Peltiers

Micro-Controller Piezoelectric vibrator

To validate our prototype and concepts of vibrotactile access to visual structures of web pages, we wanted first pretest our hypothesis: visually impaired people can explore and redraw simple grayscale shapes by using vibration motors.

PRE-TESTS PROTOCOL

We made experiments with 5 blind persons, each experiment consisted of: personal and technical questions, explanations of the test objective, a training task, and finally an evaluation task. The tested images (viewed on the tablet) contain some examples of expected results of web pages layout extraction, so success of distinguishing these shapes by blind users could be an indicator of their ability to distinguish web pages layouts. Figure 2 presents some images of evaluation phases. Image IDP1 contains 3 rectangles with matched sizes and with vertical order, and the image IDP2 contains 3 rectangles with different sizes and many relations of directions, so testing these two images could indicate the ability of distinguishing sizes, and distinguishing relations of directions. Image IDP3 contains different shapes, so it could test the ability to distinguish different shapes in the same image. We created different transitions of gray scale for each shape to analyze the reaction through a continuous decreasing/increasing intensity generated by the micro-motor.

For each image in the evaluation task, we asked users to explore it and to answer for some questions (cf. in table 1, symbol  represents a correct answer and the symbol X an incorrect one).

Table 1. Answers of questions for images IDP1, IDP2, IDP3

User-ID ID0 ID1 ID2 ID3 ID4 IDP 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 Number shapes   X  X X  X  X      sizes of shapes X X X X X X  X X X     
At the end of the test, we asked each user to draw the recognized graphical patterns founded in each touched image (we used another tablet for redrawing). Some users could redraw forms nearly identical to original evaluation images. Figure 3 views the redrawing results of the user who answered successfully to all questions.

ANALYSIS OF RESULTS

We notice from data in table 1 that the best performance is for the user with ID4, and this may be because this user is the youngest between others, and it could be because she was the only one who has already used touched devices (an iPhone, in her case working with VoiceOver), and this could be an indication that more training users more getting better results. Redrawing comparison with touched images can conclude: 1-ability of distinguishing sizes of shapes, because the degree of scaling between redrawn shapes is nearly equal to the degree of scaling between real shapes. 2-ability of distinguishing relations of directions (vertical order, left to, right to…), because relations of directions between redrawn shapes are nearly identical to relations of directions between real shapes.

CONCLUSION AND PERSPECTIVES

We have presented an embedded device dedicated for visually impaired people to explore simple grayscale shapes using vibrations. First results are promising: after a short training period, tested persons were able recognizing basic patterns. Many enhancements to be achieved such as increasing the number and quality of micro-vibrators, and more controlling frequencies and amplitudes.

7.

Figure 1 :

 1 Figure 1: "TactiNET" prototype.

Figure 2 :

 2 Figure 2: Images IDP1, IDP2, IDP3 (Evaluation Task).

Figure 3 .

 3 Figure 3. Results of redrawing images IDP1, IDP2, IDP3


Nicolas Beldiceanu

Bárbara Dumas Feris

Philippe Gravey

Sabbir Hasan

Claude Jard

Thomas Ledoux

Yunbo Li

Didier Lime

Gilles Madi-Wamba

Jean-Marc Menaud

Pascal Morel

Michel Morvan

Marie-Laure Moulinard

Anne-Cécile Orgerie

Jean-Louis Pazat

Olivier Roux

Ammar Sharaiha

The EPOC project: Energy Proportional and Opportunistic Computing system

Keywords: Data-Center, Energy-Efficiency, Virtualization, Task Placement, Optical Network

With the emergence of the Future Internet and the dawning of new IT models such as cloud computing, the usage of data centers (DC), and consequently their power consumption, increase dramatically. Besides the ecological impact, the energy consumption is a predominant criteria for DC providers since it determines the daily cost of their infrastructure. As a consequence, power management becomes one of the main challenges for DC infrastructures and more generally for large-scale distributed systems. In this paper, we present the EPOC project which focuses on optimizing the energy consumption of mono-site DCs connected to the regular electrical grid and to renewable energy sources.

INTRODUCTION

A data center (DC) is a facility used to house tens to thousands of computers and their associated components. These servers are used to host applications available in the Internet, from simple web server to multi-tier applications, but also some batch jobs. With the explosion of online services, particularly driven by the extension of cloud computing, DCs are consuming more and more energy. The growth of energy consumption by DCs is, at the same time, a technical, environmental and financial problem. Technically, in some areas (like Paris), the electrical grid has already saturated, thus preventing new DC installation or expansion of the existing ones. From an environmental point of view, the electricity production causes many CO 2 emissions, whereas financially the OPEX (Operational Expenditure) have exceeded CAPEX (Capital Expenditure). Although over the last few years, computer servers have become less expensive and highly energy efficient, the price of electricity has significantly increased even in countries known of having lower electricity price (e.g. France). To some extent, these operating costs are mainly related to the power consumption. Several actions are possible to reduce these impacts/costs. One of them consists in using a local power generation based on renewable energy, like Microsoft, Google, and Yahoo who have built new DCs close to large and cost-efficient hydroelectric power sources for instance.

However, the extension of hydroelectric power plants is severely limited by environmental issues, and other renewable energy sources provide intermittent electricity over time. In the EPOC project, we aim at focusing on energy-aware task execution from the hardware to the application's components in the context of a mono-site and small DC (all resources are in the same physical location), which is connected to the regular electric Grid and to a local renewable energy sources (such as windmills or solar cells).

Pioneering solutions have recently been proposed to tackle the challenge of powering small-scale DC with only renewable energies [START_REF] Goiri | Designing and managing data centers powered by renewable energy[END_REF]. In the context of EPOC, we are considering a hybrid approach relying on both the regular grid and a renewable energy source, like sun or wind for instance.

On the generation side, it is estimated that 10% of electric energy produced by power plants is currently lost during transmission and distribution to the consumers, with 40% of these losses occurring on the distribution network [START_REF] Feng | Smarter grids are more efficient[END_REF]. For instance in 2006, in the United-States, the total energy and distribution losses were about 1,638 billion and 655 billion kWh, respectively [START_REF] Feng | Smarter grids are more efficient[END_REF]. Most of the energy-efficient Cloud frameworks proposed in the literature do not consider electricity availability and renewable energy in their models. This is a major drawback since significant amounts of electricity are lost during transportation and storage.

In the EPOC project, the first challenge consists in developing a transparent (for users) energy proportional computing (EPC) distributed system (from system to service-oriented runtime) mainly based on hardware and virtualization capabilities. The second challenge addresses the energy issue through a strong synergy inside infrastructure-software stack and more precisely between applications and resource management systems designed to tackle the first challenge. This approach must allow adapting the Service Level Agreement (SLA) by seeking the best trade-off between energy cost (from regular electric grid), its availability (from renewable energy), and service degradation (from application reconfiguration to jobs suspension). The third challenge embarks to set energy efficient optical networks as key enablers of future internet and cloudnetworking service deployment through the convergence of optical-infrastructure layer with the upper layers. Another strength of the EPOC project is to integrate all research results into a common prototype named EpoCloud. This approach allows the pooling of development efforts, and validates solutions on common and reproducible use-cases. EPOC is an ongoing project, and the aim of this paper is to present the DC architecture designed in this context, from hardware layer to middleware layer.

EpoCloud principles

Our first goal is to design an energy-proportionalcomputing system (EPCS), which implies no energy consumption, whenever there is no activity. To date, dynamic power management has been widely used in embedded systems as an effective energy saving method with a policy that attempts to adjust the power mode according to the workload variations [START_REF] Sridharan | Reliability aware power management for dual-processor real-time embedded systems[END_REF]. Unfortunately, servers consume energy even when they are idle. For an efficient EPCS, we need to have the capability to turn on/off servers dynamically. Vary-on/vary-off (VOVO) policy reduces the aggregate-power consumption of a server cluster during periods of reduced workload. The VOVO policy turns off servers so that only the minimum number of servers that can support the workload are kept alive.

However, much of the applications running in a data center must be online constantly. To solve this problem, dynamic placement using application live migration permits to keep using VOVO policy in the on-line application context. Live migration moves a running application between different physical machines without disconnecting the client or application. Memory, storage, and network connectivity are transferred from the original host machine to the destination. Currently, the most efficient system for live migration is the use of virtualization. Virtualization refers to the creation of a virtual machine (VM) that acts like a real computer with an operating system but software executed on these VMs is separated from the underlying hardware resources. Virtualization also allows snapshots, fail-over and globally reduce the IT energy consumption by consolidating VMs on a physical machine (i.e. increasing the server utilization and thus reducing the energy footprint). Furthermore, dynamic consolidation uses live migration for effective placement of VMs on the pool of DC servers to reduce energy, increase security, etc. But live migration requires significant network resources.

Our first main objective is more concentrated on Workload-driven approach. EpoCloud adapts the power consumption of the DC depending on the application workload. Our second objective is more focused on Power-driven SLA. The Power-driven approach implies shifting or scheduling the postponable workloads to the time period when the electricity is available (from the renewable energy sources) or at the best price. For on-line application, power-driven approach implies a degradation of services when energy is at a insufficient level, while maintaining SLAs. In addition, EpoCloud takes advantage of the available energy to perform some tasks. Some of them allow limitations on application degradation. We describe our EpoCloud architecture and EpoCloud manager in section 3 and 4 respectively.

