Jacques Lacaze

Steve Dawson

Alain Hazotte

Lacaze Jacques

Dawson Steve

Hazotte Alain

Cast Iron: A Historical and Green Material Worthy of Continuous Research

Keywords: Cast irons, History, Market share, Graphite free Graphite bearing

Throughout history, cast iron is unique amongst metallic materials. No other metal can boast such a long history, together with such a wide diversity of variants, properties and applications. Arguably, no other material can claim to have such complexity. While the cast iron foundry routinely churns out myriad components, through weekends and on night shifts, a supporting cast of researchers and engineers have humbly ensured a choreography of carbon atoms that could make the busiest beehive look lethargic. We control the carbon atoms not with furnaces and wirefeeders, but with knowledge. This knowledge enables a material with a unique combination of design flexibility, mechanical properties, wear resistance, recyclability, low life cycle energy consumption, and low cost. And it will be with the continued pursuit of understanding and knowledge that tomorrow's researchers and engineers will ensure the continued growth of new material variants, with improved material properties and new applications that make the world a better place. Cast iron: thousands of years of development and progress behind us; thousands of fascinating mysteries and opportunities ahead of us.

Introduction

Cast irons are easy to shape materials whose properties have evolved over the years following technical and scientific improvements. As of 2018, cast irons represent 70% of the 110 Million tons of metals being cast each year in the world (10% for cast steels, 20% for aluminium and other alloys) [START_REF]Census of word casting production[END_REF]. Cast irons are recyclable materials at low cost and with relatively limited pollution when compared to their present-day competitors. This is schematically illustrated in Fig. 1, where so-called grey cast irons are compared with cast steels and aluminium alloys in terms of price per MPa of Yield strength, vs. embodied energy (Fig. 1-a) and CO2 footprint (Fig. 1-b). The latter two terms mean energy used and CO2 emitted, respectively, for primary production, casting and recycling of 1 kg of alloy.

Even far before dollars were established as a universal reference, and far before aluminium could be thought of, cast irons were already quite attractive for several applications in agriculture, domestic applications and decoration. Cast irons are in fact historical materials that have appeared at the Iron Age when the temperature in the furnace for processing iron ore became high enough. It is thus of first interest to have a rapid look at the evolution of cast iron materials over the time until the modern era, this is rapidly done in the historical section to follow. As with all other materials, several important steps have been taken over the last two centuries in the processing of cast iron, in casting technology and in the cast iron itself; these are mentioned in the following sections.

Nowadays, cast irons are a family of materials as depicted in Fig. 2. Two main branches can be defined depending on the carbon-rich phase, which can be either cementite and other carbides or graphite. Alloys within this former branch, also called white cast irons for the colour of the rupture surface, have good wear properties and good heat and corrosion resistance when alloyed, but they are brittle. This branch is however a minor part of the family and most of the current production consists of grey (or graphitic) cast irons in which the carbon-rich phase is graphite giving a dark contrast to rupture surfaces. The vast majority of these irons are based on Fe-C-Si alloys and thus could be called silicon cast irons. These irons are the focus of this work. Ni-resist graphitic cast irons are heat and corrosion resistant, while very high-Si alloys are corrosion resistant. Behind the sorting in Fig. 2 is a continuous evolution of cast iron alloys and their processing, as described in the section "Main Steps".

Figure 1 Grey cast irons compared with cast steels and aluminium alloys in terms of price and environment impacts; x-axes concern the ratio between price and Yield strength, while y-axes concern estimations of energy used (a) and equivalent CO2 emissions (b) for the primary production, casting and recycling of one kg of alloy. Data from [2].

Figure 2 The cast iron family with the basic microstructures indicated. They are all obtained in the as-cast state, except those appearing in italics that are subjected to specific heat-treatment (after Elliott [START_REF] Elliott | Cast iron technology[END_REF] and Stefanescu [START_REF] Stefanescu | A history of cast iron[END_REF]).

The properties of silicon cast irons are determined by their microstructure after the two following phase transformations: 1) solidification during which the graphite shape is determined; and 2) solid-state transformation of the Fe-rich matrix in ferrite or pearlite. With low levels of Ni and Mo, silicon cast iron can be austenitized (austempered) and then transformed at 300-450°C to give a matrix of ausferrite. This material is called austempered ductile iron (ADI) and has very high mechanical properties. The present market share and the associated concerns are presented in a section dedicated to production and concerns. These concerns are associated to on-going laboratory researches and are challenging, in particular because of the increasing demands for weight reduction and increased loading together with an ever-increasing number of recycling loops compounding the effect of trace elements on the parameters controlling the microstructure. Some of the up-to-date challenges are discussed in a last section.

A very brief history

Legend reports that, "approximately 4,000 years BC, scientists and sages commissioned the first iron cupolas. These early metallurgists placed a curious metal-like substance, which we now know as iron ore, in the hollow of a burnt out tree trunk. The charred inner surface of the trunk provided fuel for the fires and thermal insulation, while random holes that burned through the trunk allowed the ambient winds to enter and feed the charcoal. With the right combination of wind, rich ores and extra charcoal, the charge was reduced to a metallic iron sponge that could be rolled, using rocks, into decorative beads. It was 6000 years ago, and iron was more highly valued than gold." [START_REF] Dawson | A brief history of … foundry[END_REF]. This is more or less what historical works describe which locate the very start of iron history in Anatolia (Armenia) or Mesopotamia (Iran) [START_REF] Stefanescu | A history of cast iron[END_REF][START_REF] Coze | Purification of iron and steels, a continuous effort from 2000 BC to AD[END_REF], or else China [START_REF] Derui | An illustrated history of ancient Chinese casting[END_REF]. It is worth noting that independent development in sub-Sahara Africa at the same time was postulated and is now well documented [START_REF] Coze | Récits sidérurgiques d'hier et d'aujourd'hui, fers, fontes, aciers: 4000 ans d'affinage et de purification[END_REF][START_REF] Robion-Brunner | L'Afrique des métaux[END_REF]. A short but comprehensive presentation of the beginning of iron ore processing, and of the parallel development of iron and cast irons, has been published by Le Coze [START_REF] Coze | Purification of iron and steels, a continuous effort from 2000 BC to AD[END_REF]. Production techniques and raw materials were gradually refined as the Iron Age evolved. Iron implements and weapons began to appear in approximately 3000 BC, and the comparatively advanced technique of hardening iron weapons by heat treatment was known to the Greeks by about 1000 BC.

As the temperature of the furnaces was progressively increased and reached 970°C (Fe-C-P eutectic), 1154°C (Fe-C eutectic) and above, some liquid may have formed which was certainly mixed with scoriae. Increasing further the temperature slightly above 1200°C allowed reaching the FeO-fayalite or fayalite-SiO2 eutectic, meaning the dross and scoriae became partly liquid and could be separated more easily from the iron loupe. In most production locations, this carbon enriched liquid iron -the ancestor of cast ironswas eliminated except in China where the first crafts, tools and weapons made of cast irons date back to 600-500 BC [START_REF] Stefanescu | A history of cast iron[END_REF][START_REF] Derui | An illustrated history of ancient Chinese casting[END_REF]. This means that it was early understood that the furnaces could be conducted either to give "wrought" iron for further refining by hammering to give steel, or to give carbon-rich iron liquid that was suitable for casting. In this latter case, adding minerals (clay, shells, ...) changed the scoriae into a slag protecting the metallic liquid and thus keeping its casting ability while more of the charge was melting. It is probable that some silicon dissolved in the liquid, in an amount that increased with the temperature, and that it was early understood that solidification with fast cooling rates resulted in brittle components while slower cooling yielded enough ductility to keep the cast parts in one piece when handling them. This historic development of cast irons in China contrasts with the fact that cast irons became recognized in Europe only during the 14 th century AD.

