Pascale Bayle-Guillemaud

Emmanuel Hadji

Peter Reiss

Jacques Villain

In a nanoscience lab

Keywords: Nanoscience, nanotechnology, electron microscopy, nanowire, safety Nanoscience, nanotechnologie, microscopie électronique, nanofil, sécurité

The production, observation and manipulation of very small objects is a tour de force, but these objects, which could infiltrate anywhere without being seen, may arouse suspicion. To assess the situation at best, we describe the activity of a nanoscience research institution, some of the methods used there, the spirit of its researchers and their attitude towards risk.

Dans un laboratoire de nanosciences. La fabrication, l'observation et la manipulation d'objets très petits est un tour de force, mais ces objets, susceptibles de s'infiltrer partout sans être perçus, peuvent susciter la méfiance. Pour mieux apprécier la situation, nous décrivons l'activité d'un institut de recherche spécialisé dans les nanosciences, certaines des méthodes qui y sont utilisées, l'esprit de ses chercheurs et leur attitude vis à vis des risques.

Introduction

Since the beginning of this century, research has evolved considerably in the field of condensed matter, and nanoscience emerged as a new field of research. It is now one of the main activities in many institutions. In this article we shall visit one of them, the campus MINATEC in Grenoble. What is being done in such a complex? How does one come to make, observe, manipulate so tiny objects? This is a bit mysterious to many, including many physicists. It therefore seemed useful to ask some experts to explain in simple terms how they are doing, what are their purposes, and what place is given in their professional life to the new risks associated with these new objects.

The researchers who appear in this article belong to three laboratories. In one of these laboratories nano-objects are fabricated by physical methods of crystal growth, such as molecular beam epitaxy. In the second one, different nanoobjects are produced, in larger amounts, by chemical methods, combining reagents in solvents. In the third laboratory, the obtained structures are observed, and sometimes they are manipulated. These examples are not intended to give a comprehensive description of research in nanoscience, but rather to give a vivid and concrete picture of research.

Manufacturing nanoobjets by physical methods: the case of nanowires. Nanowires are one example of the nano-objects made in these laboratories. They are single crystals whose diameter may be a few tens of nanometers while the length can reach several micrometers. Figure 1, obtained by scanning electron microscopy, shows how perfect the faces may be, and how uniform the diameter. The laws of surface tension are responsible for this perfection. But these same laws, which tend to minimize the surface of solids, should oppose the growth of wire-like objects. How can such objects then be obtained? For the sake of simplification, we shall assume that the wire is made by depositing a material (adsorbate) on the surface of another material (substrate). In some cases, the deposit is possible only through a catalyst, typically gold. The first step is then to deposit nanometer sized droplets of gold on the substrate and then introduce the adsorbate in gaseous form. It sticks to the substrate under the gold droplets, thus formimg columns that grow, wearing their golden head (Fig. 1a). The (nanometric) size of the droplets determines the diameter of the columns. The theory of the phenomenon is described by Dubrovskii et al. [START_REF] Dubrovskii | Theoretical analysis of the vaporliquid-solid mechanism of nanowire growth during molecular beam epitaxy[END_REF].

However, nanowires can sometimes grow without catalyst. Examples are zinc oxide (ZnO) nanowires on sapphire (Fig. 1c,d). The phenomenon, which occurs in quite specific conditions (well-defined substrate temperature and molecular beam intensity) was analyzed by Périllat-Merceroz et al. [START_REF] Périllat-Merceroz | MOCVD growth mechanisms of ZnO nanorods[END_REF] but is not fully understood. Note that the ZnO crystal, unlike many conventional semiconductors, is not cubic but hexagonal. The highly anisotropic observed shape is therefore not so surprising if the surface energy and/or the growth rate are lower parallel to the hexagonal axis than in the perpendicular directions.

The realization of these objects is already an achievement, but constitutes only the first step in a program. The next step is to analyze them, i.e. to measure their physical properties (electrical conductivity, optical properties etc..) which are different from those of a bulk sample. This requires observing and manipulating nano-objects.

Manipulation is also necessary to use the nano-objects, e.g. for integration in nano-electronic devices where their purpose is to transport, either electrons or photons. The reduction in volume allows to make more sophisticated instruments in a given volume, and also to reduce heating. But in the research laboratory that we visit, the fabrication and study of the physical properties pass before applications. The knowledge obtained from this study is a prerequisite for the development of technological applications, which, however, may require to adapt the methods of fabrication. Observation and manipulation of nano-objects.

The observation of nanometer-sized objects requires suitable instruments that can see, count and locate these objects. The scanning electron microscope (SEM) as that of Figure 2 (left), is the ideal instrument. Through an electron beam which scans the surface, it produces an image with a resolution of one nanometer. A more detailed information (on the scale of the atom, i.e. 0.1nm) can be obtained from a transmission electron microscope like the one in Figure 2 (right). The electron microscope, with its resolving power, will thus replace the eye of the researcher in nanoscience. It allows to manipulate a nanowire, to drop it, to detach it from its surface, or even move it onto another substrate for a measurement. In the absence of clamps, rudimentary nanomanipulators can be used, for instance nanometric spikes driven by piezoelectric components which can move them in a strictly controlled way.

Resonators

Although the realization of nanowires is already an achievement, more complex micro-objects can be frabricated. As examples which has a definite application, optical micro resonators may be cited. These objects are capable of selectively retaining the photons of a certain wavelength, like macroscopic Fabry-Perot optical cavities do 1 .. The electromagnetic field is then locally enhanced by a resonance effect. The quality factor Q2 of the device is something like the number of times a photon crosses the cavity in both directions. Nanoresonators of Fabry-Perot can achieve a quality factor of the order of 60,000 [START_REF] Velha | Ultra-High Q/V Fabry-Perot microcavity on SOI substrate[END_REF]. To do better, the Fabry-Perot geometry must be replaced by another one, which corresponds to so-called "whispering galleries"3 .. They can be for instance disks carried by nanopillars. To improve the performance of these resonators, annealing can transform the disks into tores of a remarkable perfection (Fig. 3). Quality factors are then obtained in the range of 10 7 -10 9 --the highest values ever achieved in optical microstructures to date.

While these devices are of micrometric rather than nanometric size, Fig. 3 suggests that the surface is, even at the nanometer size, free from defects which would be the source of light scattering and loss of quality. The achievement of good micrometric devices implies a mastery of nanometric details and therefore belongs to nanoscience A "whispering gallery" as seen by scanning electron microscopy. The torus has been "polished" by annealing. The authors are grateful to Jean Baptiste Jager (INAC/SP2M/SiNaPS) for this image.