High throughput optical networks for VM migration without SAN

Recent studies on companies' data-centers show that a VM consume an average of 4 GB of Memory and 128 GB of storage. Thus, it will take a minimum of 17.5 minutes (resp. 1.75 minutes) with a 1 Gb/s (resp. 10 Gb/s) network to realize a complete VM migration. Moreover, a classical consolidation ratio in virtualized data centers is 50 VMs per server. According to the approach that we are considering in EPOC (VOVO Policy), our data center needs to be able to migrate all the VMs running on a server (7.5 TB), whenever the hypervisor requests to turn this server off in order to save power. Having one optical port per rack means that its bandwidth might be shared by the servers located in this rack. Then, is this bandwidth enough to migrate all the VMs in one server? Using 10 Gb/s this operation takes around 2 hours. However, if we consider an example, 32 servers per rack, the same operation would take about 53 hours, since now the bandwidth is being shared by the 32 servers. Consequently, increasing the bit rate of the interconnection network becomes a must.

To overcome the aforementioned problem, classical dynamic consolidation system uses live migration with a Storage Area Network (SAN). In this case, the VM storage is shared between all servers and live migration is limited to transfer VMs memory. Nevertheless, adding a SAN impacts on the global DC energy consumption. EpoCloud proposes to suppress the SAN, which is a dedicated network providing access to consolidated data storage.

Among various components of a data center, storage is one of the biggest consumers of energy. An industry report [START_REF] Inc | Power, heat, and sledgehammer[END_REF] shows that storage devices account for almost 27% of the total energy consumed by a DC. By suppressing the SAN we optimize the energy consumption but we introduce a strong hypothesis on the technical architecture : for accessing data of applications and systems, we can only use local disk servers. Turning off a server involves transferring 7.5 TB on average. Given this scenario, a high broadband network is required, but is a 100 Gb/s network card really exploitable with current server technologies? In this article, we present an innovative network architecture, detailed in section 3.1, a pre-study in section 3.2, and finally, we describe in section 3.3 architectural motives and principles for the integration of renewable energy.

Network Architecture

A classical interconnection architecture is based on a 3-Tier fat-tree topology as presented in [START_REF] Kachris | A survey on optical interconnects for data centers[END_REF]. Whereas the three main switching layers are: core, aggregation, and ToR (top-of-therack); each layer, based on Electrical Packet Switches (EPS). Servers accommodated into racks are connected through the ToR switches to the aggregation layer, and from there to the core layer using the aggregate switches. Finally, by means of the core switches, servers can be interconnected to the internet (or outside the DC).

The introduction of optical communications seems to be crucial, because it can achieve very high data rates, low latency and low power consumption [START_REF] Kachris | Power consumption evaluation of all-optical data center networks[END_REF]. This has recently become a hot research topic inside the optical net-working community. Some authors propose a direct migration to all-optical architectures, most of them based on Optical Circuit Switching (OCS) [START_REF] Singla | Proteus: A Topology Malleable Data Center Network[END_REF] that does not meet the needs of a variable traffic over time. Some hybrid architectures, involving several hierarchy levels, could have the potential to connect millions of servers in giant DC [START_REF] Gumaste | On the architectural considerations of the FISSION (Flexible Interconnection of Scalable Systems Integrated using Optical Networks) framework for data-centers[END_REF]. As already noted, EPOC aims at focusing on small/medium size data centers.

For transferring 7.5 TB, implementing a full optical interconnection architecture could be an attractive option, in terms of latency, power consumption and control complexity. This implies using Optical Packet Switching (OPS) technology, whose maturity is still highly questionable, in spite of several decades of investigation for telecom network applications [START_REF] Yoo | Optical Packet and Burst Switching Technologies for the Future Photonic Internet[END_REF]. Nevertheless, several techniques, relying on fast wavelength tunable optical emitters, have recently gained a renewed attention, in particular for metropolitan area network applications. These techniques include TWIN (Time-domain Wavelength Interleaved Networks), originally proposed by Lucent [START_REF] Sanjee | A new optical network architecture that exploits joint time and wavelength interleaving[END_REF], and POADM (Packet Optical Add and Drop Multiplexer) proposed by Alcatel-Lucent [START_REF] Chiaroni | Optical packet add/drop multiplexers for packet ring networks[END_REF].

In the EPOC project, we decided to investigate a third option, derived from TWIN, which was presented in [START_REF] Indre | POPI: A Passive Optical Pod Interconnect for high performance data centers[END_REF] under the name of POPI (Passive Optical Pod Interconnect). The main motivation for this choice is that POPI uses a purely passive optical network, with power consumption concentrated at networks edge. This architecture is simpler than the classical EPS one [START_REF] Kachris | A survey on optical interconnects for data centers[END_REF], in the sense that there is no ToR switch and the existence of racks will depend on the bandwidth assigned per server (see POPI scheme depicted at Figure 1). Therefore, there is no difference between inter-or intra-rack communications. Servers are independent and can connect to each other by means of a passive coupler. Each server i has a transmitter constituted by a tunable laser, and a receiver adjusted to wavelength i. The estimate total power consumption of POPI is around 25% of the classical EPS architecture power consumption [START_REF] Indre | POPI: A Passive Optical Pod Interconnect for high performance data centers[END_REF].

The maximum capacity of POPI in terms of number of servers (we consider one wavelength per server and no rack) is related to the limitations imposed by every component of the architecture. The tunable laser could present one of these limitations. In this paper we have taken into account the laser presented in [START_REF] Chiaroni | Novel Optical Packet Switching Nodes for Metro and Backbone Networks[END_REF], with STM64 50 GHz-spaced channels and a tuning speed of 5 ns. We reserve three wavelengths: two for the controllers and one for the gateway. Thus, if we consider one wave- length per server, the maximum number of servers shall be 61.

Server throughput capacity

In EPOC project, the DC consolidates the VMs periodically by using the live migration technology. Each server owns a 100 Gb/s transmission capacity and the live migration migrates the whole VM including the storage. It requires huge network resource to maintain the performance level therefore less degradation is employed during the process. In order to achieve a read/write speed of 100 Gb/s, we prefer SSD (Solid-State Drive) over HDD (Hard Disk Drive), since SSD is faster and less energy consuming than HDD. A single SSD of 800 GB capacity with a of PCI-E 2.1 x8 (Peripheral Component Interconnect Express) interface, can achieve 2 GB/s reading speed and 1 GB/s writing speed. This implies that we still need several SSDs in RAID (Redundant Array of Inexpensive Disks) technology in order to attain the 100 Gb/s-data rate. In the near future, a PCI-E 3.0 x16 shall offer a 15.75 GB/s network data rate, so this will be achieved by a single SSD.

Despite of higher speed, energy consumption for SSD is largely reduced to about 2 W compared to 6 W for HDD. Consequently, SSD generates less heat than HDD; making SSD more suitable to our project purposes.

Integrating Local Renewable Energy

Although several research efforts have been made to reduce energy consumption by designing/implementing server consolidation, hardware with better power/performance trade-offs, workload migration and software techniques for energy aware scheduling, still the goal for alleviating carbon footprint is being underachieved. Given the circumstances, explicit or implicit integration of renewable energy to the DC can be the only way to reduce carbon footprint at an acceptable level.

Besides that, the demand for green services is ever increasing, thus integrating renewable sources to the data center left no choice. Few green cloud providers, e.g., GreenQloud (GreenQloud, 2010), Green House Data (Green House Data, 2007) and academic researchers [START_REF] Goiri | Designing and managing data centers powered by renewable energy[END_REF] integrated renewable sources to the data-center explicitly which offers green computing services with partial SLA fulfillment.

As renewable power sources are very intermittent in nature, hence predicting the amount of renewable energy production ahead of real time might demonstrate greater error statistics in DC power management. Nonetheless, excessive production of renewable energy can imbalance the Grid as renewable energy is connected to the Grid via grid-tie device, which combines electricity produced from renewable sources and Grid. One way to overcome the challenge is to use energy storage or battery to store this superfluous green energy which can be discharged later for peak shaving of DC power demand or for fulfillment of energy aware SLA between Infrastructureas-a-Service (IaaS) and Software-as-a-Service (SaaS) provider when renewable energy needed but not available. Energy storage incurs additional costs to DCs CAPEX and OPEX, and energy losses due to battery's efficiency and finite capacity. Therefore it is not an attractive solution for small-scale data centers.

In order to avoid using storage or batteries in small-scale DC, in the EPOC project, we propose to virtualize the green energy. Virtualization of energy implies nullifying the degraded interval (lack of green energy) with the surplus interval (excessive green energy than demand), whenever the availability of green energy is over the demand. From clients or SaaS providers perspective, they realize both the interval as ideal interval (when supply meets the demand), though the green energy was not present instantaneously rather present virtually as shown in Figure 2. So, whenever the green/renewable energy is present, we use the whole portion of the available green energy and draw the other portion from the grid. In this way energy storage is not needed and neither of the portion of renewable energy is wasted. Furthermore, total expenditure of energy purchasing can be reduced since no energy goes to waste and additional cost for using storage is not needed. Even energy aware SLA between IaaS and SaaS providers can be fulfilled if there is any.

EpoCloud Manager

Architectural principles for small data centers were defined in the previous sections. They rely on innovative infrastructure where a limited number of servers (without SAN) are connected by a high speed optical network and supplied by local sources of renewable energy, composed of a limited number of server (without SAN) connected by a high speed network. To take advantage of this architecture, the EPOC project develops an innovative task management system: the EpoCloud Manager including a smart task scheduler (Section 4.1), and an energy-aware SLA oriented management system (Section 4.2).

Opportunistic energy-aware resource allocation

In the EPOC project, we propose to design a disruptive approach to Cloud's resource management which takes advantage of renewable energy availability to perform opportunistic tasks. Let's recall that, the considered EpoCloud is mono-site (i.e. all resources are in the same physical location) and performs tasks (like web hosting or MapReduce tasks) running in virtual machines. The EpoCloud receives a fixed amount of power from the regular electrical grid. This power allows it to run usual tasks. In addition, the EpoCloud is also connected to renewable energy sources (such as windmills or solar cells) and when these sources produce electricity, the EpoCloud uses it to run more, less urgent, tasks.