Cast iron was such an "easy" material to process that it soon allowed gigantism. This applies to the use of cast iron that was selected to erect impressive sculptures such as the Cangzhou lion in Fig. 3 or the famous pagoda in the Yuquan temple. With some delay, the same happened in Europe with the 35 km of cast iron pipes and other components that were installed during the second part of the 17 th century to bring water to the dozens of jets in the garden of the Versailles castle [10] so as to satisfy the egocentrism and megalomania of one single human being.

The development of cast iron followed the progress of iron processing which is nowadays known in detail with the evolution in Europe of the Catalan forges to larger and larger blast furnaces. The state of the art at the beginning of the 19 th century has been described by Karsten [START_REF] Karsten | Manuel de la métallurgie du fer[END_REF], including a detailed description of the work by de Réaumur at the beginning of the 18 th century on the solid-state heat treatment of cast irons to give malleable cast irons and cemented iron. The two routes for making steel, i.e. the direct reduction through a pasty state that gave "wrought" iron and remelting and oxidizing of cast iron, were both used until the development of the Bessemer's converter. Although the metallurgy was not understood, and Bessemer wrongly concluded that air was fuel as many before him [START_REF] Coze | Récits sidérurgiques d'hier et d'aujourd'hui, fers, fontes, aciers: 4000 ans d'affinage et de purification[END_REF], the breakthrough was realised and steelmaking could leap from kilograms per day to tonnes per hour. This is only with the publication of the first Fe-C phase diagram at the very end of the 19 th century that it was clearly understood that "wrought" iron, steel and cast irons belong to the same family.

Iron ore processing used huge amounts of charcoal so that optimizing it became a necessity certainly very early, and this implied increasing the maximum furnace temperature. To achieve this, blast furnaces have become increasingly taller with better and better control of the air blown onto the charge. In the early 18 th , charcoal started to be replaced by coke for its better mechanical properties with the drawback of adding sulphur to the melt. For later processing, the liquid cast iron could be tapped into a trough made in the sand floor of the foundry and flowed into impressions made by the foundryman's foot on either side of the trough. Because the orientation of the foot-sized ingots along the trough resembled piglets feeding at a sow, the small ingots became known as pig iron.

Even though cast iron metallurgy was not yet mastered, casting technology was already in an advanced stage. This is illustrated with Berlin iron (fer de Berlin) jewellery that came in fashion during the early 1800's when Prussia needed to finance their war against Napoleon. The Prussians were asked to turn in their gold jewellery to raise funds for the war and were given iron jewellery in return. Berlin iron was sand-cast and then lacquered black. Many such pieces can be found in museums in Continental Europe, but also in Brazil where the fashion was exported at the same time the country built its first blast furnaces; see Fig. 4. At the same time, cupolas and other kinds of furnaces for melting and processing irons and steels developed. In present days, blast furnaces define facilities providing pig iron for immediate use, particularly in integrated steel mills. Cupolas, in contrast, were designed for manufacturing stocks of cast iron to be processed later. They evolved along the time and became the most usual means to melt the charge in foundries. Today, high volume foundries use a combination of cupola and electric induction furnaces to melt the iron. The melting costs of both techniques are similar, but the cupola has a productivity advantage. Large cupola furnaces can melt more than 100 tonnes per hour from a single furnace while the largest induction furnaces can only produce approximately 20 tonnes per hour. Despite the productivity advantages, the cupola will eventually be relegated to the history books. In Europe, the cupola will likely be phased out by 2050 as part of the Net Zero European Green Deal. Good for the environment, but potentially devastating for the European foundries who must continue to face international competition.

The main steps during the modern era (Anthropocene)

Being optimistic, scientific and technical understandings are intermixed with industrial progresses, though these latter may often be leading for economic concerns. As for iron and steel, this was put in an historical perspective by Le Coze who stressed the technical evolution between 400 BC and 1400 AD in contrast to the explosive and parallel scientific and technical changes since then [START_REF] Coze | Purification of iron and steels, a continuous effort from 2000 BC to AD[END_REF]. The advent of chemical analysis, microscopic observations and other metallographic tools such as X-rays supported the improvements of materials properties.

Until the 19 th century, the composition of the cast parts was given by that of the iron ore [START_REF] Wittmoser | Ein halbes Jahrhundert Giessereitechnik in Deutschland[END_REF] and also, though not understood at that time, by the minerals added to generate the slag. The importance of the slag composition for limiting iron loss (as FeO) and to remove sulphur from the coke and phosphorus from the ore has developed progressively, mostly in the 20 th century. Slag is most generally described by its basicity which is the ratio of basic oxides (CaO, MnO, FeO, Na2O) over acid oxides (SiO2, P2O5). By controlling the composition of the slag and the temperature at the metal/slag interface, it is possible nowadays to get out of the blast furnace a cast iron with the desired silicon content. Beyond the cast iron industry, modern steelmakers have taken slag science to new heights, leading to the expression: "take care of the slag and the iron will take care of itself".

Until the mid of the 20 th century, the main concern was low ductility and associated moderate tensile strength because of the lamellar shape of graphite, see the bottom-left micrograph in Fig. 5. In the 1940s, following earlier attempts in Germany, Morrough and Williams in England and Millis and co-workers in America sought a way to ball graphite. The British (at BCIRA) found that additions of less than 0.1% cerium were sufficient to cause the graphite to grow in the form of individual spheroids instead of flakes. The Americans (at Inco) found that additions of approximately 0.05% magnesium would do the same thing. The Inco magnesium process eventually won the battle on technical grounds and the annual worldwide production of spheroidal graphite (aka nodular graphite or ductile) iron (SGI) has since grown to more than 25 million tonnes.

Figure 5 Evolution of the elongation at rupture of cast iron used for water pipes (adapted from Lesoult et al. [START_REF] Lesoult | Les fontes à Pont-à-Mousson[END_REF]) and of their microstructure. Graphite appears in dark contrast in the micrographs. The samples were etched so as to reveal the constitution of the matrix, which is pearlite in the historical material (with some coarse eutectic cementite) and ferrite in the two centrifugally cast pipes. Note that centrifugally cast LGI and SGI are both heat treated for full graphitization just after casting.

The modification of the graphite shape from flakes to nodules had a profound affect on the properties and the market potential of cast iron. In comparison to grey iron, the tensile strength tripled, the stiffness increased by 50% and ductility or elongation changed from nearly 0% to more than 5%, and up to more than 25% with a ferritic matrix. This quantum step in cast iron technology is illustrated in Fig. 5 for the case of cast iron pipes manufactured by the Pont-à-Mousson company that was created in the mid of the 19 th century. This figure presents the parallel evolution of the elongation at rupture and of the microstructure of the cast irons. The microstructure at the turn of the 20 th century was certainly much alike that of Versailles' pipes that is illustrated here. Brittleness of cast iron Year parts was a concern, not only because of the lamellar shape of graphite, but also because the parts contained large precipitates of eutectic cementite and some phosphides. The treatment of the melt or the selection of iron ores allowed halving the phosphorus content and thus suppressed phosphides. At about the same time was introduced the centrifugal process in which the melt is poured in a water-cooled and rotating metallic die. This led to a significant refining of the microstructure of lamellar graphite irons (LGI), in particular by shortening the graphite lamellae. The dramatic change in the 1950 is due to the switch to SGI instead of LGI.