Chemical fabrication of nanoobjets.

By chemical methods, nano-objects can be obtained in large quantities as colloids in solution. Through a sufficiently rapid injection of the appropriate reactants into a heated solvent, a rapid nucleation results, followed by particle growth. One can thus obtain nanocrystals of a few nanometers in diameter, which consist of a few hundred to several thousand atoms, also called "quantum dots" (objects in which the charge carriers are confined in all three dimensions of space). They will be formed for example by a II-VI (such as CdSe), III-V (such as InP), or I-III-VI semiconductor (such as CuInS 2 ) [START_REF] Li | Highly Luminescent CuInS 2 /ZnS Core/Shell Nanocrystals: Cadmium-Free Quantum Dots for In Vivo Imaging[END_REF]. These nanocrystals are covered by suitable surfactant molecules (amphiphilic molecules with a polar head and apolar tail), which prevent their aggregation while allowing them to grow at high temperature. To further protect the core of CdSe, InP or CuInS 2 nanocrystals, it can capped by an inorganic shell, for example of zinc sulfide ZnS (Figure 4d). The diameter can be adjusted with reasonable precision by adjusting the synthesis conditions (temperature, type and concentration of reactants and surfactants,...) and suspensions can thus be obtained whose colors depend on the size of the particles (Fig. 4 / b). Photoluminescence gives information on the average size (by the line position) and the dispersion of sizes (through the linewidth). 4Such quantum dots with a core/ shell structure have, when compared to organic dyes, excellent photostability, i.e. their emission and absorption properties deteriorate much more slowly in the air when exposed to light. Their fluorescence efficiency, expressed by the quantum yield, can exceed 50%.

The structure of core/ shell nanocrystals can be determined by transmission electron microscopy (Fig. 4c) which also allows analyzing the chemical composition of thin objects by electron energy loss spectroscopy (EELS).

Researchers and applications.

The objects we describe here will probably have applications, but most of them have not yet reached the industrial stage. For instance, the resonators described above are not currently used in the instruments of our everyday life. Using them in stimulated emission, we could certainly make lasers, but lasers that read our CDs are not nanoscale.

For the researchers who contribute to this article, the immediate concern is to design and manufacture new objects and study their physical properties. They think that their work will constitute the germs of new useful applications.

Nanotechnology can also be useful for fuel cells. These devices generate electricity by using combustion. The usual method to generate mechanical or electrical energy using combustion is through a heat engine, whose performance is limited by the Carnot principle. Fuel cells escape the Carnot principle, but they require a very expensive catalyst (platinum), whose life is not very long. By using catalyst particles of very small dimensions (i.e. nanoscale), we can hope to save it. On the other hand, the investigation of platinum nanoparticles by electron microscopy may lead to new ideas how to prolong the catalyst life.

Photovoltaic electricity production is another potential application of nanoparticles. The photovoltaic way is attractive for countries as Europe, which do not have many energy resources, however it is currently more expensive than other methods of energy production. Semiconductor nanocrystals, for example dispersed in suitable conjugated polymers5 such as poly (3-hexylthiophene), are a possible solution to this situation [START_REF] Jiu | Hybrid Solar Cells Based on Blends of CdSe Nanorods and Poly(3-alkylthiophene) Nanofibers[END_REF]: these "hybrid" materials can be manufactured at low cost and deposited over large supports, which may even be flexible. Nanoparticles provide in that case an improvement of the light absorption by solar cells. Electron tomography, a variation of transmission electron microscopy, allows to control the structure of hybrid materials in three dimensions. The homogeneous distribution of nanocrystals in the polymer matrix is very important for the proper functioning of the solar cell.

A last example of application is the generation of white light by combining nanocrystals emitting in the blue, green and red. It might be an appreciable improvement with respect to current LEDs Researchers and security.

The nanoparticles have a particularly strong chemical activity 6 .. This makes their interest for catalysis. This can also make them toxic; for instance the silver nanoparticles, used as bactericides, are perhaps not safe for humans [START_REF] Senjen | Nano and biocidal silver[END_REF].

Another case is that of carbon nanotubes, suspected to present risks similar to those of asbestos fibers. These objects are not present in the laboratory that we visit today, but CdSe nanoparticles are produced there and they are dangerous, since both components, even in its bulk state, are toxic. Notoriously or potentially toxic nanoparticles are specifically treated in the laboratories. There is indeed a "nano-risk" as there is a risk related to chemical products handled in laboratories.

Risk is a familiar companion for experimental researchers. Chemical risks, particularly related to acids and organic solvents, are probably the most frequently present. In a large research institution, professional hazards are identified and evaluated by specialists (including physicians). At the end of the assessment procedure, a protection regulation is defined by the management, namely that of the Commissariat à l"Energie Atomique et aux Energies Alternatives (CEA) in the case of the laboratory visited to-day. Researchers have to comply with this regulation, and at the CEA the control is made by a Safety Engineer (one per department) whose exclusive task is security and by the Committee of health and safety (one per centre). This check is done here in a climate of mutual confidence which is a condition for productive research. The interest of the researcher is clearly that the risk be as low as possible. However, systematic checks are carried out. The measurements that have been made in the laboratories that we visited did not reveal any contamination by nano-particles.

As a matter of fact, the labs we have presented are not among the most exposed ones. Indeed, the objects studied are present in very small quantities and usually solidary with a substrate of millimetric dimension. The nanowires are attached to the substrate on which they have grown, while the nanoparticles prepared by chemical means are not volatile (because of the organic coating) and inhalation is not to be feared. Concerning the nano-risk, appropriate means of protection are implemented, such as masks, hoods, glove boxes, specific bins and appropriate channels to collect waste.

What will happen when the nanoparticles leave the laboratory to be incorportated into a commercial product? Then it will be necessary to reassess the risks that may or may not appear, depending on the use made of these nano-objects.

The relationship between the user and a product is indeed very different from the relationship between the same product and the researcher, ready to hide behind masks and hoods.

Conclusion

Nanoscience is a great conquest of the late twentieth century and early twenty-first. To make, observe and manipulate objects whose dimensions are not much larger than an atom, that is a feat which moves researchers to enthusiasm. The potential applications are very important, but the prime mover of researchers expressed here is the desire to promote knowledge. Nano-risks are taken into account taking care to neglect neither safety nor efficiency.