The proposed resource management system integrates a prediction model to be able to forecast these extra-power periods of time in order to schedule more work during these periods. Given a reliable prediction model, it is possible to design a scheduling heuristic that aims at optimizing resource utilization and energy usage, problem known to be NP-hard. So, the proposed heuristics will schedule tasks spatially (on the appropriate servers) and temporally (over time, with tasks that can be planed in the future).

In order to achieve this energy-aware resource allocation, we distinguish two kinds of jobs to be scheduled on the data center: the web jobs which represent jobs requiring to run continuously (like web server) and the batch jobs which represent jobs that can be delayed and interrupted, but with a deadline constraint. The second type of jobs are the natural candidates of the opportunistic scheduling algorithm. Additionally for reducing further energy consumption in the EpoCloud, we are taking advantage of consolidation algorithms and on/off mechanisms to optimize the number of powered-on resources. These consolidation algorithms also relies on VM suspend/resume mechanisms for the batch jobs and live migration mechanisms of VMs for the web jobs. However, such mechanisms have a cost in terms of both time and energy, and so, the algorithms take these costs into account to optimize the overall energy utilization.

Energy-aware SLA

In cloud systems, applications are embedded in VMs. A VM must be executed on a single server. So for an application, a transparent elastic system, also called vertical scaling (or scale up), consists in adapting resources to the VM needs, typically involving the addition of CPUs or memory. Vertical scaling relies on cloud computing models and virtualization techniques to scale up/down applications based on their performance metrics. Although those proposals can reduce the energy footprint of applications and by transitivity of cloud infrastructures, they do not consider the internal characteristics of applications to finely define a trade-off between the application's Quality of Service (QoS) and their energy footprint.

Contrary to the previous approach, Horizontal scaling (or scale out) implies to add more resources for an application, such as adding a new VM to a distributed software application. Nonetheless, one needs to change the application configuration. For example an Apache Web application is scaled out by adding/removing VM apache workers and by modifying the mod proxy balancer file configuration.

One of the main challenges is to consider both the application internals and the global system to reduce the energy footprint in our cloud infrastructure. More precisely, we focus on adding the usual scaling up/down by considering all application's internal to be able to use several and different application configurations (corresponding to different quality of service (QoS) level). Each application is equipped with one autonomic loop in charge of determining the minimum amount of resources required to provide the best QoS possible while an additional loop manages the physical resources at the infrastructure level. The autonomic loop may switch from one configuration to another according to the incoming charge, the QoS expectations and the infrastructure constraints.

Conclusion

In this paper, we have presented the EpoCloud principles, architecture and middleware components. EpoCloud is our prototype, which will tackle three major challenges: 1) To optimize the energy consumption of distributed infrastructures and service compositions in the presence of ever more dynamic service applications and ever more stringent availability requirements for services; 2) To design a clever cloud's resource management, which takes advantage of renewable energy availability to perform opportunistic tasks, then exploring the trade-off between energy saving and performance aspects in large-scale distributed system; 3) To investigate energy-aware optical ultra high-speed interconnection networks to exchange large volumes of data (VM memory and storage) over very short periods of time.

In order to achieve these ambitious goals, we propose: 1) To determine energy-aware SLA management policies considering energy as a first class resource and relying on the concept of virtual green energy to better utilize renewable energy; 2) To evaluate energy-aware task scheduling algorithms based on the distinction of two kinds of tasks (web tasks and batch tasks) and leveraging renewable energy availability to perform opportunistic tasks without hampering performance; 3) To assess the ability of a specific OPS-based interconnection architecture to support the exchange of large data volumes (about 7.5 TB for the migration of all VMs hosted by a single server while allowing background traffic exchange between servers).

Figure 1 :

 1 Figure 1: POPI architecture (from (Indre et al., 2014)).

Figure 2 :

 2 Figure 2: Concept of Virtualized Green Energy

ACKNOWLEDGMENTS

The author acknowledges the support of the Comin-Labs Labex.


Thomas Bonald
email: thomas.bonald@orange-ftgroup.com

Mathieu Feuillet
email: mathieu.feuillet@telecom-paristech.fr

Alexandre Proutiere
email: alexandre.proutiere@microsoft.com

Is the "Law of the Jungle" Sustainable for the Internet?

In this paper we seek to characterize the behavior of the Internet in the absence of congestion control. More specifically, we assume all sources transmit at their maximum rate and recover from packet loss by the use of some ideal erasure coding scheme. We estimate the efficiency of resource utilization in terms of the maximum load the network can sustain, accounting for the random nature of traffic. Contrary to common belief, there is generally no congestion collapse. Efficiency remains higher than 90% for most network topologies as long as maximum source rates are less than link capacity by one or two orders of magnitude. Moreover, a simple fair drop policy enforcing fair sharing at flow level is sufficient to guarantee 100% efficiency in all cases.

I. INTRODUCTION

It is commonly accepted that the surprisingly high robustness of the Internet to the traffic growth of the last two decades is mainly due to the presence of congestion control. Since the introduction of congestion control algorithms in TCP in the late 1980s, we haven't witnessed any congestion collapse similar to that of the young Internet. However, it is unclear whether the IETF will be able to enforce the use of TCP in the future. As a matter of fact, many applications already apply their own congestion control, if any, without complying with the "TCP-friendliness" principle [START_REF] Floyd | Promoting the use of end-to-end congestion control in the internet[END_REF], [START_REF] Handley | TCP Friendly Rate Control (TFRC): Protocol Specification[END_REF].

In this paper we seek to characterize the behavior of the Internet in the absence of any congestion control. As pointed out in [START_REF] Raghavan | Decongestion control[END_REF], the induced packet losses may easily be recovered thanks to erasure coding. It would also be necessary to limit buffer sizes so as to maintain acceptable end-to-end delays. Under these assumptions, there is no clear reason why the rate of data transfers should be controlled. We thus go one step beyond the "decongestion control" scenario of Raghavan and Snoeren [START_REF] Raghavan | Decongestion control[END_REF] and assume that all sources send data at their maximum rate without any control. Clearly, the network would then reach a totally different equilibrium point from today's, with significant packet loss rates and the "Law of the Jungle" as a sharing principle. Most applications could adapt and survive. It remains to quantify the potential loss of efficiency in the utilization of network resources.

To this aim, we consider a traffic scenario with random arrivals of finite size flows and analyse the stability of the stochastic process formed by the number of flows in progress.

Flows are represented as fluid streams that thin on their path to the destination due to packet losses. Efficiency is assessed through the maximum traffic load the network can sustain under the stability condition. Surprisingly, there is generally no congestion collapse, even under usual FIFO scheduling and tail dropping. The worst scenario where all routers transmit packets that are eventually lost on their path to the destination occurs for cyclic networks only. In acyclic networks, our results show that efficiency remains generally higher than 90% as long as maximum source rates are less than link capacity by one or two orders of magnitude. Moreover, a simple fair drop policy enforcing fair sharing at flow level is sufficient to guarantee 100% efficiency in all cases.

While somewhat in contradiction with common belief that TCP plays a key role in the stability of the Internet, these results rely on the strong assumption that all applications use perfect erasure coding or selective retransmission schemes that guarantee the usefulness of any received packet. Another concern about the relevance of the considered scenario is the sensitivity of some key applications like voice and media streaming to high loss rates and varying available bandwidth. These issues are beyond the scope of the present paper, however, whose primary goal is to show that the absence of congestion control is not, in principle, an unviable option for the Internet.

The rest of the paper is structured as follows. Related work is presented in the next section. Section III describes the fluid model used to represent bandwidth sharing in the considered scenario without congestion control. The flow-level dynamics and the associated stability issues are introduced in Section IV. These results are used in Section V to estimate the loss of efficiency due to the absence of congestion control and validated through packet-level simulations in Section VI. Section VII concludes the paper.

II. RELATED WORK

Since the introduction of congestion control algorithms in TCP [START_REF] Jacobson | Congestion avoidance and control[END_REF], a number of scheduling and dropping policies have been proposed to alleviate the problem of unresponsive flows, see [START_REF] Mckenney | Stochastic fairness queueing[END_REF], [START_REF] Ott | SRED: stabilized RED[END_REF], [START_REF] Pan | CHOKe -A stateless active queue management scheme for approximating fair bandwidth allocation[END_REF], [START_REF] Shreedhar | Efficient fair queueing using deficit round robin[END_REF], [START_REF] Stoica | Core-stateless fair queueing: achieving approximately fair bandwidth allocations in high speed networks[END_REF], for instance. These policies have been designed with the aim of enforcing fair bandwidth sharing. In the presence of congestion control, they typically achieve max-min fairness [START_REF] Massoulié | Bandwidth sharing: objectives and algorithms[END_REF]; we shall see that the absence of congestion control leads to a different allocation, that does not belong to the usual class of α-fairness [START_REF] Mo | Fair end-to-end window-based congestion control[END_REF].