Figure 6 Evolution of graphite shape and UTS (pearlitic matrix) with Mg content, illustrated for a base iron sulphur content of 0.013% [START_REF]Microstructure and Porosity Control[END_REF].

Millis et al. also received a patent in 1949 for a grey iron having improved properties because graphite assumed a more compact shape than usual flakes. The compact graphite shape was achieved by adding a controlled amount of magnesium, and such irons became later known as compacted graphite irons (CGI). Interestingly, the patent was applied for in 1948 on the same day as Millis' more famous ductile iron patent, and the two patents were granted on the same day in 1949, with US patent number US2485760 for SGI and US2485761 for CGI. CGI has better mechanical properties than LGI and is much less prone to microporosity than SGI. Critically, for modern engine applications, CGI also has higher thermal conductivity than ductile iron. However, it was only in the 1960s that it was found interesting to purposely generate this intermediate graphite form, see the review by Nechtelberger et al. [START_REF] Nechtelberger | Cast iron with vermicular graphite -State of the art. Development, production, properties, applications[END_REF]. CGI has thus been in use for a long time but this was only for niche applications until the development of a proper thermal analysis control of melt preparation just before casting. The first series production of CGI engine for passenger vehicles was launched by Audi in 1999 and since the beginning of the 21 st century CGI is breaking the market of automotive engines and components, though not yet appearing as such in production trends that differentiate data only considering LGI, SGI and malleable irons [START_REF] Stefanescu | The meritocratic ascendance of cast iron: from magic to virtual cast iron[END_REF].

SGI CGI LGI

Nowadays, graphite is industrially spheroidized by simultaneous addition of magnesium and rare earths (RE). In contrast, there are several ways to generate compacted graphite as reviewed by Nechtelberger et al. [START_REF] Nechtelberger | Cast iron with vermicular graphite -State of the art. Development, production, properties, applications[END_REF]. However, the most usual is by limited additions of Mg and RE when compared to SGI. Figure 6 shows the change in graphite nodularity as function of Mg content [START_REF]Microstructure and Porosity Control[END_REF], with indication of the domains and typical examples of graphite shape for LGI, CGI and SGI. Note that the nodularity was set negative for lamellar graphite to emphasize that the industrial nodularity scale is defined for compacted and spheroidal graphite and not for lamellar graphite for which letters (A-E) are used to differentiate the various shapes [START_REF] Dawson | Cast iron alloy and method making the same[END_REF]. The evolution of ultimate tensile stress (UTS) with nodularity for 85-100% pearlitic cast irons is also illustrated. The composition of most of the silicon cast irons is selected on the basis of the graphite shape as illustrated with Fig. 7 adapted from Elliott [START_REF] Elliott | Cast iron technology[END_REF]. This graph shows the projection of the iron-rich corner of the Fe-C-Si diagram with the solubility limit of carbon in austenite (at eutectic temperature) that defines the lower limit of the cast irons domain. The upper limit is at carbon contents slightly larger than the eutectic line. White cast irons are low-silicon hypo-eutectic alloys and are more and more replaced by high-Cr cast irons for wear properties. Their use is mostly limited to manufacture malleable irons by heattreatment. LGI is mostly hypoeutectic, with the austenite dendrite network guaranteeing a minimum rupture elongation. If LGI could be made with higher carbon content, the industry would do it to maximise thermal conductivity. But there is a need to restrict the carbon content to ensure (i) that the graphite doesn't cause brittleness, and (ii) that there is enough austenite to carry the load and compensate for the ill-effects of the graphite. CGI and SGI are near-eutectic alloys for minimizing microporosity.

The trend for higher silicon content for SGI and CGI started in the early 1980s with the development of the so-called SiMo cast iron for exhaust manifolds. Higher Si content raises the eutectoid temperature and thus increases the maximum service temperature while Mo is added as it gives rise to a network of eutectic carbides that improve creep properties. However, as exhaust temperatures continue to rise to satisfy emissions requirements, SiMo exhaust manifolds and turbocharger housings are being widely replaced by stainless steels that do not incur a phase transformation. These higher temperature capabilities outweigh the fact that oxide scales are generally better adhering on cast irons than on steels.

Since the 2000s, high-silicon SGI has developed as the yield stress and ultimate tensile stress are linearly increasing with Si concentration because of partial ordering of the ferritic matrix. Mechanical properties similar to pearlitic irons may thus be obtained with the advantage of decreasing tool wear during machining because the matrix is more homogeneous. More generally, there is a wealth of research for improving mechanical properties of the matrix of cast irons. Amongst the many possibilities that have been considered, austempered cast irons are slowly showing great industrial potentialities.

An overview of present cast iron production and concerns

Cast irons are found in pipes, machinery, railway, automotive and energy industries, as well as in cooking and ornaments in public areas, etc. Figure 8 differentiates the market share for SGI (about 25 million tonnes per year) and LGI (about 50 million tonnes per year) based on the 2018 USA data [START_REF]Metalcasting Forecast and Trends[END_REF] that are certainly representative of the world market.

Figure 8

Market distribution in USA in 2018 for SGI (to the left) and LGI (to the right) [START_REF]Metalcasting Forecast and Trends[END_REF]. CGI data is included in the LGI data, as the primary application for CGI is cylinder blocks and heads.

After the short panic the COVID-19 pandemic created, things appear to run again following the same steps as before. In particular, there is no more any question on the fact that the demand for goods transportation must be met, which led to the development of gigantic cargos that require corresponding engines (Fig. 9). These engines are run with high-sulphur oils that pollute air to an incredible level, but significant development is ongoing in this field as well, with low-sulphur fuel mandated since 2020, longer-stroke engines that provide efficiency improvements by reducing rpm, and the likely introduction of ammonia as an alternate fuel in the near-term. Stronger cast irons are required to ensure that this demand can be satisfied with more efficient engines and with decreased pollution.

