(γi)) f (γi)
The inequality (20) becomes:

A|g i | 2 b|g i | 2 +γ i σ 2 B + T s=T -T min +1 Eg C|g i | 2 b|g i | 2 + γ i( σ 2 + 1 L j =i p max j |g j | 2 ) D ≤ G|g i |

Fig. 1 .

 1 Fig. 1. Pareto inefficiency of the NE.

Fig. 2 .

 2 Fig. 2. Pareto dominance and Pareto efficiency of the proposed OP regarding the NE.

Fig.

  Fig.3. Improvement of the social welfare in finite repeated game vs the Nash equilibrium. While the efficiency of the RG while using the traditional metric defined in[START_REF] Goodman | Power control for wireless data[END_REF] seems to be higher, it requires a longer game than in the cross layer model.

  utilities max and minmax are expressed respectively as follows: ūi = maxp -i maxp i ui(pi,p-i) ui = minp -i maxp i ui(pi,p-i)

From [ 2 ]

 2 , we have:ūi (p(t)) + T s=T -Tmin+1 E g { u i (p(s))} ≤ ũi (p(t)) + T s=T -Tmin+1 E g {u * i (p(s))}