The phenomenon of congestion collapse has rarely been considered under realistic traffic conditions with random flow arrivals. A notable exception is the paper by Massoulié and Roberts [START_REF] Massoulié | Bandwidth sharing and admission control for elastic traffic[END_REF] that pointed out the potential instability of cycles. It was also shown in [START_REF] Bonald | Impact of fairness on Internet performance[END_REF], [START_REF] Veciana | Stability and performance analysis of networks supporting elastic services[END_REF], [START_REF] Ye | Stability of data networks under an optimization-based bandwidth allocation[END_REF] that fairness stabilizes the network whenever possible, while the bias introduced by discriminatory scheduling policies may lead to loss of capacity. It turns out, however, that these critical flow-level dynamics are ignored in all papers that have previously considered performance in the absence of congestion control, see [START_REF] Akella | Selfish behavior and stability of the internet:: a game-theoretic analysis of tcp[END_REF], [START_REF] Floyd | Promoting the use of end-to-end congestion control in the internet[END_REF], [START_REF] Kelly | Patterns of congestion collapse[END_REF], [START_REF] López | A game theoretic comparison of tcp and digital fountain based protocols[END_REF], [START_REF] Raghavan | Decongestion control[END_REF], [START_REF] Zhang | TCP connection game: a study on the selfish behavior of TCP users[END_REF] for instance.

III. BANDWIDTH SHARING

We first present a simple fluid model that captures the way flows may share bandwidth in the absence of congestion control. This model will be validated through packet-level simulations in Section VI.

A. Network model

Consider a network of L links. Denote by C l the capacity of link l (in bit/s). A number of flows compete for access to these links. These flows are categorized into an arbitrary set of K classes. There are n k class-k flows; each of these flows has an access rate to the network denoted by a k and follows a route of length d k in the network defined as an ordered set of distinct links

r k = {r k (1), r k (2), . . . , r k (d k )}.
Let ϕ k be the throughput of each class-k flow. We refer to the vector ϕ = (ϕ 1 , . . . , ϕ K ) as the bandwidth allocation. It must satisfy the rate constraints:

∀k, ϕ k ≤ a k and ∀l, k:l∈r k n k ϕ k ≤ C l .
(1)

A number of allocations have been considered in the literature to model the way flows actually share bandwidth in the Internet. These include max-min fairness, proportional fairness and α-fair policies [START_REF] Hahne | Round-robin scheduling for max-min fairness in data networks[END_REF], [START_REF] Kelly | Rate control in communication networks: shadow prices, proportional fairness and stability[END_REF], [START_REF] Massoulié | Bandwidth sharing: objectives and algorithms[END_REF], [START_REF] Mo | Fair end-to-end window-based congestion control[END_REF]. It turns out that, for all these allocations, the order in which class-k flows go through the set of links r k does not matter. This is because flows are typically assumed to be rate-controlled, so that packet losses are neglected beyond the feedback they provide to the sources for rate adaptation. In particular, flows are generally considered as fluid streams and the rate of each class-k flow is assumed to be either equal to its access rate a k or limited by the capacity of some link l ∈ r k , independently of its position on route r k ; there is no bandwidth waste, a property known as Pareto efficiency.

In the absence of congestion control, packet losses play a key role. Thus, while retaining the necessary abstraction of fluid streams, we need to describe the evolution of the rate of each flow going through the network. Specifically, we assume that each class-k flow initially transmits at full rate a k and we denote by θ k (i) its rate at the output of the i-th link on its route r k , for all i = 1, 2, . . . , d k . The actual throughput ϕ k of each class-k flow corresponds to its rate at the output of the last link on its route, θ k (d k ).

Due to potential fluid loss at each link, we have:

ϕ k = θ k (d k ) ≤ θ k (d k -1) ≤ . . . ≤ θ k (1) ≤ a k . (2)
Moreover, the total rate at the output of each link cannot exceed the capacity of this link, so that:

∀l, k,i:r k (i)=l n k θ k (i) ≤ C l . ( 3
)
We say that a link is saturated if the corresponding inequality (3) is an equality. Note that the rate constraints (1) follow from ( 2) and (3). We shall see that, unlike usual allocations, the order in which class-k flows go through the set of links r k does matter. Additionally, losses may result in bandwidth waste: the allocation is generally not Pareto efficient.

In order to characterize the allocation achieved in the absence of congestion control, it remains to determine how output rates depend on input rates at each link. This depends on the buffer management policy. In the considered fluid model, losses occur on saturated links only, when the total input rate exceeds capacity; the total output rate is then assumed to be equal to capacity. In the following, we consider two buffer management policies: tail dropping and fair dropping.

B. Tail dropping

The tail drop policy simply consists in dropping each incoming packet when the buffer is full. Assuming that all packets have the same probability of being dropped at the input of any given link, we deduce that, in the considered fluid model, the proportion of fluid lost is the same for all flows going through that link.

Specifically, let R l be the total rate at the input of link l:

R l = k:r k (1)=l n k a k + k,i:r k (i+1)=l n k θ k (i).
The output rate of any class-k flow satisfies:

θ k (1) = a k min C l R l , 1 if l = r k (1), (4)
and for all i = 2, 3, . . . , d k ,

θ k (i) = θ k (i -1) min C l R l , 1 if l = r k (i). ( 5
)
The definition is non-ambiguous for acyclic networks, i.e., if links can be numbered in such a way that each route consists of an increasing sequence of link indices. The allocation then directly follows from applying (4) and ( 5) for all l = 1, . . . , L, successively. For general networks, the definition remains consistent in view of the following result, proved in the Appendix:

Theorem 1: The output rates obtained under tail dropping are uniquely defined by ( 4) and [START_REF] Veciana | Stability and performance analysis of networks supporting elastic services[END_REF].

Consider the triangle network of Figure 1 for instance. This network has three links of unit capacity, and is shared by three flows whose routes are {1, 2}, {2, 3} and {3, 1}, respectively. All flows have the same access rate a > 1/2. By symmetry, the common rate at the output of the first link of each route, denoted by θ( 1), is the solution to the fixed point equation:

θ(1) = a a + θ [START_REF] Akella | Selfish behavior and stability of the internet:: a game-theoretic analysis of tcp[END_REF] .

We get θ(1) = 1 2 ( √ a 2 + 4a -a)
and the rate at the output of the second link of each route θ(2) = θ( 1)/(a + θ( 1)), that is:

θ(2) = √ a 2 + 4a -a √ a 2 + 4a + a . ( 6
)
This is the common throughput of all three flows. Note that the allocation is not Pareto-efficient since θ(2) < 1/2. This bandwidth waste is due to the phenomenon of "dead" packets, a term coined by Kelly, Floyd and Shenker [START_REF] Kelly | Patterns of congestion collapse[END_REF]: each flow uses some bandwidth of the first link of its route to transmit data that is lost at the second link.

C. Fair dropping

Under the fair drop policy, a packet belonging to that flow having the largest data volume in the buffer is dropped in case of buffer overflow. This is intended to guarantee to each incoming flow its fair share of bandwidth. In the considered fluid model, we shall assume that the output rates are max-min fair shares of the input rates at each link.

Specifically, let S l be the fair share of any saturated link l. This is the unique rate such that the output rate of any class-k flow satisfies:

θ k (1) = min (S l , a k ) if l = r k (1), (7)
and for all i = 2, 3, . . . , d k ,

θ k (i) = min (S l , θ k (i -1)) if l = r k (i), (8)
and the total rate at the output of link l is equal to C l . These rates follow from the following modified waterfilling procedure: (i) start from null rates;

(ii) increase the rate of each flow at the same speed until some rate constraint (1) is reached; (iii) when some access rate constraint is reached, freeze the rate of the corresponding flows and keep increasing the rate of all other flows, if any; when the capacity of some link is reached, freeze the downstream rate of all flows going through this link and keep increasing the upstream rate of these flows and the rate of the other flows, if any;

(iv) stop when the rate of all flows is frozen. This is similar to the standard waterfilling procedure leading to max-min fairness, except that one keeps increasing the upstream rate of those flows going through a saturated link. This defines a unique allocation, whose throughputs may be higher or lower than the max-min fair shares. Consider for example the network of Figure 2, consisting of three unit capacity links and four classes with a common unit access rate. For n = (1, 1, 2, 3), we get under max-min fairness:

ϕ mm 1 = 1 2 , ϕ mm 2 = 1 2 , ϕ mm 3 = 1 5 , ϕ mm 4 = 1 5 ,
while under fair dropping:

ϕ 1 = 2 3 , ϕ 2 = 1 3 , ϕ 3 = 1 5 , ϕ 4 = 1 5 .
In the modified waterfilling procedure, keeping increasing the upstream rate of class 3 harms class 2 which, in turn, benefits to class 1. The allocation ϕ obtained under fair dropping cannot be compared with max-min fairness but with the min allocation defined by:

ϕ min k = min a k , min l∈r k C l j:l∈rj n j . ( 9
)
For the example of Figure 2 with n = (1, 1, 2, 3), we get:

ϕ min 1 = 1 2 , ϕ min 2 = 1 3 , ϕ min 3 = 1 5 , ϕ min 4 = 1 5 .
The throughput of class-k flows under the min allocation follows from a waterfilling process after removing all links other than those of route r k . It is easy to see that the resulting allocation is worse than that obtained with the modified waterfilling process. We deduce:

Lemma 1: Let ϕ be the allocation obtained under fair dropping. We have ϕ k ≥ ϕ min k for all classes k. This result will prove useful for the stability analysis of the allocation achieved under fair dropping.

IV. STABILITY ANALYSIS

In the rest of the paper, we consider a traffic scenario with random arrivals of finite size flows. We first analyse the stability of the stochastic process formed by the number of flows in progress. These results are used in the next section to estimate the ability of flows to effectively utilize network resources.