In the introduction, we abruptly stated that cast irons are 'green' materials with respect to others cast alloys (cf. Fig. 1). This is certainly not all the truth. Similar to other engineering materials, the production of cast irons has steadily increased since the 19 th century, with the associated increase of environment impacts (pollution, energy and primary resource consumption, green-house gas -mostly CO2-emissions, …) to reach an overall level that has become unacceptable. However, sustained efforts have long been made to decrease pollution from fumes during melt preparation (both blast furnaces and other foundry furnaces). Because the cupola will be banned in Europe in 2050, the full More recently, fine and complete life cycle analyses (LCA) have been carried out, which trace the environment impacts at different steps of the whole extraction and production process (see for instance [START_REF] Mitterpach | Environmental evaluation of grey cast iron via life cycle assessment[END_REF][START_REF] Jolly | Primary Manufacturing, Engine Production and on-the-road CO2: How can the Automotive Industry Best Contribute to Environmental Sustainability?[END_REF][START_REF] Jhaveri | Life cycle assessment of thin-wall ductile cast iron for automotive lightweighting applications[END_REF][START_REF] Salonitis | Life-Cycle and Energy Assessment of Automotive Component Manufacturing: The Dilemma Between Aluminum and Cast Iron[END_REF][START_REF] Stefana | A Review of Energy and Environmental Management Practices in Cast Iron Foundries to Increase Sustainability[END_REF] and an example in Fig. 10). These tedious works are necessary guides for still reducing energy consumption and CO2 footprint of casting industry. Recyclability is also a positive point for cast irons. The percentage of recycled and downcycled material in the total worldwide supply of grey cast irons has reached about 70%, compared to less than 50% for cast steels or Al-alloys [2]. Finally, a particular feature of aluminium and cast iron castings is the use of huge amounts of sand, with green sands that may be recycled while sands that have been chemically bound cannot. It is noted that the gravity sand casting of passenger vehicle aluminium cylinder blocks uses approximately four times more chemically bonded sand than an equivalent cast iron cylinder block.

Stronger materials, like CGI, also reduce the weight of conventional LGI components, contributing to the environment by reducing the melting demand. If CGI reduces the weight of a grey iron cylinder block by 15%, the saving corresponds to 7,500 tonnes per year of castings and 10,000 tonnes per year of liquid metal for every one million 50 kg cylinder blocks. In turn, this saving corresponds to a reduction of 100 MJ of energy for electric melting and 500 tonnes of CO2 per year.

In transportation industry, the weakest point of cast irons is clearly their high density, which evidently suggests to substitute them by lighter materials, as Al-alloys or even Mg alloys. However, this solution needs to be validated for each component, since the gain in energy consumed (or CO2 generated) during service can not always counterbalance the excess of energy used for the production of lighter alloy (cf. Fig. 1 and Fig. 10-a). As an example, Fig. 10-b (from [START_REF] Salonitis | Life-Cycle and Energy Assessment of Automotive Component Manufacturing: The Dilemma Between Aluminum and Cast Iron[END_REF]) shows that the break-even distance for paying back the substitution of cast iron by Al-alloy in a given engine block is of the similar or even longer than the service life of the vehicle.

While considerable attention is given to the electrification of passenger vehicles, the market penetration to date -and the likely development over the next decade -is not as rapid as most media reports suggest. The pie charts to the right of Fig. 11 show passenger car sales in 2020 for Europe, the US and China according to fuel, namely petrol, diesel, hybrid, electrified (plug-in hybrids plus battery electric vehicles) and alternate fuel (propane, natural gas, etc). Globally, passenger vehicle electrification in 2020 was less than 10% and, as shown in Fig. 11, global sales are expected to be approximately 30% in 2030. Many of these vehicles will be hybrids, also containing an internal combustion engine. On the commercial vehicle side, there is no foreseeable alternative to the diesel engine for long-haul heavy duty goods transport and there is a widespread consensus in the industry that the diesel engine will dominate until at least 2040. Even if the transformation to electrification becomes complete, the internal combustion engine will remain on global road for decades after production stops. Clearly, there is a need for continued research in cast iron to make the internal combustion as environmentally effective as possible.

Future outlooks: a material worthy of further research

At the end of the 20 th century, progress in metallurgy has benefited from the development of computers and small-scale analytical methods. Computer capabilities have been first used for developing simulation tools for describing the successive processes of melt preparation, filling and casting, and several commercial software packages are nowadays available. Another important application is in the control of melt preparation just before pouring that makes use of thermal analysis with dedicated rapid curve analysis so that a green or red light is provided prior to casting, enabling the foundry to adjust the iron and to ensure that only good iron is poured. However, the most exciting advances for metallurgists are the capabilities of modern apparatus to investigate matter at small-scale, going down to atom scale as with the so-called Atomic Resolution Microscopes. This certainly opens ways to improve the understanding of microstructure formation in cast irons.

In general, it is desirable for lowering the costs to use cast iron parts in their as-cast state, which means avoiding heat-treatments. This implies that the graphite shape and the matrix constitution that were sought for are obtained, which may be challenging in parts having complex shapes. As a matter of fact, the shape of graphite is highly sensitive to small changes in both melt preparation and cooling rate. Furthermore, changes brought about by these factors are closely inter-related; see Fig. 12. After standard preparation, a cast iron melt yields flake graphite, while a full or partial spheroidizing treatment after desulphurisation is used to get spheroidal or compacted graphite, respectively. However, in a casting with various section thicknesses, the graphite shape depends on the local cooling rate. The following changes can be noticed as the cooling rate increases:  Lamellar graphite to undercooled graphite.  Compacted graphite with low nodularity to nearly or fully spheroidal graphite.  Irregular spheroidal graphite to well-rounded one.

 Figure 12 The interplay between chemical and cooling rate routes (after [START_REF] Lacaze | From atom scale to casting: a contemporary monograph on cast irons microstructure[END_REF]).

Inoculation helps controlling graphite shape, e.g. avoiding undercooled graphite in LGI that would conduct to a ferritic matrix while fully pearlitic LGI are preferred for mechanical properties. Furthermore, obtaining a fully graphitic SGI upon casting definitely relies on appropriate inoculation of the melt for avoiding the formation of metastable eutectic. Understanding the bases of the formation of graphite nuclei [START_REF] Skaland | A model for the graphite formation[END_REF][START_REF] Loper | Inoculation of cast iron-summary of current understanding[END_REF] has allowed defining new inoculant compositions in particular to fight against fading and thus against porosity in SGI. This is an area of fierce competition on an industrial point of view, and of sharp stimulation for scientists. The main concern is understanding early precipitation and growth of graphite, in particular in relation with the presence of additives in the inoculants. Recent results on hypereutectic alloys strongly suggest that both primary and eutectic growth of graphite needs a significant undercooling to proceed, and that this undercooling depends on melt chemistry [START_REF] Castro-Roman | Revisiting thermal analysis of hypereutectic spheroidal graphite cast irons[END_REF]. This should lead to revisit thermal analysis records as well as modelling approach of cast iron solidification.

The additives mentioned above are elements prone to react with oxygen, sulphur and nitrogen to precipitate compounds that are active in the process on graphite nuclei formation. However, at the same time, many of these elements also affect graphite shape and are then seen as impurities leading to graphite "degeneracy" -i.e. the development of unwanted graphite forms -when present in the melt at a level above a critical limit that may be so low that these so-called poisoning elements can often be classified as trace elements. These degeneracies are illustrated in Fig. 12 with mesh and spiky graphite in the case of SGI, but the phenomenon is also known for LGI and CGI. While the role of these impurities have been described since a long time as can be seen in several reviews [START_REF] Sawyer | Effects and neutralization of trace elements in gray, ductile and malleable iron[END_REF][START_REF] Javaid | Quality control of heavy-section ductile cast irons[END_REF], understanding how they actually proceed to induce graphite degeneracy is still in its infancy [START_REF] Lacaze | Surface tension, interfacial segregation and graphite shape in cast irons[END_REF]. In practice, rare earths are used to fight the deleterious effects of trace elements, but again very little has yet been made to describe the kinetics of the very many precipitation and dissolution processes behind rare earths additions [32,[START_REF] Lekakh | Effect of Non-metallic Inclusions on Solidification of Inoculated Spheroidal Graphite Iron[END_REF].