Appendix. Electron microscopy

Scanning electron microscopy has been present in laboratories since 1965. It uses electrons from 1 to 40 keV. Its resolution can reach the nanometer. It is not so good as that of the scanning tunneling microscope, but its depth of field is incomparably greater (Figures 1 and3).

Transmission electron microscopy was developed in Germany before and after the Second World War. It requires very thin samples, less than a micrometer thick. It is therefore particularly suited to the observation of nanoscale objects. With an appropriately designed transmission microscope, one can make energy-loss spectroscopy and thus perform the chemical analysis of a material. The energy loss indeed measures the energy required to extract an electron from a deep layer, which is a known feature of each atom.

Apart from energy-loss spectroscopy, specific uses of electron microscopy are electron holography and atomic scale imaging.

Figure 1 :

 1 Figure 1: a) Image obtained by scanning electron microscopy (SEM) of a forest of Si nanowires with catalysts at the top; b) atomic resolution image of the edge of a nanowires obtained by transmission electron microscopy (TEM) ; c) and d) SEM image of ZnO nanowires without a catalyst obtained on sapphire. We are grateful to Pascal Gentile (INAC/SP2M/SiNaPS) for providing us with Fig. 1a and to P.H. Jouneau for Fig. 1d. Fig. 1c is taken from ref. 2 with the kind permission of the publisher and the authors.

Figure 2 :

 2 Figure 2 : A scanning electron microscope (a) and a transmission electron microscope (b) at MINATEC/ Reproduced with the kind permission of Artechnique, CEA (a) and P. Stroppa, CEA (b)

Figure 3 :

 3 Figure 3 : A "whispering gallery" as seen by scanning electron microscopy. The torus has been "polished" by annealing. The authors are grateful to Jean Baptiste Jager (INAC/SP2M/SiNaPS) for this image.

Figure 4 :

 4 Figure 4 : CuInS2/ZnS nanocrystals of approximately 2 nm (yellow) to 4 nm (red) in diameter. a) Ambient Light b) under UV light, c) High resolution transmission electron microscopy image. d) Structure of a core / shell nanocrystal.

The term "Fabry-Perot"corresponds to the geometry (a transparent medium between two mirrors) that is common to the micro-resonator and interferometer built by Fabry and Perot. The transparent medium is different, its thickness is different and the use is different

defined as the ratio Q=, where  is the wavelength and  the resonance width.

According to Wikipedia, "a whispering gallery is a gallery beneath a dome, vault, or enclosed in a circular or elliptical area in which whispers can be heard clearly in other parts of the building". That of Saint Paul Cathedral in London is particularly famous

The emitted color corresponds to a photon whose energy is that of an exciton, i.e. close the gap of the semiconductor. This gap depends on particle size because of quantum confinement. When the size is of the same order of magnitude as the de Broglie wavelength of electrons, the energy gap increases as the nanocrystal radius decreases[START_REF] Brus | Electron-electron and Electron-Hole Interactions in Small Semiconductor Crystallites: The Size Dependence of the Lowest Excited Electronic State[END_REF].

These polymers are, in the neutral (undoped) state, organic semiconductors due to their characteristic of being conjugated, i.e. their main chain consists of alternating single and double bonds

Presumably related to the high proportion of atoms situated on a surface or an edge or a tip.


Jean-Philippe Georges
email: jean-philippe.georges@cran.uhp-nancy.fr

Thierry Divoux

Éric Rondeau

Network Calculus: Application to switched real-time networking

Keywords: C, 2 [Computer Systems Organization]: Computer Communication Networks-Network Architecture and Design, Local and Wide-Area Networks; C, 4 [Computer Systems Organization]: Performance of Systems-Modeling Techniques

In this paper, we show how Network Calculus can be used to determine whether a switched network may satisfy the time constraints of a real-time application. If switched architecture are interesting in the sense that they offer flexible design and may eliminate collisions in Ethernet-based network, they are not guaranteeing end-to-end performances (in particular in terms of delay), especially when cross-traffic are present. We illustrate Network Calculus usefulness by showing how the internal switching structure of an Ethernet switch simplify the analysis and which kind of traffic inter-dependencies are problematic.

INTRODUCTION

Historically, industrial communications were mainly based on specific networks called fieldbuses such as Profibus or CAN. Those networks interconnect programmable controllers, CNC, robots, etc. to exchange technical data for monitoring, controlling, and synchronizing industrial processes. Their protocols ensure that the end-to-end delays of messages remain limited, compared with the time constraints of the applications. Thus, these networks are deterministic and it is possible for such networks to obtain directly time performances.

During the last decade, a trend to replace these dedicated networks by widely-used standardized solutions like Ethernet has been going on. The expected benefits are less costly network installations, because equipment is available off-the-shelf, and the avoidance of interoperability problems, because Ethernet technology is broadly used. Other advan-tages are that Ethernet is a well-known protocol, which is widely implemented, and its performance improves continuously with technological evolution (especially bandwidth). However, access to the Ethernet medium relies on the nondeterministic CSMA/CD algorithm, which applies a stochastic method for resolving collisions. In bus topology, Ethernet cannot guarantee that messages will be received in a bounded time. The idea is to ensure that the data flow is sufficiently quickly handled, compared with the time-cycle of the industrial applications. For [START_REF] Alves | Ethernet goes real-time: a survey on research and technological developments[END_REF][START_REF] Decotignie | Ethernet-based real-time and industrial communications[END_REF] the micro-segmentation with full-duplex switches combined with the appropriate real-time scheduling and fault-tolerance techniques also must enable the use of Ethernet in safety-critical applications with hard real-time constraints. Switched Ethernet networks are employed in a large scale of systems even in avionic aircrafts [START_REF] Boyer | Tightening end to end delay upper bound for afdx network calculus with rate latency fifo servers using network calculus[END_REF][START_REF] Grieu | Analyse et évaluation de techniques de commutation Ethernet pour l'interconnexion des systèmes avioniques[END_REF] and spatial launchers [START_REF] Robert | Ethernet networks for real-time systems: application to launchers[END_REF]. Figure 1 shows a switched topology with full-duplex links and based on the micro-segmentation principle where collisions cannot appear but are shifted to potential congestion into switches. But all these considerations do not prove that all the packets are effectively received under a predefined bound. Figure 2 shows variations of end-to-end delays for a flow competing for access of an Ethernet switch with another flow. It highlights the interest for determining an upper-bound delivery time (defined in the IEC 61784 standard) which corresponds to the time required to send application data (message payload) from one node to another.