A. Flow-level dynamics

Let n = (n 1 , . . . , n K ) be the vector of the numbers of flows in progress, which we refer to as the network state. This is now a stochastic process, that evolves as new flows are generated by users and ongoing flows cease upon completion. We assume that class-k flows are generated according to a Poisson process of intensity λ k and have independent, exponentially distributed sizes with mean 1/μ k (in bits). We define the traffic intensity of class-k flows as ρ k = λ k /μ k (in bit/s). In any state n, we denote by ϕ k (n) the throughput of each class-k flow and by

φ k (n) = n k ϕ k (n) the total throughput of class-k flows.
Under the above assumptions, the network state defines a Markov process with transition rates λ k from state n to state n + e k and φ k (n)μ k from state n to state ne k , where e k denotes the vector with 1 in the k-th component and 0 elsewhere. We say that the network is stable when this Markov process is ergodic, so that all flows have finite durations.

The usual stability condition is that traffic intensity is less than capacity for all links: ∀l,

k:l∈r k ρ k < C l . ( 10
)
This condition is known to be necessary for any allocation that satisfies the capacity constraints [START_REF] Akella | Selfish behavior and stability of the internet:: a game-theoretic analysis of tcp[END_REF], see e.g. [START_REF] Bonald | A queueing analysis of max-min fairness, proportional fairness, and balanced fairness[END_REF]. It is also sufficient for α-fair allocations, that typically represent the bandwidth sharing achieved by rate-controlled flows [START_REF] Bonald | Impact of fairness on Internet performance[END_REF]. This result tends to show that congestion control stabilizes the network whenever possible.

In the following, we consider a network without any congestion control and derive the stability condition under tail dropping and fair dropping. The corresponding allocations are characterized by (4), ( 5) and ( 7), [START_REF] Hahne | Round-robin scheduling for max-min fairness in data networks[END_REF], respectively.

B. Tail dropping

It turns out that the usual stability condition [START_REF] Jacobson | Congestion avoidance and control[END_REF] is generally not sufficient under tail dropping. We successively consider trees, cycles and lines to illustrate the sensitivity of the stability condition to the network topology.

1) Downstream trees: Downstream trees are defined as follows: (i) a common root link, say link 1, so that r k (1) = 1 for all classes k; (ii) any two classes j, k that have m links in common satisfy r j (i) = r k (i) for all i = 1, 2, . . . , m. An example of downstream tree is presented in Figure 3. It may be easily verified that:

∀k, φ k (n) = min n k a k , min l∈r k C l n k a k j:l∈rj n j a j .
Thus the allocation coincides with weighted max-min fairness, with weights proportional to the access rates. This allocation is known to be stable under the usual stability condition (10) [START_REF] Veciana | Stability and performance analysis of networks supporting elastic services[END_REF]. We deduce that the usual stability condition ( 10) is both necessary and sufficient for downstream trees.

2) Upstream trees: Upstream trees are defined as follows: (i) a common root link, say link 1, so that r k (d k ) = 1 for all classes k; (ii) any two classes j, k that have m links in common satisfy r j (

d j + 1 -i) = r k (d k + 1 -i) for all i = 1, 2, . . . , m.
An example of upstream tree is presented in Figure 3.

We first consider a homogeneous upstream tree consisting of a unit capacity root link and K branches of capacity C. Each route goes through one branch and the root link. Traffic distribution is uniform; we denote by ρ the traffic intensity of each class.

Proposition 1: A homogeneous tree is stable under the usual stability condition ρ < min(C, 1/K).

Proof. The proof simply consists in observing that φ k (n) ≥ min(C, 1/K) in all states n. Thus the network state is upper bounded by the state of K independent M/M/1 queues, each of load ρ/ min(C, 1/K) < 1.

Unlike homogeneous trees, the usual condition (10) may be not sufficient for the stability of heterogeneous trees. Consider a tree consisting of a unit capacity root link and K -1 branches, each of capacity C. A route, corresponding to class 1, is directly connected to the root link, i.e., d 1 = 1; each of the K -1 other routes goes through one branch and the root link. Classes 2, . . . , K have a common access rate a 2 and a common traffic intensity ρ 2 . The usual stability condition is:

ρ 1 + (K -1)ρ 2 < 1, ρ 2 < C. ( 11
)
Now let ψ 1 (n 1 ) be the minimum total rate of n 1 class-1 flows, obtained by letting the number of other flows grow to infinity:

ψ 1 (n 1 ) = lim n2,...,nK →∞ φ 1 (n), = min n 1 a 1 , n 1 a 1 n 1 a 1 + (K -1)C .
Define the corresponding total rate of any other class:

ψ 2 (n 1 ) = lim n2,...,nK →∞ φ 2 (n), = min C, C n 1 a 1 + (K -1)C .
Finally, consider the evolution of the number of class-1 flows, assuming these flows always get their minimum rate. Provided ρ 1 < 1, this is a stable birth-and-death process with stationary distribution:

π 1 (n 1 ) = π 1 (0) ρ n1 1 ψ 1 (1)ψ 1 (2) . . . ψ 1 (n 1 )
.

Proposition 2: The considered heterogeneous upstream tree is stable if and only if ρ 1 < 1 and ρ 2 < κ, with:

κ = ∞ n1=0 π 1 (n 1 )ψ 2 (n 1 ).
Proof. The result is a direct consequence of Theorem 2 and Theorem 3 of [START_REF] Borst | Stability of parallel queueing systems with coupled service rates[END_REF], since the allocation ϕ is partially decreasing in n and has an asymptotically uniform limit as n tends to infinity.

Note that, when (K -1)C ≥ 1, the stability condition coincides with the usual one; we have in this case:

κ = ∞ n1=0 π 1 (n 1 ) C n 1 a 1 + (K -1)C , = ∞ n1=0 π 1 (n 1 ) 1 -ψ 1 (n 1 ) K -1 , = 1 -ρ 1 K -1 , ( 12
)
from which [START_REF] Kelly | Rate control in communication networks: shadow prices, proportional fairness and stability[END_REF] follows. When (K -1)C < 1, the stability condition is stricter than the usual one. In the limiting case a 1 → ∞ for instance, class-1 flows have strict priority over other flows and the stability condition becomes ρ 1 < 1 and ρ 2 < (1ρ 1 )C. The reduction of the stability region turns out to be slighter for low access rates, as illustrated by Figure 4 for K = 2 and C = 0.5. 3) Cycles: We refer to a cycle as a network such that (i) for each class k and all i < d k , r k (i +1) = r k (i)+1 mod L; (ii) for all l = 1, . . . , L, there exists some k such that route r k contains links l and l + 1 mod L. The latter property ensures connectivity. The triangle of Figure 1 is an example of cycle.

Cycles are the worst network topologies for the tail drop policy, due to the phenomenon of "dead packets" described at the end of §III-B: whatever the initial condition, the network evolves towards a state with an infinite number of flows in progress, all links transmitting data that are eventually lost:

Proposition 3: A cycle is unstable whenever ρ k > 0 for all classes k. Proof. Let C = max l C l , a = min k a k and ρ = min k ρ k . Choose m such that: C ma + C C < ρ.
Let K ⊂ {1, . . . , K} be a minimal set such that the network restricted to classes K is a cycle. Now assume that n k (t 0 ) ≥ m for all k ∈ K at some time t 0 . Then there is some positive probability that n k (t 0 ) ≥ m for all k ∈ K and all t ≥ t 0 . This is because the number of flows of each of these classes behaves as a transient single server queue with arrival rate λ k and service rate:

φ k (t)μ k ≤ C ma + C Cμ k < λ k ,
where the first inequality follows from the existence of some index i ≥ 2 and some class j ∈ K such that r k (i) = r j (1).

The Markov process n(t) is transient.

4) Lines:

Lines are "broken" cycles. They are formally defined by the following two properties: (i) for each class k and for all i = 1, 2, . . . , d k -1, r k (i + 1) = r k (i) + 1; (ii) for all l = 1, . . . , L -1, there exists some k such that route r k contains links l and l + 1. An example of line is the following: there are K = L flow classes with r k = {k} for all k = 2, . . . , L and r 1 = {1, 2, . . . , L}. Such a line with L = 3 links is presented in Figure 5. In the limiting case where the access rates are infinite, the tail drop policy gives strict priority to short route flows, i.e., flows of classes k = 2, . . . , L. The network behaves as that considered in [START_REF] Bonald | Impact of fairness on Internet performance[END_REF]. The stability condition is given by ρ k < C k for all k = 2, . . . , L and: Again, this condition is more constraining than the usual one, which is here ρ 1 < C 1 and ρ 1 + ρ k < C k for all k = 2, . . . , L. As for upstream trees, the reduction of the stability region is actually marginal for low access rates. This is illustrated by Figure 6 for L = 3 links with

ρ 1 < (min l C l ) × L k=2 1 - ρ k C k . 0 0.
C 1 = ∞, C 2 = C 3 = 1 and ρ 2 = ρ 3 .

C. Fair dropping

We have seen that the absence of congestion control may reduce the stability region for some network topologies under tail dropping. A key result of the paper is that the fair drop policy stabilizes the network whenever possible: Theorem 2: Under fair dropping, the network is stable under the usual stability condition [START_REF] Jacobson | Congestion avoidance and control[END_REF].

Proof. Let n min (t) be the Markov process associated with the min allocation described in §III-C. For any state n, denote by ϕ(n) and ϕ min (n) the vectors of per-flow rates under the fair dropping allocation and the min allocation, respectively. Denote by ≤ the componentwise order on R K . We have for any two states n, n such that n ≤ n :

ϕ(n) ≥ ϕ min (n) ≥ ϕ min (n ),
where the first and second inequalities follow from Lemma 1 and expression [START_REF] Handley | TCP Friendly Rate Control (TFRC): Protocol Specification[END_REF], respectively. By coupling, this implies that n(t) ≤ n min (t) a.s. for all time t ≥ 0, provided n(0) ≤ n min (0). The proof then follows from the stability of the min allocation under the usual stability condition [START_REF] Jacobson | Congestion avoidance and control[END_REF], cf. [START_REF] Fayolle | Best-effort networks: Modeling and performance analysis via large networks asymptotics[END_REF].