The precise mechanism of the action of the spheroidizers is still unknown, or more precisely, still controversial. The first effect of magnesium and rare earths has long been associated to their high affinity with oxygen and sulphur, i.e. considering that change in graphite shape is controlled by tightening oxygen and sulphur as oxy-sulphides. Subramanian et al. [START_REF] Subramanian | Compacted graphite morphology control[END_REF] therefore suggested associating the different forms of graphite with the thermodynamic activity of residual oxygen and sulphur. The presence of some residual magnesium and RE dissolved in the liquid of a melt dedicated to SGI and CGI appears however essential for achieving the desired graphite shape with the associated challenge of understanding the exact way these elements interact with graphite.

In the same way as for graphite shape, the constitution of the matrix in the as-cast state depends both on the chemical composition and on the cooling rate, with a strong effect of graphite fineness. Increased inoculation in SGI and decrease of graphite size in LGI eases the transformation of the matrix in the stable system, i.e., increases the amount of ferrite. On the other hand, increased cooling rate favours the transformation in the metastable system with precipitation of pearlite. Adding Sn and Sb is used industrially to avoid the formation of ferrite, with the risk of graphite degeneracy if added at too high levels. It is thus usual to add also Mn and Cu which are less potent for pearlite formation but are also easier to tolerate when considering the avoidance of deleterious graphite shapes. An important feature of the eutectoid transformation is the effect of microsegregations that are issued from the solidification step. While they can be easily described [START_REF] Eiken | 3D phase-field computations of microsegregation in nodular cast iron compared to experimental data and CalPhad-based Scheil-prediction[END_REF], their mastering for controlling the eutectoid transformation has not yet given the expected results. Microsegregation is also a concern for achieving controlled austempering of ADI parts as well as for mastering so-called dual-phase microstructures. Here, it is worth noting that recent works have demonstrated that 1-3 wt.% Al added to a silicon cast iron inverses silicon segregation [START_REF] Muhmond | Graphite growth morphologies in high Al cast iron[END_REF][START_REF] Franzen | Influence of aluminium on silicon microsegregation in solution strengthened ductile iron[END_REF]. This finding may soon be exploited to decrease microsegregation and thus improve matrix properties of cast irons.

Improvements of melt and casting processes as well as mastering of microstructure formation are triggered by the need for better service properties within a framework of competition between materials. For dynamic properties, and in particular in the case of SGI, it is not only the matrix constitution that is important but also the casting defects such as porosity. Some progress has been made in describing porosity formation [START_REF] Kweon | Computational modelling of shrinkage porosity formation in spheroidal graphite iron: a proof of concept and experimental validation[END_REF] though still not accounting for the mechanical effect due to graphite bulging that has long been recognized but has rarely been accounted for [LES09]. Also, the mechanical behaviour of cast iron is intrinsically elasto-visco-plastic with a strong effect of damage which has led to a wealth of complex modelling approaches that are hardly introduced in foundry software packages. Ad hoc models fitted to experimental overall mechanical behaviours are often preferred.

Beyond the metallurgy itself, significant improvements continue to be introduced to the mechanical engineering side of the foundry. Over the last 25 years, complex components like cylinder blocks and heads have progressively evolved from green sand casting to fully-enclosed core packages. In parallel, minimum wall thickness for passenger vehicle cylinder blocks has decreased from 4.5 mm (+/-1.0 mm) to the current state-ofthe-art of 2.7 mm (+/-0.5 mm). And the development has not stopped. Until recently, 3D sand printing was preserved for rapid prototyping, but many series production foundries are now embracing additive manufacturing for the production of highly complex cores. Without draft angles, these complex cores can add to weight reduction and eliminate costly -and energy consuming -post processing of the castings.

Conclusion

We continue to judge the iron foundry by its grey walls, despite the significant progress in cleaning up the dust. The real image should be the atoms, not the walls. Cast iron is the first composite material, and it remains one of the most versatile composites available today from a technical point of view, and one of the most fascinating from a scientific point of view. In the future, iron foundries will produce castings with different graphite shapes in different areas of the component to optimise specific properties where they are needed. While the iron foundry world may struggle for image, our present-day ability to control alloy additions to within 10 grams per tonne will soon seem rudimentary. The real iron-age is just beginning and the next iron will build its own legend.

  and high-Si cast irons)

Figure 4

 4 Figure 4 Brazilian cast iron necklace following the fashion known as Berlin iron, early 19 th century [12]. With permission from Irina Aragão and Anais do Museu paulista (Brazil).

Figure 7

 7 Figure 7 Carbon and silicon composition ranges of silicon cast irons (adapted from Elliott [3]).

  only blast furnaces and cupolas to only electric -will be closed in about two centuries.

Figure 9 .

 9 Figure 9. A large ship engine (Wärtsilä X62 engine), the steps on the left side can be used for an estimate of the scale. (https://www.wartsila.com/media/news/08-01-2015-wartsila-x62-enginenow-fully-approved-and-available-to-the-market

Figure 10 .

 10 Figure 10. a) Energy for manufacturing a 1.6 L diesel engine block made of cast iron (38 kg) or Al alloy (27 kg); b) Total (manufacturing + use) energy as a function of driving distance for the same engine block in an automotive vehicle of 1200 kg with an average consumption of 7 L/100 km; vertical dotted lines mark break-even distances between cast iron and Al-alloy produced by different casting processes; the average life of such a vehicle is approximately 220 000 km.(adapted from[START_REF] Salonitis | Life-Cycle and Energy Assessment of Automotive Component Manufacturing: The Dilemma Between Aluminum and Cast Iron[END_REF]).

Figure 11 .

 11 Figure 11. Penetration of electrically charged vehicles is poised to remain below 50% for at least the next decade (adapted from Automotive News, 17 May 2021). The pie charts show passenger car sales in 2020 for Europe, the US and China according to fuel.

Acknowledgments

F. Landgraf, Uni. Sao Paulo, and W. Menk kindly provided historical references.


Keywords: TCP, Comopund TCP, Bernoulli losses, Markovian Models

We analyze the performance of a single, long-lived, Compound TCP (CTCP) connection in the presence of random packet losses. CTCP is a new version of TCP implemented in Microsoft Windows to improve the performance on networks with large bandwidth delay-products. We derive a Markovian model for the CTCP sending window and compute the steady state distribution of the window and the average throughput of a CTCP connection. We observe that the previous approximation, using a "typical cycle," underestimates the average window and its variance while the Markovian model gives more accurate results. We use our model to compare CTCP and TCP Reno. We notice that CTCP gives always a throughput equal or greater than Reno, while relative performance in terms of jitter depends on the specific network scenario: CTCP generates more jitter for moderate-high drop rate values, while the opposite is true for low drop rate values.