According to [START_REF] Jasperneite | Limits of increasing the performance of industrial ethernet protocols[END_REF], Ethernet-based products can be summarily classified into three main categories: the native Ether- net standard (Ethernet/Industrial Protocol, Modbus/TCP), Ethernet solutions using the priorities defined in IEEE 802.1 D/Q, and Ethernet-based solutions that incorporate new scheduling features in ASIC/FPGA (EtherCAT, Profinet IRT). The last approach provides a priori known communication modes such that it simplifies transmission time estimation. For such protocols, when Ethernet is used as a dedicated network, [START_REF] Jasperneite | Limits of increasing the performance of industrial ethernet protocols[END_REF] gives direct analysis of delays.

The time performance evaluation remains an issue when only native switched Ethernet architecture are considered. Rather than using specific electronics, it means that stations, sensors, actuators, etc. are not synchronized and might try accessing the medium in a pure CSMA/CD manner (non TDMA, non polling and non clock synchronization strategy). For example, the Ethernet/IP runs without introducing other concepts than those defined in IEEE 802.1D/Q standards. As a consequence, several flows might simultaneously sending frames on the network and create varying congestion situations as shown in Figure 2. Moreover, this issue will be more sensitive when the switched Ethernet networks is shared with other non time constrained applications. Next sections will detail how the Network Calculus has been applied for such networks and how the switching networking knowledge could be taken into account.

MOTIVATIONS 2.1 Network Calculus framework

The network calculus theory is based on the (min, +) algebra and models flows and services in a network with nondecreasing functions taking their values in the (min, +) semiring. Defined as a min-plus system theory for deterministic queuing systems, the network calculus has been used for DiffServ architectures on Internet, industrial Ethernet networks [START_REF] Georges | Strict priority versus weighted fair queueing in switched ethernet networks for time-critical applications[END_REF], avionics networks, wireless sensor networks [START_REF] Schmitt | A comprehensive worst-case calculus for wireless sensor networks with in-network processing[END_REF] or networks on chip [START_REF] Thiele | Embedded software in network processors -models and algorithms[END_REF]. Main developments and results about network calculus can be found in [START_REF] Chang | Performance Guarantees in Communication Networks[END_REF][START_REF] Cruz | A calculus for network delay, part I: Network elements in isolation[END_REF][START_REF] Boudec | Network calculus, a theory of deterministic queueing systems for the Internet[END_REF]. In addition to the elementary calculus, several filtering operations are also translated in the (min, +) setting. The min-plus convolution respectively deconvolution of two functions f and g are hence defined to be:

(f ⊗ g) (t) = inf 0≤s≤t {f (t -s) + g (s)} (f ⊘ g) (t) = sup u≥0 {f (t + u) -g (u)}
Network modeling in network calculus is achieved by nondecreasing functions, characterising an amount of data at a time t. The set of such functions is given by:

F = f : R + → R + , ∀t ≥ s : f (t) ≥ f (s) , f (0) = 0
The first use of such function is for the input R (t) and the output function R * (t), which cumulatively count the number of bits that are input to respectively output from a system S. This corresponds to the real movements of data. Throughout the paper, we assume these functions to be continuous in time and space. This is not a major restriction as there are transformations from discrete to continuous models [START_REF] Boudec | Network calculus, a theory of deterministic queueing systems for the Internet[END_REF].

The second use is for the constraints that the real movements of data satisfy. Given a flow with input function R, a function α ∈ F is an arrival curve for R if:

∀t, s ≥ 0, s ≤ t : R (t) -R (t -s) ≤ α (s) ⇔ R ≤ R ⊗ α
The third use is for the service offered by a system to a flow defined by an input function R which results in an output function R * . Then, the system is said to provide a minimum service curve β ∈ F if:

R * ≥ R ⊗ β
In addition, β will be called a strict service curve for the system S if during any backlogged period of duration u, at least β (u) data is served.

Let us turn now to the performance characteristics of flows that can be bounded by network calculus. Assume a flow with input function R that traverses a system S resulting in the output function R * . The backlog of the flow at time t is defined as:

b (t) = R (t) -R * (t)
Assuming first-in-first-out delivery, the delay for an input at time t is defined as:

d (t) = inf {τ ≥ 0 : R (t) ≤ R * (t + τ )}
One may now consider the arrival and service curves definitions. It gives the bounds defined below.

Theorem 1 ( [START_REF] Chang | Performance Guarantees in Communication Networks[END_REF][START_REF] Boudec | Network calculus, a theory of deterministic queueing systems for the Internet[END_REF]). Consider a system S that offers a service curve β and that stores input data in a FIFOordered queue. Assume a flow R traversing the system that has an arrival curve α. Then we obtain the following performance bounds for the backlog b, delay d and output arrival curve α * for R * :

b (t) ≤ sup {t ≥ 0 | α (t) -β (t)} = (α ⊘ β) (0) d (t) ≤ inf {d ≥ 0 | ∀t ≥ 0, α (t) ≤ β (t + d)} α * ≤ α ⊘ β
Finally, the consideration of the service offered to a flow along a given path may be studied by considering the two following results.

Lemma 1 (Tandem Systems [START_REF] Chang | Performance Guarantees in Communication Networks[END_REF][START_REF] Boudec | Network calculus, a theory of deterministic queueing systems for the Internet[END_REF]). Consider a flow crossing two nodes in tandem with respective service curves β1 and β2. Then the concatenation of the two nodes offers a minimum service curve β1 ⊗ β2.

Lemma 2 (Residual Service [START_REF] Chang | Performance Guarantees in Communication Networks[END_REF][START_REF] Boudec | Network calculus, a theory of deterministic queueing systems for the Internet[END_REF]). Consider a node offering a strict service curve β and two flows entering that server, with respective arrival curves α1 and α2. Then a service curve for flow 1 is β1 = (β -α2) + .

Next sections show how this theory has been applied until now in the framework of the performance evaluation of Ethernet based real-time communications.