V. RESOURCE UTILIZATION

We now apply the results of the previous section to quantify the loss of efficiency due to the absence of congestion control.

A. The Price of Anarchy

We refer to this loss of efficiency as the price of anarchy, a term that has been coined by Papadimitriou [START_REF] Papadimitriou | Algorithms, games, and the Internet[END_REF] to characterize the inefficiency of Nash equilibria in routing games. Here the game consists for each user in adapting their transmission rate; since each user has interest in increasing their transmission rate, the unique Nash equilibrium of our game is obtained when all users transmit at their respective maximum rates. The cost of this greedy strategy is a potential reduction of the stability region: flows are unable to effectively use the whole network capacity.

In the following, we define the price of anarchy P as the maximum reduction of traffic intensity imposed by the absence of congestion control. Specifically, we define:

P = max u∈R K + : u =1 α(u) -β(u) α(u) , ( 13
)
where • denotes the L 2 -norm on R K and α(u) and β(u) denote the maximum values of α and β such that the vectors of traffic intensities α × u and β × u satisfy the stability condition in the presence and in the absence of congestion control, respectively. Note that 0 ≤ P ≤ 1. The price of anarchy depends on the network topology, the access rates and the considered buffer management policy.

B. Tail dropping

Under tail dropping, it follows from the previous results that the price of anarchy is equal to 0 for downstream trees and homogeneous upstream trees and equal to 1 for cycles. In general, P belongs to (0, 1) and is very sensitive to the access rates. This is because the proportion of lost data strongly depends on the maximum source rates. Figures 4 and6 suggest for instance that, for trees and lines, the price of anarchy vanishes when the access rates tend to 0. Proposition 4 below shows that this is indeed the case for the heterogeneous upstream trees considered in §IV-B. We believe that the result is in fact valid for any acyclic network, as defined in §III-B.

Proposition 4: Consider the heterogeneous upstream tree of §IV-B. The price of anarchy P tends to 0 as the class-1 access rate a 1 tends to 0.

Proof. In view of [START_REF] Kelly | Patterns of congestion collapse[END_REF], it is sufficient to prove the result in the case (K -1)C < 1. Let m be such that:

ψ 1 (n 1 ) = n 1 a 1 if n 1 ≤ m, ψ 1 (n 1 ) = n 1 a 1 n 1 a 1 + (K -1)C otherwise. Define p = m n1=0 π 1 (n 1 )
. It may easily be verified that, as a 1 tends to 0, p tends to 1 if ρ 1 < 1 -(K -1)C and to 0 otherwise. In the former case, we have:

κ ≥ m n1=0 π 1 (n 1 )ψ 2 (n 1 ) = pC → C,
while in the latter:

κ ≥ ∞ n1=m+1 π 1 (n 1 )ψ 2 (n 1 ), ≥ ∞ n1=0 π 1 (n 1 ) 1 -ψ 1 (n 1 ) K -1 -p, = 1 -ρ 1 K -1 -p → 1 -ρ 1 K -1 .
In view of Proposition 2, the network becomes stable as a 1 tends to 0 provided the usual stability condition ( 11) is satisfied.

Figures 7 and8 illustrate the decrease of the price of anarchy as a function of the assumed common access rate for the heterogeneous upstream tree of §IV-B2, with branch capacity C = 1/K, and the homogeneous line of §IV-B4, respectively. We observe that P is typically less than 0.1 for access rates less than 0.01, meaning that the efficiency in bandwidth utilization is higher than 90% in this practically interesting case.

Thus the loss of efficiency due to the absence of congestion control is significant in the presence of cycles only. It is worth observing, however, that the scenario described in §IV-B3 where all links transmit data that are eventually lost, leading to P = 1, is unlikely to occur in practice due to the presence of links at the input of the considered cycle. Consider for Price of anarchy for a heterogeneous upstream tree under tail dropping. instance the triangle of Figure 9, with access links of capacity A ≥ 1/2 (the three other links have unit capacity):

Proposition 5: The price of anarchy associated with the considered triangle is given by:

P = 3 -1 + 4/A 1 + 1 + 4/A .
Proof. It follows from ( 6) that, for a large number of flows of each class, each class is served at rate:

ϕ = 1 + 4/A -1 1 + 4/A + 1 .
We deduce as in Propositions 1 and 3 that the stability condition is that the traffic intensity of each class is less than ϕ. The price of anarchy is then given by P = 1 -2ϕ.

The impact of the access link capacity is illustrated by Figure 10: the price of anarchy decreases significantly with the access link capacity. This is also true for larger cycles, for which we use the following extension of Proposition 5. Consider a network consisting of L = 2N links: N links forming a cycle, each of unit capacity, and N access links, each of capacity A. There K = N (N -1) classes, if we exclude the routes containing the N links of the cycle. We assume that A ≥ 1/d, where d = N -1 is the number of routes per link on the cycle. We have: Proposition 6: The price of anarchy associated with the considered cycle with access links is given by P = 1 -dAδ d , where δ is the unique solution on (0, A] to the equation:

δ + δ 2 + . . . + δ d = 1 A . ( 14
)
Proof. First observe that the worst case is obtained when all routes have length N (the access link and d links on the cycle). As in the proof of Proposition 5, it remains to calculate the rate of each class, ϕ, in the presence of a large number of flows of each class. Denoting by θ(i) the rate of each class at the output of the i-th link on the cycle, we get:

θ(i) = θ(i -1)δ with δ = 1 A + θ(1) + . . . + θ(d -1)
, using the convention θ(0) = A. We deduce that θ(i) = Aδ i , from which [START_REF] Massoulié | Bandwidth sharing: objectives and algorithms[END_REF] follows. The rate of each class is then given by ϕ = θ(d) and the price of anarchy by P = 1dϕ.

C. Fair dropping

In view of Theorem 2, the price of anarchy associated with fair dropping is equal to 0. This simple buffer management policy is sufficient to guarantee the full utilization of network resources in all cases.

VI. PACKET-LEVEL SIMULATIONS

This section is devoted to the validation of the previous theoretical results through packet-level simulations.

A. Simulation setting

All packets have the same unit size. Flows are generated according to a Poisson process and have random sizes in number of packets, geometrically distributed with mean 100. The source associated with a class-k flow of size n sends packets at rate a k along route r k until the destination has received n packets; the source then immediately stops sending packets. There are 10 6 flow arrivals per simulation run; the network is declared stable if the average number of flows in each class is less than 500.

Each link is equipped with a buffer of 100 packets. To speed up the simulation, packets are served in random order instead of the FIFO discipline. Note that the fluid model described in Section III depends on the buffer management policy and not on the scheduling policy, as long as the latter is not biased against any class. Recall that the tail drop policy consists in dropping any incoming packet in case of buffer overflow, while the fair drop policy consists in dropping a packet from that flow having the largest number of packets in the buffer (in the presence of several such flows, one is chosen at random).

B. Results

First, we verified that under fair dropping, the network is stable under the usual stability condition for all topologies considered in Section IV: trees, cycles, lines. This is in line with Theorem 2. We also verified that, under tail dropping, downstream trees are stable under the usual stability condition whereas cycles are always unstable.

For those topologies where the price of anarchy is neither 0 nor 1, we obtain the results of Figures 11 and12. The former shows the price of anarchy with respect to the number of classes for the heterogeneous upstream tree of §IV-B2 with branch capacity C = 1/K; the latter gives the price of anarchy with respect to the number of links for the homogeneous line of §IV-B4. In both cases, flows have a common access rate equal to 0.01, 0.1 or 1. Packet-level simulations are compared with flow-level simulations based on the allocation derived from the fluid model of Section III. We observe a good match of the results, showing that the complex packet-level dynamics are well captured by the simple fluid model we consider.

VII. CONCLUSION

Our results suggest that, somewhat surprisingly, the absence of congestion control does not lead to the well-known "tragedy of the commons". The price of anarchy turns out to be moderate under tail dropping, at least for acyclic networks and low access rates, and null under fair dropping. Of course, these results are based on a number of simplifying assumptions whose impact should be quantified. In particular, we have assumed some perfect erasure coding scheme while all practical schemes involve some overhead. Other concerns about the practical relevance of the considered scenario include the phenomenon of local starvation induced by high-rate flows, that may increase the mean duration of low-rate flows, and the assumed ability of voice and media streaming applications to adapt to significant bandwidth variations. Again, these issues are alleviated by the fair drop policy, which tends to impose fair bandwidth sharing.

It is therefore not absurd to envisage the eventual abandon of congestion control, that may anyway be imposed by users and applications. This "ultimate" best-effort paradigm opens new perspectives regarding MAC layers, that may take advantage of the robustness of applications with respect to packet losses to further increase bandwidth utilization.

APPENDIX PROOF OF THEOREM 1

Denote by A k = n k a k the total access rate of class-k flows. Without any loss of generality, we assume that n k > 0 for all k, so that A k > 0.

y k (1) = A k min C l R l , 1 if l = r k (1),
and for all i = 2, 3, . . . , d k ,

y k (i) = x k (i -1) min C l R l , 1 if l = r k (i),
where R l = k:r k (1)=l

A k + k,i:r k (i+1)=l
x k (i).

We must prove that f has a unique fixed point, which then corresponds to the total output rate of each class. Note that there exists some interval To prove uniqueness, we need the following intermediate result. We denote by || • || the L ∞ -norm on R N . Lemma 2: Let x be any fixed point of f . For any r > 0, denote by B(r) the ball of center x and radius r. For any sufficiently small r > 0, there exists ε > 0 such that the function f ε = (1ε)f has a fixed point in B(r).