Introduction

With the increasing popularity of faster access links like Fiber To The Home [START_REF]Fiber to the home council[END_REF], the current standard TCP is not always ideal. As indicated by Floyd [START_REF] Floyd | HighSpeed TCP for Large Congestion Windows[END_REF] the current standard is not able to reach high rates in realistic environments, i.e. with typical packet loss rates. Many new transport protocols have been proposed and are currently being studied to replace it. Some of them are already implemented in the latest versions of some operating systems, like Compound TCP (CTCP) on Windows, and Cubic (and others) on Linux. For a survey and comparative analysis of several high speed TCP versions see, for example, [START_REF] Kherani | Comparative study of different adaptive window protocols[END_REF][START_REF] Li | Evaluating the performance of TCP stacks for high-speed networks[END_REF][START_REF] Li | Experimental evaluation of tcp protocols for high-speed networks[END_REF]. For the next few years, the new high speed TCP versions will play an increasing role in resource sharing among flows in the Internet. Yet the behavior, the performance, and the impact on the network of these protocols are not well-known. In particular, there is no comprehensive analytical study of CTCP.

CTCP has been presented by Microsoft Research in [START_REF] Tan | Compound TCP : A scalable and TCP-friendly congestion control for high-speed networks[END_REF] and [START_REF] Tan | A compound tcp approach for high-speed and long distance networks[END_REF] in 2006. It is currently submitted as a draft to the IETF Network Working group with minor differences [START_REF] Sridharan | Compound TCP : A new TCP congestion control for high-speed and long distance networks[END_REF]. CTCP is enabled by default in computers running Windows Server 2008 and disabled by default in computers running Windows Vista [START_REF] Davies | Performance enhancements in the next generation TCP/IP stack[END_REF]. It is also possible to add support for CTCP to Windows XP. An implementation of CTCP, based on [START_REF] Tan | A compound tcp approach for high-speed and long distance networks[END_REF][START_REF] Sridharan | Compound TCP : A new TCP congestion control for high-speed and long distance networks[END_REF], is also available for Linux [START_REF] Andrew | Compound TCP Linux module[END_REF]. The main objective of the authors of CTCP [START_REF] Tan | A compound tcp approach for high-speed and long distance networks[END_REF] is to specify a transport protocol which is efficient, using all the available bandwidth, fair and conservative, limiting its impact on the network. They propose to combine the fairness of a delay-based approach with the aggressiveness of a loss-based approach. As the proposal of CTCP is still recent, there are only a few published evaluations of it. The only analytical model of CTCP in [START_REF] Tan | A compound tcp approach for high-speed and long distance networks[END_REF] is based on a de facto deterministic model.

In the present work we study the performance of CTCP under random losses. There are at least two important motivations to analyze TCP performance under random losses : random losses harm the performance of the current New Reno TCP on high speed optical links and random losses are inherent in wireless networks (WiMax can provide significantly high transmission rates in wireless networks).

The outline of the paper and of our results is as follows. In Section 2 we give a brief overview CTCP. In Section 3 we present a two-dimensional Markov chain model for CTCP. With the help of this model we obtain the long-run average throughput of CTCP and the distribution of its congestion window at congestion events. Then, in Section 3.2 we use Palm calculus to obtain the distribution of the congestion window at arbitrary time moments. In Section 3.3 we propose some heuristics and compare them with the accurate two-dimension Markov chain model. Finally, in Section 4 we provide numerical and simulation results which confirm our theoretical findings. In particular, we conclude that CTCP provides a higher throughput than TCP New Reno and , even if more aggressive, it causes less traffic jitter than TCP New Reno on high speed links with small random losses.

Due to space constraints, some technical details are provided in a companion technical report [START_REF] Blanc | Compound tcp with random losses[END_REF].

Compound TCP Overview

In this section we give a very brief overview of CTCP (see [START_REF] Tan | A compound tcp approach for high-speed and long distance networks[END_REF] for a complete description). The main idea of CTCP is to quickly increase the window as long as the network path is not fully utilized, then to keep it constant for a certain period of time and finally to increase it by one MSS (Maximum Segment Size) per round trip time just like TCP Reno. We will often use the term "phases" for these three different behaviors.

During phase 1 the sender computes the value at the (i + 1)-th round trip as w i+1 = w i + αw k i (as suggested in [START_REF] Tan | A compound tcp approach for high-speed and long distance networks[END_REF] we use α = 1/8 and k = 3/4). At each round trip the sender estimates the bandwidth-delay product and the amount of data backlogged in the network using the same method adopted by TCP Vegas [START_REF] Brakmo | Tcp vegas : end to end congestion avoidance on a global internet[END_REF]. If the amount of backlogged data is greater than a certain threshold (γ, usually set to 30 [START_REF] Tan | A compound tcp approach for high-speed and long distance networks[END_REF][START_REF] Tan | CTCP-TUBE : Improving TCP-friendliness over low-buffered network links[END_REF]) the sender switches to phase 2 and keeps the window constant. This constant value corresponds to the sum of the estimated bandwidth-delay product and the estimated amount of backlogged data. We consider an ideal behaviour of CTCP, assuming that such estimates are correct. In such case the window in phase 2 is equal to θ µτ +γ, where µ is the capacity of the bottleneck link and τ is the round trip propagation delay. In reality any queue size estimate available at the sender is outdated due to feedback delays, this fact combined with the CTCP algorithm presented in [START_REF] Tan | A compound tcp approach for high-speed and long distance networks[END_REF] causes the window to oscillate during this phase as we analyzed in [START_REF] Blanc | Oscillations of the sending window in Compound TCP[END_REF]. The length of phase 2 is dictated by the "congestion component" of the window. In fact in CTCP the congestion window w is the sum of two components : the delay window w d and the congestion window w c . The congestion component is incremented by one every round trip (just like the TCP Reno congestion window) and when this component reaches θ phase 2 ends. The delay component is set such that the value of the total window (w i = w ci + w di ) at the i-th round trip time follows the following evolution in absence of packet losses :

w i =      w i-1 + δ i , if w i-1 + δ i < θ θ , if w i-1 + δ i ≥ θ and w c0 + i < θ w i-1 + 1 , otherwise (1)
where δ i = max αw k i-1 , 1 . Figure 1 shows the three phases of the window evolution. When a loss occurs both the total window and the congestion window are halved.

Performances with Random Losses

We consider a single long lived TCP compound flow using a path with µτ bandwidth delay product and buffer size equal to b. The flow will experience a loss every time that its window size reaches the value µτ + b. For this reason, we can consider w max = µτ + b as an upper bound for the the window size. Beside the deterministic losses due to buffer overflow, we consider also that each packet can be dropped with some probability p, independently from all other packets, i.e. according to a Bernoulli process. In what follows we derive the throughput and the window distribution in steady state. We are going to assume that w can only take integer values.

Throughput calculation

We define a cycle as the time interval between two consecutive losses. We denote as w t cn (respectively w t dn ) the congestion (respectively delay) window at the begin of the (n + 1)-th round trip time of the t-th cycle. We will omit the superscript t whenever it is clear which cycle is being considered.

We observe that in our framework the evolution of the window in each cycle t depends from previous cycles only through the window value at the begin of the cycle, or, more precisely, through the two initial values w t c0 and w t d0 , which can be determined by the final value of the windows at the t -1-th cycle. This also implies that it is possible to use the renewal reward theorem to compute the average throughput as (see [START_REF] Padhye | Modeling tcp reno performance : a simple model and its empirical validation[END_REF]) :

E[λ] = E[S] E[T ] ( 2
)
where λ is the throughput (in MSS/s), S is the total number of packets sent during a cycle and T is the duration of the cycle.