Network Calculus interests

Network calculus is an interesting tool for the design of industrial/embedded networking where applications require hard quality of service guarantees, in particular bounded end-to-end delays. This theory has been hence used for switched Ethernet networks, in particular when the network is shared by several applications. Initial Network Calculus developments [START_REF] Chang | Performance Guarantees in Communication Networks[END_REF][START_REF] Cruz | A calculus for network delay, part I: Network elements in isolation[END_REF][START_REF] Boudec | Network calculus, a theory of deterministic queueing systems for the Internet[END_REF] (which were more considered for resources reservation in Internet) were used in a short time for Ethernet networks. In [START_REF] Jasperneite | Deterministic real-time communication with switched ethernet[END_REF], the Network Calculus theory is employed in order to bound delays by simply assuming that the service provided by switches is following rate-latency functions. It enables to compare efficiency of line vs star topologies. In other studies, for instance [START_REF] Georges | Evaluation of switched ethernet in an industrial context by using the network calculus[END_REF], a functional analysis of switching principles extends the switch modeling in a more detailed manner rather the simple indication of a β (t) = R (t -T ) + switch service curve. Results have been next used in order to optimize the topology by using the end-to-end delays formula as the objective function [START_REF] Georges | A design process of switched ethernet architectures according to real-time application constraints[END_REF]. Switched Ethernet topologies in a real-time context are not only appropriate to the factory area, but covers also embedded systems like the EADS A380 aircraft. In [START_REF] Grieu | Analyse et évaluation de techniques de commutation Ethernet pour l'interconnexion des systèmes avioniques[END_REF] the modeling of AFDX networks (switched Ethernet with admission control techniques) was achieved thankfully the Network Calculus theory. Extensions of these preliminary works were dedicated to tackle strict priority scheduling strategies in order to reduce maximum delays for time constraints applications.

These first results allowed to validate the method to determine time performances bounds in spite of best effort protocols. However, it was only limited to simple networks. In fact, first results mainly address the tandem systems shown in Figure 3(a) thankfully Lemma 1. The pay bursts only once principle enables here to tight the bounds rather than computing each time the output arrival curve (as defined in Theorem 1). For rate-latency services curves, [START_REF] Boudec | Network calculus, a theory of deterministic queueing systems for the Internet[END_REF] shows that the composition of rate-latency curves (the most employed) correspond to another rate-latency one. It gives also delays bound expression for affine arrival curve.

Next step consisted in considering the aggregation issue shown in Figure 3(b) for acyclic networks. If the service curve β is for the whole service offered by a system (i.e. a switch) and not dedicated to a given flow, it means that bounds could be only computed for the aggregated traffic (α1 + α2). Due to inter-flow dependencies, multiplexing of a flow depends on other flow and vice versa. To obtain bounds for a given flow (for instance α1), the scheduling policy of the system should be taken into account in order to derivate an appropriate service curve β ′ from the initial curve β. For this topic, Lemma 2 gives an interesting result known as blind multiplexing or residual service. It is equivalent to consider that in the worst case, a given flow will be served in an equivalent manner as a low priority traffic for a strict priority scheduler. [START_REF] Bouillard | Optimal routing for end-to-end guarantees using network calculus[END_REF][START_REF] Lenzini | A methodology for computing end-to-end delay bounds in fifo-multiplexing tandems[END_REF][START_REF] Schmitt | Delay bounds under arbitrary multiplexing: When network calculus leaves you in the lurch[END_REF] detail next how this lemma can be used in conjunction with the pay bursts only once principle in order to catch the pay multiplexing only once principle. For affine function and rate-latency functions, it is shown that the path service curve offered to the traffic α1 is given by (β1 ⊗ β2 ⊗ β3 -α2) + . A complete study is given in [START_REF] Bouillard | Optimal routing for end-to-end guarantees using network calculus[END_REF] about the conditions required to use this result and also how to compute more tight bounds. Associated network topology are related to the feed-forward property (see Figure 3(c)) defined by [START_REF] Schmitt | Performance bounds in feed-forward networks under blind multiplexing[END_REF] as a network in which nodes can be labeled in such a way that the path of every flow is composed of an increasing sequence of node labels. Different methods have been proposed as the least upper delay bound in [START_REF] Lenzini | A methodology for computing end-to-end delay bounds in fifo-multiplexing tandems[END_REF]. Here the end-to-end service curve in a tandem is computed by iteratively removing interfering flows. Recent developments about this topic can be found in [START_REF] Bouillard | Tight performance bounds in the worst-case analysis of feed-forward networks[END_REF]. It gives an algorithm which computes the maximum end-to-end delay for a given flow for any feed-forward networks under blind multiplexing with concave arrival curves and convex service curves.

These new theoretical developments have been taken into consideration in the context of performance evaluation based on Network Calculus for switched Ethernet architectures. For instance, [START_REF] Boyer | Tightening end to end delay upper bound for afdx network calculus with rate latency fifo servers using network calculus[END_REF] identify in the special case of AFDX networks, advanced knowledges about the aggregation (call grouping) of several flows. It enables to obtain more precise arrival curve functions and hence more tight delays. This paper shows the relation between theoretical results and applied results in local networking where additional a priori knowledge information might be extracted. However communications in a realistic network might be more complex than the acyclic network illustrated in Figures 3(b). Figure 3(d) show more complicated studies. This one is typical in an industrial context where a calculator might send a control update to an actuator/sensor while the sensor is sending the feedback. In such case, previous results can not be applied such that it might be not possible to identify the service guaranteed by a path for a given flow.

As a consequence, it is not possible to use the pay bursts only once principle for instance, and only local upper-bounds delays (i.e. the delay for crossing one switch) will be computed according output arrival functions given in Theorem 1. It means here that studies deal with total flow analysis.

Since computed upper-bounded end-to-end delays are in an industrial context the key point to decide if a switched Ethernet topology might be employed or should be optimized, or even to decide to degrade the global applications performances, the tightness of these bounds is an important topic. Furthermore, it may be noticed here that switched Ethernet networks corresponds to a particular case study: paths are a prori known (defined by spanning tree algorithms), routing issue like the one illustrated in Figure 3(c) are not possible due to the routing along the spanning tree and switches have multiple input/output ports which means that two level of aggregation should be discussed: between flows entering by different ports and between flows entering by the same input port. Moreover, flows entering by the same input port are not necessarily exiting a switch by the same output port.

In following sections, this paper introduces which kind of additional information could be integrated in the application of the Network Calculus theory in switched Ethernet networks. It shows also where blind multiplexing issues are remaining.