Proof. Let R l be the input rate of link l in state x :

R l = k:r k (1)=l A k + k,i:r k (i+1)=l x k (i).
Without any loss of generality, we can assume that all links are saturated in state x in the sense that R l > C l (otherwise, it is sufficient to restrict the analysis to non-saturated links). Using the fact that x ∈ [a, b] N , we deduce that f is locally contracting in x : for all classes j, k and all indices i = 1, 2, . . . , d k -1 such that l = r j (1) = r k (i + 1), we have:

∂y j (1) ∂x k (i) = A j C l R l 2 ≤ A j R l < A j A j + a .
Moreover, for all classes j, k and all indices h = 2, 3, . . . , d j , i = 1, 2, . . . , d k -1 such that l = r j (h) = r k (i + 1),

∂y j (h) ∂x k (i) = C l R l -x k (i) R l 2 ≤ R l -x k (i) R l < N (A + b) a + N (A + b) if j = k and i = h -1, and
∂y j (h) ∂x k (i) = x h-1 C l R l 2 ≤ x h-1 R l < b a + b
otherwise. Thus there exists some α < 1 such that f (B(r)) ⊂ B(αr) for sufficiently small r > 0. For all x ∈ B(r), we have:

||f ε (x) -x || = ||(1 -ε)f (x) -x || ≤ (1 -ε)||f (x) -x || + ε||x || ≤ (1 -ε)αr + ε||x ||,
which is less than r for sufficiently small ε > 0. We deduce that f ε (B(r)) ⊂ B(r), so that f ε has some fixed point in B(r).

To prove uniqueness, assume that f has two different fixed points, x (1) and x (2) . In view of Lemma 2, there exist two nonoverlapping balls B (1) and B (2) centered on x (1) and x (2) , respectively, and some ε > 0 such that the function f ε has at least one fixed point in each of these two balls. This is in contradiction with the fact that f ε is a contracting function, which has a unique fixed point in R N + .

1 2 3 Fig. 1 .

 31 Fig. 1. A triangle network.

Fig. 2 .

 2 Fig. 2. A 3-link 4-class network.

Fig. 3 .

 3 Fig.3. A downstream tree (left), an upstream tree (right).

4 .

 4 Stability region of a heterogeneous upstream tree under tail dropping.

Fig. 5 .

 5 Fig. 5. A 3-link line.

Fig. 6 .

 6 Fig. 6. Stability region of a homogeneous line under tail dropping.

  Fig. 7.Price of anarchy for a heterogeneous upstream tree under tail dropping.

Fig. 8 .

 8 Fig. 8. Price of anarchy for a homogeneous line under tail dropping.

2 3 1 Fig. 9 .

 219 Fig. 9. A triangle with access links.

Fig. 10 .

 10 Fig. 10. Price of anarchy for a homogeneous cycle with access links under tail dropping.

Fig. 11 .Fig. 12 .

 1112 Fig. 11. Comparison of packet-and flow-level simulations for a tree.

  Let A = max k A k , d = max k d k and N = K k=1 d k , where d k is the route length of class k. Let f : R N + → R N + be the function defined by y = f (x), with:

  [a, b] ⊂ (0, +∞) such that f ([a, b] N ) ⊂ [a, b] N and f d (R N + ) ⊂ [a, b] N . Since f is a continuous function on R N+ , it follows from Brouwer's theorem that f has at least one fixed point. In addition, any fixed point belongs to [a, b] N .


Jérôme Jourde
email: jjourde@univ-lr.fr

Leandro Sampaio

Ruth Barnich

Paulo Bonifácio

Céline Labrune

Victor Quintino

Pierre-Guy Sauriau

Malmgrenia louiseae sp. nov., a new scale worm species (Polychaeta: Polynoidae) from southern Europe with a key to European Malmgrenia species

Keywords: Polychaeta, Polynoidae, Taxonomy, Malmgrenia, Mediterranean, North-east Atlantic, Associations

come   L'archive ouverte pluridisciplinaire

INTRODUCTION

Five programs monitoring benthic communities, independently carried out in the Gulf of Lions, Mediterranean, in Portuguese waters and in the Pertuis Charentais part of the Bay of Biscay led to the collection of several specimens of an unknown polynoid species. The review of these specimens during the 4 th RESOMAR Benthos Taxonomic Workshop held in June 2013 in La Rochelle revealed that they belong to a hitherto unknown Malmgrenia species.

So far 10 species of Malmgrenia McIntosh, 1874 have been reported from the Mediterranean and the North-east Atlantic to which the new species M. louiseae sp. nov., described herein, has to be added (Table 1). They were either attributed to the genus Malmgrenia McIntosh, 1874 or Malmgreniella Hartman, 1967 by their respective original author and there has been some controversy in the literature regarding the correct generic name to be used [START_REF] Barnich | The Mediterranean species of Malmgreniella Hartman, 1967 (Polychaeta: Polynoidae: Polynoinae), including the description of a new species[END_REF][START_REF] Muir | Case 3417 Malmgrenia McIntosh, 1874 (Annelida, Polychaeta, Polynoidae): proposed conservation of usage by designation of Malmgrenia andreapolis McIntosh, 1874 as the type species[END_REF]. Following ICZN Opinion 2233(2009), which ruled that the usage of the generic name Malmgrenia McIntosh, 1874 is to be conserved, at least the North-east Atlantic and Mediterranean species should now be attributed to this genus.

Most of these species are known to live in association with echinoderms and other invertebrates such as tubicolous or terricolous species [START_REF] Barel | A survey of the echinoderm associates of the North-east Atlantic area[END_REF][START_REF] Pettibone | Scaled Polychaetes (Polynoidae) associated with Ophiuroids and other invertebrates and review of species referred to Malmgrenia McIntosh and replaced by Malmgreniella Hartman, with description of new taxa[END_REF]. The potential associates are reported and a key to all Malmgrenia species found in the area is given.

MATERIAL AND METHODS

The specimens were collected from subtidal grab samples (Van Veen or Smith-McIntyre) in the following surveys or inventories of benthic macrofauna communities: CARTHAM in the Mediterranean (ASCONIT, 2012), Guia marine outfall monitoring program, ACOSHELF and MeshAtlantic in Portuguese waters, OBIONE in Bay of Biscay (Table 2; Figure 1).

The samples were washed with sea water onboard on a 1 mm mesh size and fixed in a 4% formalin-sea water solution. They were sorted in the laboratory and the specimens preserved in a 70% ethanol solution. All observations and measurements were carried out on fixed specimens. The animals were very fragile and most of the elytra were lost and bodies fragmented during the washing steps. Also, body fragmentation and elytra losses occurred when live specimens were added to freshwater or alcohol.

The preserved specimens were studied and photographed using a stereomicroscope Leica M205C coupled to a digital camera Leica IC80HD and the Leica Application Software.

Details of elytra and parapodia needed the use of a compound microscope (Leica DMIRB, coupled with a digital camera Olympus DP70 and the DP Controller software). The photographs of the holotype were used as basis for drawings of the animal with the free vector graphics editor Inkscape. Length (L) was measured from the anterior margin of the prostomium to the posterior border of the last segment (pharynx not included, if everted) and width (W) was taken at the widest segment, including parapodia but excluding chaetae.

The type material is deposited in the collections of the Muséum National d'Histoire Naturelle de Paris, France (MNHN), the Museu Nacional de História Natural e da Ciência de Lisboa, Portugal (MNHNC-UL) and the Senckenberg Museum Frankfurt, Germany (SMF). , 4, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 26, 29, 32. Prostomium bilobed, without distinct cephalic peaks, with three antennae; lateral antennae with ceratophores inserted terminoventrally; two pairs of eyes, anterior pair usually dorsolaterally in front of widest part of prostomium, posterior pair dorsally near hind margin of prostomium. Parapodia biramous, noto-and neuropodia with elongate acicular lobe; tips of noto-and neuroacicula penetrating epidermis; neuropodia with or without supra-acicular or sub-acicular process. Notochaetae with rows of spines and blunt or pointed tips; neurochaetae more numerous, with rows of spines only distally and one or two kinds of tips: bidentate with secondary tooth subdistally and/or unidentate with pointed or knob-like tip.

SYSTEMATICS

Malmgrenia louiseae

DIAGNOSIS

Elytral surface smooth, microtubercules totally absent, outer lateral elytral margin with few small scattered papillae, posterior margin with fewer short papillae; neuropodia with an infraacicular process in addition to the supra-acicular process; two types of notochaetae: upper ones stout with blunt tips and lower ones slender with very pointed tips; neurochaetae all unidentate, upper tapering to long, pointed tips.

DESCRIPTION (based on holotype)

Prostomium bilobed, without cephalic peaks; median antenna with ceratophore in anterior notch, style papillate, tapering to filiform tip; lateral antennae with ceratophores inserted terminoventrally and with papillate, tapering styles; palps smooth, long, tapering; anterior pair of eyes dorsolaterally in front of widest part of prostomium, posterior pair dorsally near hind margin (Figure 2A). Tentaculophores inserted laterally to prostomium, without chaetae, with a pair of papillate dorsal and ventral tentacular cirri, tapering to filiform tip. Second segment with first pair of elytra, biramous parapodia, ventral buccal cirri obviously longer than the following ventral cirri, papillate. 15 pairs of elytra for 33 chaetigers; elytra delicate, surface smooth; outer lateral and posterior elytral margin with few short papillae; surface near the outer lateral margin with very few scattered surface papillae of variable length (some as long as the largest marginal papillae); faint pigmentation in form of isolated spot near place of attachment of elytrophore and on the inner lateral part (Figure 2B-C). Styles of dorsal cirri papillate, tapering to filiform tip, extending beyond tips of neurochaetae; styles of ventral cirri with few papillae, tapering, shorter than neuropodia (Figure 2D). Parapodia biramous, both rami with single aciculum penetrating epidermis; notopodia with short, inconspicuous rounded preacicular lobe and longer, pointed acicular lobe; neuropodia with subconical prechaetal acicular lobe with longer, digitiform supra-acicular process and shorter, but conspicuous sub-acicular process, postchaetal lobe rounded (Figure 2D). Notochaetae with distinct rows of spines and of two kinds: upper ones stout with blunt tip and lower ones slender, tapering to very fine tip, (Figure 2

HABITAT

The species is currently known from muddy substrates, between 34 to 110 m depth (Table 2).