Both E[T ] and E[S] can be evaluated starting from the knowledge of the distribution size of the two (correlated) random variables W c0 and W d0 . We denote g(w c0 , w d0 ), the probability mass function (pmf) of these random variables. We first show how to derive the distribution of cycle duration from g(w c0 , w d0 ) and then derive the pmf g() itself. E[S] can be evaluated similarly to E[T ].

A cycle has length equal to n if there is a loss at the n-th round trip time. As we are assuming that there is a loss whenever w = w max , all cycles have a finite length. Let m(w 0 ) = min {n|w n ≥ w max } be the maximum possible length (in round trips) of a cycle starting at w 0 . For n < m(w 0 ) the probability of a cycle having length equal to n can be derived from the Bernoulli loss process as :

P [T = n|W c0 = w c0 , W d0 = w d0 ] = (1 -p) Vn-1(w0) -(1 -p) Vn(w0) (3)
a n (w c0 , w d0 ), where

V n (w 0 ) n-1 i=0 w i , V 0 0, RR n°6736
and w i is computed as in (1) so that V n is the number of packets sent during the n-th round trip of a cycle starting with w = w 0 . Both V n and a n , as most quantities used in this section, depend on the initial window w 0 = w c0 + w d0 as highlighted by the notation a n (w 0 ) and V n (w 0 ), even though we will also use the simplified notation a n and V n .

The expression of a n follows from the cumulative distribution function of the geometric random variable F (x) = 1 -(1 -p) k and corresponds to the probability of at least one packet being dropped during the n-th round trip.

The probability that a loss occurs at the m(w 0 )-th round trip can be evaluated simply considering that

m i=1 P [T = i] = 1 : P [T = m ( w 0 )|W c0 = w c0 , W d0 = w d0 ] = 1 - m ( w0)-1 i=1 a i (w c0 , w d0 ).
Finally, being that the support of the discrete random variables W c0 and W d0 is finite, we can use a finite sum to compute P [T ] :

P [T = n] = wc 0 ,w d 0 a n (w c0 , w d0 )g(w c0 , w d0 ). (4)
In order to compute g(w c0 , w d0 we model the evolution of the window (at the beginning of each cycle) with a Markov chain. The evolution of the window of TCP Reno at the begin of a cycle (w t 0 ) has been modeled in other works as a Markov chain (see, for example, [START_REF] Lakshman | The performance of tcp/ip for networks with high bandwidth-delay products and random loss[END_REF][START_REF] Padhye | Modeling tcp reno performance : a simple model and its empirical validation[END_REF]). In fact w t 0 is equal to half of the window value at the end of the (t -1)-th cycle, which depends only on w t-1 0 and on packet loss probability p.

In order to model CTCP we use a two-dimensional discrete Markov chain X t to account for w t c0 and w t d0 . For each state (i, j) the first index represents w t c0 and the second w t d0 . For any pair of states it is possible to compute the transition probability as :

P [X t+1 = (k, l)|X t = (i, j)] = n∈B a n (i, j) with w t c0 = i, w t d0 = j, B = n| w t cn /2 = k, w t dn /2 = l
, where w t cn and w t dn are evaluated according to [START_REF] Altman | A stochastic model of tcp/ip with stationary random losses[END_REF]. The sum on the right hand side is needed because different pairs (w t cn ,w t dn ) can originate, after a loss, the same pair (w t+1 d0 ,w t+1 d0 ) as we use integer values for the window. As w ≤ w max we have that w c0 ≤ w max /2 N and, if θ is the value of the window during the constant window phase, w d0 ≤ θ/2 M (as w d ≤ θ). Combining these two bounds we obtain that the number of states in the Markov chain is N M . Using the ARPACK implementation of the Arnoldi method [17] it is possible to efficiently calculate the steady state distribution of the Markov chain X t even for large values of N M . The more time consuming step is actually to compute the transition matrix for X. The complexity of the algorithm we used is O(M N 2 ) ; we believe that it is not possible to decrease the complexity of the algorithm given that it has to compute all the possible transitions and these grow like M N 2 . Note that the number of possible transitions for a Markov chain with N M states is N 2 M 2 therefore we already take into account that, in this case, some transitions are not possible.

INRIA

Once the steady state distribution of the Markov chain X t (g(w c0 , w d0 )) is derived, we can use it to compute E[T ], E[S] and then the average throughput using [START_REF] Andrew | Compound TCP Linux module[END_REF].

If the queueing delays are negligible with respect to the propagation delays it is possible to convert round trip times into seconds by simply multiplying the number or round trip times by the total propagation delay. If this approximation is not acceptable we can account for the queueing delays by estimating the queue size q as q = wµτ (this is true as long as w > µτ and the source has been sending date at a rate greater than µ over the last round trip). Using this estimate we have that the round trip time is w/µ whenever w ≥ µτ .

For a given n, w c0 and w d0 , the right hand side of (4) corresponds to a cycle starting with w = w c0 + w d0 and lasting n round trip times. For each value of n we know the window size w and we can compute the length in seconds of each round trip and, hence, the total length of each cycle in seconds. Summing over all the possible cycles we can compute the expected value of T in seconds, and, by using (2), the average throughput in MSS/s.

Steady State Distribution of the Window

In the previous section we have described how to compute the steady state distribution of w c0 and w d0 , and consequently also the value of the window w 0 = w c0 + w d at the beginning of each cycle. In this section we are interested in the steady state distribution of the window as a function of time. We denote as Y n the value of the window at the begin of the (n -1)-th round trip time. Note that Y n is different both from X t , which is the value of the window after a packet loss, and from w t n , which is the value of the window at the begin of the (n -1)-th round trip time in the t-th cycle. Clearly X t represents a subsequence of the sequence Y n . We observe that Y n can also be modeled as a discrete time Markov chain where a transition occurs every round trip. Also this Markov chain is ergodic, hence it admits a steady state and we assume that it is in steady state at time 0.

Using Palm calculus, we first compute P [Y n = k] starting from P [W 0 = w 0 ], where Y n represents the window after n round trip times starting from some arbitrary value (given that all the Markov chains involved are ergodic, the initial value is irrelevant).

Let Z n be the (discrete) time of the n-th packet drop after time 0. Using the intensity and inversion formulas of Palm calculus [START_REF] Boudec | Understanding the simulation of mobility models with palm calculus[END_REF] we can compute P [Y n = k] as a function of P [W 0 = w 0 ] :

P [Y n = k] = E 1 {Yn=k} = P [Z 0 = 0]E 0 Z1 s=1 1 {Ys=k} (5) = ηE 0 Z1 s=1 1 {Ys=k} (6) = η w0,l P [W 0 = w 0 ]P [Z 1 = l|W 0 = w 0 ] l s=1 1 {Ys=k|W0=w0} (7)
RR n°6736

where E 0 is the Palm expectation, 1 {Yn=k} is the indicator function for the event Y n = k and η is the intensity of the process Z n . The second (5) and third [START_REF] Boudec | Understanding the simulation of mobility models with palm calculus[END_REF] equalities follow from the inversion and intensity formulas, respectively, while [START_REF] Brakmo | Tcp vegas : end to end congestion avoidance on a global internet[END_REF] follows from the total probability theorem, conditioning on all the possible values of W 0 and l. Given that Z n is an ergodic process P [Z 0 = 0] can be computed, using the intensity formula, as the inverse of the expected value of

T d = Z n -Z n-1
that is as the average length of a cycle (in round trips) so that η = 1/E[T ] where E[T ] can be computed using [START_REF] Blanc | Modelling an isolated Compound TCP connection[END_REF].