SERVICE OFFERED BY SWITCHES 3.1 Arrival curve

Since the objective is to guarantee some deterministic performances of the network, the incoming traffic has to be bounded. For this, the leaky bucket controller concept [START_REF] Boudec | Network calculus, a theory of deterministic queueing systems for the Internet[END_REF] is mainly used to model the arrival constraints representing each traffic [START_REF] Georges | Evaluation of switched ethernet in an industrial context by using the network calculus[END_REF][START_REF] Grieu | Analyse et évaluation de techniques de commutation Ethernet pour l'interconnexion des systèmes avioniques[END_REF][START_REF] Jasperneite | Deterministic real-time communication with switched ethernet[END_REF].

R(t) σ ρ b(t)= σ+ ρt α (t) = σ + ρt ∀x, y; y ≥ x, x ≥ 0, then R ∼ α ⇔ y x R (t) dt ≤ σ + ρ (y -x)
The burstiness constraint [START_REF] Cruz | A calculus for network delay, part I: Network elements in isolation[END_REF] imposes the traffic generation to be bounded by an affine function α (t), in which a variable burst value σ is associated to a constant rate ρ. R (t) represents here the instantaneous rate of the stream, C the capacity of the links on the network, σ is the maximum amount of traffic that can arrive in a burst, i.e. the maximum length of the frames and ρ is an upper bound on the long-term average rate of the traffic flow : the data amount sent each time cycle. To take into account the capacity of the links, the previous affine function is completed with a stability constraint α (x) ≤ Cx. It means that the arrival of data cannot be greater than the capacity C of the link. So, we have:

α (t) = min {Ct, σ + ρt} (1)
In industrial communications in which controls are often time-triggered, it might be noticed here that affine function are interesting for modeling periodic arrivals.

Service curves

In switched networks, a key topic is relative to the identification of a service curve for each switch. The challenge is here to determine the service curve parameters and to not directly assume that a rate-latency curve is enough. Whereas the use of rate-latency curve consists in a black box strategy [START_REF] Jasperneite | Deterministic real-time communication with switched ethernet[END_REF], other models that have been proposed try to isolate the elementary functions of the switching principle as proposed in [START_REF] Boyer | Tightening end to end delay upper bound for afdx network calculus with rate latency fifo servers using network calculus[END_REF][START_REF] Cruz | A calculus for network delay, part II : Network analysis[END_REF][START_REF] Georges | Evaluation of switched ethernet in an industrial context by using the network calculus[END_REF].

Figure 4(a) is the result of previous studies (see [START_REF] Georges | Strict priority versus weighted fair queueing in switched ethernet networks for time-critical applications[END_REF]) in which several models are built and compared according to the IEEE 802.1D standard. It is constituted of a sequence of three basic elements: one multiplexer, one queue and one demultiplexer. In order to take into account the internal speed of a switch, the three following capacities should be used: C, the throughput inside the switch, Cin, the throughput of arrival of data on input ports and Cout, the output capacity. Since the study is about Ethernet networks, the scheduling policy has to be non preemptive. Moreover, it is considered that the components are work-conserving systems, i.e. they cannot have vacations and their forwarding policy is the best-effort. In this paper, the FIFO forwarding policy is chosen in order to respect the fairness of Ethernet: a FIFO memory to model the switching process and FIFO queues to model the transmission function of the output ports.

This model has been improved in [START_REF] Georges | Strict priority versus weighted fair queueing in switched ethernet networks for time-critical applications[END_REF]. The model is extended with the priority management. The implementation of priorities inside a switch mainly consists in adding for each switch output port as many buffers as priorities. These buffers are modeled by FIFO queues (component 6 on the figure 4(b)). The classification operation of frames belonging to a same switch output port is achieved by a demultiplexer (component 5). It enables to select the relevant queue from the frames priority. Finally the goal of the last multiplexers (component 7; one per switch output port) is to apply a scheduling policy between their different queues. The output capacity of the component 7 corresponds to the network bandwidth. Figure 4(b) represents a two-ports switch model which is able to manage the frames forwarding with two levels of priorities. In order to take into account the internal speed of a switch, the three capacities Cin, C and Cout are used.

Based on this model, the minimal service of a switch depends hence on the service curve defined for each elementary components. While elements like demultiplexer correspond to electronic latencies (and hence burst delay curves) and may be neglected, others like multiplexers are related to the scheduling strategies. The service curve offered by a switch might be directly obtained by considering the switch as a tandem systems and hence Lemma 1. For Figure 4(a), it gives then β = β1 ⊗ β2 ⊗ β3 ⊗ β4.

Consider firstly the switch model defined in Figure 4. The first element is a FIFO multiplexer. The service by this multiplexer is related to the internal switch capacity C and to the switching mode latency (store & forward, fragmentfree, etc.), such that a rate-latency service curves might be defined. For instance, for store & forward, it gives β (t) = C (Rt -L/Cin) + . It means here that results like blind multiplexing may be applied in order to determine a separate flow analysis. It is important to note here that two aggregation levels have to be considered: between input ports and between flows sharing the same input port. In switched networks with full-segmentation, it means also that the task scheduling in end devices (local multiplexing) should be also considered. Nevertheless, it is important to note here that the switching fabric capacity is larger than the sum of the input port capacities C ≫ Cin (backplane property). It means that only the switching mode has a real impact and the service consists mainly in a burst delay. The second el-ement, represents electronic latencies, and stands too for a burst delay. Demultiplexers identify the associated output port and/or the associated queueing policy. It corresponds to a burst delay service (but it is generally neglected).

Finally, it is important to note here how the output buffering in switches is managed. Modeling should be able to catch the capacity for a switch to simultaneously forward frames on different output ports. The output buffering strategy (which is selected to avoid the head of line blocking) reflects this point. (Output) multiplexing in switch is mainly related to flows trying to access the same output port. Without classification of service mechanisms (i.e. FIFO multiplexing), blind multiplexing strategies have to deal not with the whole traffic entering in the switch, but only with the amount of traffic exiting the switch by the same port (which limits the aggregation issue). For FIFO multiplexers, the service is hence defined from the output port capacity. With classification of service, it corresponds to a locally FIFO scheme. Globally, the service offered will be relative to a strict priority or a fair queue scheduler and flows belonging to the same class (or priority) and forwarded to the same output port will be served according to a FIFO strategy. The main interest of strict priority or fair queueing policies is to not depend on cross-traffic parameters, but only on static priorities and weights. It enables to compute specific curves for super flows gathering all traffic with the same class at the same output port as shown in the following section. Strategies like blind multiplexing will hence only be used for the FIFO scheme.