Several potential hosts were found in the stations where the new species was collected. Thus, most of the Portuguese specimens were caught with the echiurid Thalassema thalassemum (Pallas, 1766) and one with the synaptid holothurian Leptosynapta inhaerens (O. F. Müller, 1776). However, specimens of Malmgrenia louiseae sp. nov. were never observed in immediate contact with the echiurid or the holothurian. In the Bay of Biscay, all the specimens were collected with Leptosynapta cf. bergensis (Östergren, 1905)

ETYMOLOGY

The species is named in honor of Louise Jourde, first author's daughter, born a few months before the beginning of this work.

REMARKS

Malmgrenia louiseae sp. nov. is unique due to its neuropodial sub-acicular process present in addition to the supra-acicular process which is known from several other Malmgrenia and many other polynoid species. It might be confused with Malmgrenia lilianae [START_REF] Pettibone | Scaled Polychaetes (Polynoidae) associated with Ophiuroids and other invertebrates and review of species referred to Malmgrenia McIntosh and replaced by Malmgreniella Hartman, with description of new taxa[END_REF], a species originally described from the South-west Atlantic [START_REF] Pettibone | Scaled Polychaetes (Polynoidae) associated with Ophiuroids and other invertebrates and review of species referred to Malmgrenia McIntosh and replaced by Malmgreniella Hartman, with description of new taxa[END_REF], then reported for the Mediterranean [START_REF] Barnich | The Mediterranean species of Malmgreniella Hartman, 1967 (Polychaeta: Polynoidae: Polynoinae), including the description of a new species[END_REF][START_REF] Barnich | The Aphroditoidea (Annelida: Polychaeta) of the Mediterranean Sea[END_REF] and now also recorded from North-east Atlantic off Portugal (unpublished) and in Bay of Biscay (unpublished). In both species elytra are devoid of microtubercles, with marginal papillae, and neurochaetae are exclusively unidentate. However, in M. lilianae there is only one kind of notochaetae (stout with pointed tip) and the sub-acicular process is absent. The identification key given below highlights further differences to other species in Europe. Table 1. Malmgrenia species known to occur in the Mediterranean and North-East Atlantic (cf. [START_REF] Barnich | The Mediterranean species of Malmgreniella Hartman, 1967 (Polychaeta: Polynoidae: Polynoinae), including the description of a new species[END_REF][START_REF] Barnich | The Aphroditoidea (Annelida: Polychaeta) of the Mediterranean Sea[END_REF]Barnich 2011;[START_REF] Pettibone | Scaled Polychaetes (Polynoidae) associated with Ophiuroids and other invertebrates and review of species referred to Malmgrenia McIntosh and replaced by Malmgreniella Hartman, with description of new taxa[END_REF].

  sp. nov. (Figure 2) TYPE MATERIAL Holotype: 1 complete specimen (cs) (MNHN POLY TYPE 1559), L 13.7 mm W 4.3 mm for 33 segments (fragmented); Gulf of Lions, Côte Catalane, CARTHAM B50, 24 August 2010, 3°09'57''E 42°35'19,6''N; 56 m, coastal mud, leg. C. Labrune and J-M. Amouroux. Paratypes: 1 cs (MNHN POLY TYPE 1560), L 10.5 mm W 3.7 mm for 32 segments (fragmented); Bay of Biscay, Pertuis Charentais, OBIONE, Antioche, 22 August 2011, 1°18'30''W 46°05'03"N, 35 m, coastal muddy sand, leg. J. Jourde and P-G. Sauriau. 1 cs (SMF 23918), L 12.5 mm W 3.8 mm for 34 segments (fragmented); Bay of Biscay, Pertuis Charentais, OBIONE, Antioche, 28 March 2012, 1°18'30''W 46°05'03"N, 35 m, coastal muddy sand, leg. J. Jourde and P-G. Sauriau. 1 cs (MB29-000340), L 22 mm W 5 mm for 36 segments (fragmented); off Portugal, Cascais-Guia G29(2), October 2008, 9°24'58.50"W 38° 39' 37.86''N, 34 m, mud, leg. L. Sampaio and V. Quintino. 2 anterior fragments (MB29-000340), L 5.5 mm W 4.5 for 12 segments and L 3 mm W 2.5 mm for 11 segments; off Portugal, Cascais-Guia G29(2), October 2008, 9°24'58.50" W 38° 39' 37.86''N, 34 m, mud, leg. L. Sampaio and V. Quintino.

  E1-E2); neurochaetae with rows of spines only in distal part; upper tapering to long, pointed, unidentate tip, lower ones with short bent enlarged distal part ending in blunt tip, middle ones of intermediate shape with blunt distal part (Figure 2 E3-E5).

  and one specimen was observed in contact with the holothurian. The Mediterranean specimens were collected with the synaptid Oestergrenia digitata (Montagu, 1815). DISTRIBUTION Currently known from type locality in the Western Mediterranean (Gulf of Lions) and Northeast Atlantic: Portuguese coasts (Cascais-Guia, Costa da Caparica and Figueira da Foz) and Bay of Biscay (Pertuis Charentais).

  Figure captions

Figure 1 .

 1 Figure 1. Sampling locations of Malmgrenia louiseae sp. nov. between 2002 and 2012.

Figure 2 .

 2 Figure 2. Malmgrenia louiseae sp. nov., holotype (MNHN POLY TYPE 1559): (A) anterior end, dorsal view; (B) left middle elytron (9 th ); (C) detail of lateral margin of same; (D) right cirrigerous parapodium of chaetiger 12, posterior view; (E) chaetae, E1: upper notochaetae, E2: lower notochaetae, E3: upper neurochaetae, E4: middle neurochaetae, E5: lower neurochaetae. Scale bars: A, 1 mm; B, D, 500 µm; E1, E2, C, 100 µm; E3, E4, E5, 50 µm.

  Body flattened dorsoventrally, short, up to 46 segments, more or less covered by elytra or short tail uncovered (large specimens). Elytra 15 pairs on segments 2

	Family POLYNOIDAE Kinberg, 1856
	Genus Malmgrenia McIntosh, 1874
	TYPE SPECIES Malmgrenia andreapolis McIntosh, 1874
	DIAGNOSIS (emended to include new species described herein)

Table 2 .

 2 Localities, geographical coordinates (WGS 84), sediment, depth, number of specimens collected and sampling dates of Malmgrenia louiseae sp. nov.

Table 1 .

 1

ACKNOWLEDGMENTS JJ was financially supported by the Région Poitou-Charentes through CPER funding, the Université de La Rochelle and CNRS. LS and VQ were supported by the research projects "ACOSHELF" (POCI/MAR/56441/2004-PPCDT/MAR/56441/2004), "Monitorização Ambiental do Emissário Submarino e da ETAR da Guia do Sistema de Saneamento da Costa do Estoril", funded by SANEST, S.A., "MeshAtlantic" (with the support of the European Union ERDF-Atlantic Area Program 2009-1/110) and by European Funds through COMPETE and National Funds through the Portuguese Science Foundation (FCT) within project PEst-C/MAR/LA0017/2013. LS benefited from a Post-doc grant (SFRH/BPD/72997/2010), given by FCT (Fundação para a Ciência e Tecnologia). The Mediterranean data used in this paper are part of the PhD Thesis of PB funded by IFREMER (convention 09/3211321) and the Agence de l'Eau Corse-Méditerranée (convention n°2010 0871). They were obtained during the monitoring program CARTHAM funded by the Agence des Aires Marines Protégées and managed as a part by the GIS Posidonie. LS and VQ would like to thank Susan Chambers, National Museums of Scotland (UK) for commenting on the identity of the new species at an earlier stage. JJ and PGS gratefully thank Fabien Aubert, Nicolas Lachaussée, Philippe Pineau and the captain of the coastal N.O. "Estran" Vincent Ottmann for their help during the sampling sessions. JJ also thanks Hélène Agogué and Eric Pante for their valuable comments. CL and PB thank the captains and crew members of N.O. Thetys II and Alicia Romero Ramirez for assistance during samplings and Jean-Michel Amouroux for help during identification.

Authors are grateful to the two anonymous referees for their corrections and valuable comments that improved the manuscript.

Species

Mediterranean North-East Atlantic M. andreapolis McIntosh, 1874 X X M. arenicolae (de Saint Joseph, 1888) X M. castanea McIntosh, 1876 X X M. darbouxi [START_REF] Pettibone | Scaled Polychaetes (Polynoidae) associated with Ophiuroids and other invertebrates and review of species referred to Malmgrenia McIntosh and replaced by Malmgreniella Hartman, with description of new taxa[END_REF] X X M. lilianae [START_REF] Pettibone | Scaled Polychaetes (Polynoidae) associated with Ophiuroids and other invertebrates and review of species referred to Malmgrenia McIntosh and replaced by Malmgreniella Hartman, with description of new taxa[END_REF]