OUTPUT SCHEDULING

The evolution of Ethernet to segmented architectures and the definition of the Virtual Local Area Networks (VLAN) have led to the birth of a new standards set (802.1D/p, 802.1Q) in which new encapsulation fields are added to the native frame format. One of these fields defines a priority level (8 levels are supported). These levels are related to 8 types of applications (voice, video, network management, best effort, etc.). The number of classes of service may be different to the number of priority levels, and also different for each port. Two scheduling policies are supported: the strict priority (SP) and the weighted round robin (WRR).

In the following, it is assumed that the size of a frame is upper-bounded by L. The notion of flow corresponds here to the set of traffic sharing the same priority and exiting the switch by the same output port (i.e. the traffic entering by a common input port of an output multiplexer on Figure 4(b). We assume also that the capacity of the inputs and outputs are relatively fixed at Cin and C b/s. Each flow i ∈ N * is (σi, ρi) upper-constrained with n j=1 ρj < C where n ∈ N * is the number of priority/class and a weight φi is given to each flow. Now, we analyze in particular the delays for each flow according to whether the policy of the node is SP or WRR.

Strict priority

In the strict priority policy, none guarantee is offered to one flow. The selection order will simply depend on the priority (weight) order. Also, we have to distinguish the service curve offered to each flow. The strict priority policy guarantees to the packets of the flow with the highest priority (here, the flow 1) to be selected first. But, since the forwarding of a packet on the network cannot be preempted, packets of the flow 1 might have to wait the complete forwarding of a packet with a lower priority. Also, the service curve associated to the flow with the highest priority is given by :

β1 (t) =R (t -T ) + T = L/C, R = C (2)
Obviously, packets of the flow 2 are served before packets of the flow 3, but they will have to wait that none packets of the flow 1 is waiting before to be served. So there is a first latency corresponding to the processing time of the initial bursty period of the flow 1. Moreover, like above, it will not be possible to preempt the forwarding of a packet of the flow 3 in order to serve a packet of the flow 2 that has just arrived. Finally, since the node will serve first packets of the flow 1, the forwarding rate will be limited to C -ρ1. A packet of the flow 3 will be served only if none packet of the flows 1 and 2 are waiting. That is to say that the service has a first latency period defined by the processing time of the initial bursty period of the union of the two others flows. Moreover, the service rate will be limited in this case to C -ρ1 -ρ2. The service curve of a flow i > 1 will be hence defined by for a flow :

βi (t) =R (t -T ) + (3)
T = j=i-1 j=1 σj C -j=i-1 j=1 ρj + L/C, R = C - j=i-1 j=1 ρj
Equations show that with the strict priority, the service offered to one flow depends on the other flows and the service offered to the flow with the lowest priority may tend towards zero.

Fair queueing and weighted round robin

The weight associated to a given port i is defined by φi (with φi ∈ N * ). In other words, it means that no more φi flits might be successively served per round robin cycle for a port i. The maximum number of frames served during a round robin cycle is defined by Φn = n j=1 φj.

The Weighted Fair Queueing is also known as the Packetized Generalized Processor Sharing (PGPS). It is based on the conceptual algorithm called the Generalized Processor Sharing (GPS). A GPS server is characterized by n positive real numbers φ1, φ2 . . . φn. It operates at a fixed rate C and is work conserving. Each flow i has a guaranteed service rate c such as:

c = φi Φn C
The GPS policy is interesting since it uses fairness, it is flexible (the number φi enables to modify the service offered to a given flow, and consequently to other flows) and finally, it is analyzable and bounded. For example, for the case where n = 2, supposing that R2 is (σ2, ρ2) upper-constrained, [START_REF] Chang | Performance Guarantees in Communication Networks[END_REF] shows that the service curve (number of bits served at time t) for the first flow β1 can be defined by :

β1 (t) = max (C -ρ1) t -σ, φ1 φ1 + φ2 Ct (4)
At all, in opposition to the strict priority, the service offered to one flow only depends on the weights of the flows and on its properties: it respects the fairness queueing. The definition of the minimal service curve offered to the flow will use the properties given in ( 4), but has also to consider that the worst case. When a frame just misses its slot, it will have to wait its next slot (at the next round).

The frame will have to wait up to j =i φ j

C

. A rate-latency service curve can be hence formulated as in [START_REF] Chang | Performance Guarantees in Communication Networks[END_REF] and shown in Figure 5.

βi (t) =R (t -T ) + R = C φi Φn , T = j =i φj C (5)
It means that we retrieve here a rate-latency function as for SP. An upper-bound end-to-end delay can be hence computed as given in Theorem 1. We note τi

= σ i C in -ρ i . di (t) ≤ max t≥0 inf ∆≥0 min {Cint, σi + ρit} = R (t + ∆ -T ) + ≤T ∨ max 0<t<τ i inf ∆ ≥ 0 : Cint = R (t + ∆ -T ) + ∨ max t≥τ i inf ∆ ≥ 0 : σi + ρit = R (t + ∆ -T ) +
Since for all t > 0, min (Cint, σi + ρit) > 0, R ≤ Cin and the stability conditions implies

j ρj ≤ R di (t) ≤T ∨ (Cin -R) τi + RT R ∨ (ρi -R) τi + σi + RT R ≤ (T -τi) + σi + ρiτi R (6)
Using the equation [START_REF] Cruz | A calculus for network delay, part I: Network elements in isolation[END_REF], it is now possible to compare delays provided by strict priority and weighted round robin. Considering n = 3. First, consider the flow 1 (with the highest priority). It is interesting to note here that if φ2 → 0 and φ3 → 0, the weighted round robin will be very closed to the strict policy. But if we consider now the flow 3 (with the lowest priority), if ρ1 + ρ2 → C, the forwarding rate offered to the packet with a lower priority will tend to zero in the strict priority scheduling. This problem does not appear in the weighted round robin policy, since the forwarding rate will only depends on the weights values.

Figure 5 hightlights that rate-latency service curves introduce conservativeness in round robin principle since the forwarding rate of a packet does not correspond to the output port capacity. A local improvement of the delays bounds deals with the definition of the second service curve shown in dashed in Figure 5. The main idea is hence to take into account that it is the access to the switching fabric which is shared between the input ports, not directly the forwarding capacity. In fact, this service can be seen as the composition of the service offered by each cycle. Indeed, for a cycle n, the curve corresponds to a constant plus a rate latency nK + βR,nT (t) as shown by Figure 6. We propose then to replace (5) by:

t bits 1st 2nd 3rd β ′ i (t) φiL 2φiL 3φiL (Φn -φi) L/C φiL/C
β ′ i (t) = (1st) ∧ (2nd) ∧ (3rd) ∧ . . . = inf k∈N {kK + β R,kT (t)} = inf k∈N kφiL + C (t -Φn (k + 1)) + (7)
Different properties might be given for equation [START_REF] Cruz | A calculus for network delay, part II : Network analysis[END_REF]. To simplify, flows are constrained by an affine arrival curve αi (t) = σi + ρit. Firstly, assume here that the service curve β ′ i is a strict service curve offered by a server to the aggregate of two flows incoming by the same input port. Then if we have:

β1 (t) := inf k∈N K + R (t -T ) + -α2 (t) +
Since the distributivity of + with respect to ∧, it gives:

β1 (t) = inf k∈N K + R (t -T ) + -α2 (t) + = inf k∈N K + R (t -T ) + -α2 (t) +
Then it is well known that the residual service between a rate-latency service curve and an affine arrival curve leads to a new rate latency service curve [START_REF] Boudec | Network calculus, a theory of deterministic queueing systems for the Internet[END_REF][START_REF] Schmitt | Performance bounds in feed-forward networks under blind multiplexing[END_REF]. The result is similar when a rate latency plus a constant service curve (K + R (t -T ) + ) is considered, such that :

β1 (t) = inf k∈N K ′ + R ′ t -T ′ + with R ′ = R -ρ2 (it is assumed that R > ρ2), T ′ = σ 2 +T ρ 2 R-ρ 2
and K ′ = (K -σ2) + . It can be finally noticed that the curve β1 is wide-sense increasing.

Definition 1. Consider a switch serving two flows, 1 and 2 entering into the switch by the same input port i with some unknown arbitration between the two flows. Assume that the switch guarantees a strict service curve βi to the aggregate of the two flows. Assume that flow 2 is α2-smooth and that R > ρ2. The service curve offered to the flow 1 is defined by:

β1 (t) = inf k∈N K ′ + R ′ t -T ′ + with R ′ = R -ρ2, T ′ = σ 2 +T ρ 2
R-ρ 2 , K ′ = (K -σ2) + and K, R and T defined as in equation [START_REF] Cruz | A calculus for network delay, part II : Network analysis[END_REF].

Secondly, according to the Lemma 1, we have:

β1 ⊗ β2 (t) = inf 0≤s≤t {β1 (t -s) + β2 (s)} = inf k 1 ,k 2 ∈N K1 + R1 (t -T1) + ⊗ K2 + R2 (t -T2) +
By considering K1 and K2 as two constant regarding t, it gives:

= inf k 1 ,k 2 ∈N K1 + K2 + R1 (t -T1) + ⊗ R2 (t -T2) +
and by integrating the rate-latency convolution results, it finally gives:

= inf k 1 ,k 2 ∈N K1 + K2 + min (R1, R2) (t -T1 -T2) +
Definition 2. Assume a flow traverses switches 1 and 2 in sequence. Assume that each switch offers a service curve of βi, i = 1, 2 to the flow. Then the concatenation of the two switches offers a service curve:

β1 ⊗ β2 (t) = inf k 1 ,k 2 ∈N {K + R (t -T )} + with K = K1 + K2, R = min (R1, R2) and T = T1 + T2.

Conclusion

In this section, service curves have been presented for the output multiplexing into the switch model in Figure 4(b). When no classification of service mechanism is used, the service offered by the output buffer is directly linked to a FIFO multiplexer with a rate related to the output port capacity. To obtain service curve for a particular flow, blind multiplexing should be considered.

END-TO-END UPPER-BOUNDS COMPU-TATION

The objective of this section is to emphasize how Network Calculus results are then applied to compute end-to-end delays in switched Ethernet architectures. Figure 7 introduced the inter-depencies of the flow on such topology for models given in Figure 4.

In switched network, an important topic is not loose the ability for a switch to forward flows with input/output ports, simultaneously in parallel. This impossibility might occurs if the switch is simply modeled by a global service curve β on which blind multiplexing is applied. The kind of fullduplex communications shown in Figure 3(d) is typical for such situation. Considering Figure 7, it may be noticed that even if flows entering in a switch are multiplexed in a common shared memory, it is done with a rate larger than the total input capacities. Hence, the additional delay added by this FIFO multiplexing remains limited compared to the output forwarding.

In such network, aggregation issues are limited between flows exiting a switch by the same output port. In addition, this issue is restricted when classification of service mechanisms are used for flows exiting a switch by the same output port and with the same class/priority level. Here current developments in the Network Calculus theory framework present a real interest.

The service offered by a switch to a given flow will hence be formulated for each output port. Considering Figure 7, if we neglect latencies and constant delays, it means that each switch may be represented by two distinct service curves related to the output forwarding. Hence, no aggregation issues are required as shown in Figure 8.

CONCLUSION

This paper highlights how the Network Calculus has been applied for switched real-time communications when applications require time performance guarantees even if the network does not provide explicitly such guarantees. Computed end-to-end delays bounds are interesting to decide if the network will fulfilled the applications needs, to optimize the network topology (on which port on which switch an end devices should be interconnected) such that delays are minimized, to optimize classification of service parameters. This paper shows also how the last developments of this theory are improving the tightness of the end-to-end delay bounds.

In the same time, it highlights that some issue are not applying to this kind of networks.

Figure 1 :

 1 Figure 1: A hierarchical switched topology.

Figure 2 :

 2 Figure 2: Measurement of end-to-end delays.

Figure 3 :

 3 Figure 3: Flows mapping on switched topology.

  Prioritised switching model.

Figure 4 :

 4 Figure 4: The prioritisation mechanism modifies the output buffering at output ports.

Figure 5 :

 5 Figure 5: Service curve weighted round robin.

Figure 6 :

 6 Figure 6: Composition of service for each roundrobin cycle.

Figure 7 :

 7 Figure 7: Synoptic to a full duplex communication between two node interconnected by a three switches linear topology.

3 Figure 8 :

 38 Figure 8: Flows mapping equivalent to synoptic of Figure 7.

This paper talks about switched network, mainly switched Ethernet networks. One may considers other switched topologies like networks on chip. It is important to note here that aggregation issue will be different when frames queueing will be achieved at the input (and not at output as in this work).