If the queueing delays are non-negligible with respect to the propagation delays it is possible to re-normalize π k P [Y n = k] as follows in order to take this into account : πk π k q k wmax i=1 π k q k where q k max τ , k µ .

A Simple Approximation and the Deterministic Response Function

The method described in the previous section provides an exact solution, but it can be computationally expensive for medium and large values of w max and θ. Using the same method as in [START_REF] Lakshman | The performance of tcp/ip for networks with high bandwidth-delay products and random loss[END_REF] it is possible to quickly find an approximate solution for the average window size. The idea is to consider a sequence of "typical" or "average" cycle. If p is the probability that a packet is dropped, the average cycle has exactly 1/p packets (provided the probability of reachind w max ). Let us denote w 0 the initial window of an average cycle and w n (w 0 ) the final window, after n round trip times during which 1/p packets are transmitted. Being that the following average cycle has to be identical and that CTCP, as Reno, halves the window at the end of each cycle, it has to be :

w n (w 0 ) = 2w 0 . (8)
Imposing the constraint V n (w 0 ) = 1/p (V n (w 0 ) is defined in section 3.1), we can identify the unique possible value of w 0 and then the unique possible window evolution corresponding to p. Once the window evolution is known, the average window can be obtained and can be plotted as a function of the drop rate. As previously noted in [START_REF] Altman | A stochastic model of tcp/ip with stationary random losses[END_REF] this approach corresponds to the calculation of what is usually called the "deterministic response function." This function is often used in the literature on TCP. For example, [START_REF] Floyd | HighSpeed TCP for Large Congestion Windows[END_REF] defines the response function as "the function mapping the steady-state packet drop rate to TCP's average sending rate in packets per round-trip time." Where the drop rate is simply the inverse of the number of packets sent during each cycle. In this case the term "drop rate" always refers to this quantity and not to any probabilistic model, while this usage is not the most appropriate one it is, nonetheless, common in the literature. From another point of view, we observe that the average cycle corresponds to the actual evolution of the window under our loss model, when there is no random loss, but the bandwidth delay product and the buffer size are such to cause a deterministic loss every 1/p packets.

INRIA

When the window update rule operates on a round trip time basis -as in CTCP but also in Reno and HighSpeed for example-the deterministic response function does not depend on physical parameters like capacity or propagation delay. In other words it can be considered as an intrinsic property of the specific growth function used to increment the window. On the contrary, for TCP Cubic the growth of the window depends on real time and hence also on link capacities and propagation delays, so that a comparison with the TCP versions indicated above would need special care.

Figure 2 shows the response function for different values of θ, between 100 MSS and 1000 MSS (Maximum Segment Size). The two lines correspond to two different ways to express the window evolution in [START_REF] Davies | Performance enhancements in the next generation TCP/IP stack[END_REF]. The dotted line corresponds to a fluid model where the growth of the window is approximated with a continuous function (see [START_REF] Blanc | Modelling an isolated Compound TCP connection[END_REF]). More precisely

w n (w 0 ) = (1 -k)αn + w * 1-k 0 1 1-k .
The solid line, instead, considers only integer values of the window and computes the increment of the window as according to (1) with w c0 = w 0 . In this case the last round trip time of the cycle -the n-th one-is evaluated as n = min{i : w i ≥ 2w 0 }. In both cases we have considered w 0 as the independent value. For each value of w 0 we have then computed S, the total amount of packets sent during a cycle starting with w = w 0 and then we have plotted the average window as a function of 1/S = p. Clearly the total number of packets sent during such a cycle is a monotonically increasing function of w 0 (as the TCP window is monotonically increasing during each cycle) so that increasing values of w 0 correspond to increasing values of S, and decreasing values of p = 1/S.

Regarding the integer approximation we observe that, for α = 1/8 and k = 3/4 (values suggested in [START_REF] Tan | A compound tcp approach for high-speed and long distance networks[END_REF]), αw k ≥ 2 only if w > 30. That is the CTCP window grows by one each round trip, the same as Reno, as long as w < 30. This is somewhat consistent with what suggested in [START_REF] Tan | A compound tcp approach for high-speed and long distance networks[END_REF] where the authors call for the delay component to be used (that is to increment the window by αw k ) only if the window is larger than lowwnd which they set equal to 41. This observation explains why for small values of w the fluid and integer approximation are different, with the integer approximation giving a larger average value of the window.

In the case of the integer approximation and for large drop rates (more than 10 -3 ) the response function is the same as Reno. The same is true, regardless of the approximation, for small drop rates (less than 10 -6 ) given that, in this latter case, the evolution of the window is the same in CTCP and Reno. For drop rates roughly between 4 • 10 -4 and 10 -3 only the rapid increase phase of CTCP is used so that the response function is steeper. For drop rates between 10 -6 and 4 • 10 -4 the "constant" window phase is used along with all the other phases and this explains why the response function increases more slowly until it reaches the Reno response function.

While for Reno and HighSpeed TCP the response function is only a function of the drop rate, for CTCP the response function is also a function of θ (the value of the window during the constant window phase). The larger the value of θ the longer the rapid increase phase can be. In the extreme case of θ = ∞ the other phases would not take place at all and the response function would be much steeper. Note that in [START_REF] Tan | A compound tcp approach for high-speed and long distance networks[END_REF] the authors compute the response function for exactly this case (θ = ∞). Given that they are interested in limiting the aggressiveness of CTCP, they are in effect considering the worst case, by only considering the rapid increase phase. At the same time it can be argued that, as a consequence, CTCP is less aggressive than HighSpeed TCP as, for small values of p, CTCP is as aggressive Reno, which is less aggressive than HighSpeed TCP.

Finally it is worth noting that, as p decreases, the difference between the fluid and integer models for the rapid increase phase becomes negligible.

Numerical Results

Using the models presented in the previous sections we can compute the average throughput and the average window size for different values of the drop probability p. Figure 3 shows the CTCP response function computed using the two deterministic models presented in section 3.3 and the probabilistic model discussed in section 3.1, for θ = 250 MSS. For the same "drop probability" the deterministic model with integer approximations gives a smaller average window than the probabilistic model, which uses the very same integer approximations, as already observed in [START_REF] Altman | A stochastic model of tcp/ip with stationary random losses[END_REF]. The fluid model, instead, does agree with the proba- bilistic model, for some values of p, but we believe that this is just a coincidence. One should take some care in comparing these two models : in the case of the deterministic model the buffer size at the bottleneck is fixed (so that all the cycles have the same size) while in the case of the probabilistic model the buffer size at the bottleneck link is much larger (in theory infinite, set to 1600 MSS for the numerical results) and allows the window to reach larger values. If we used the same buffer size in both models the average window would be smaller in the probabilistic case.

Figure 4 shows the average throughput for CTCP and TCP Reno. For CTCP we have used the probabilistic model introduced in section 3.1 while for Reno we have used an equivalent model but with a "one dimensional" Markov chain as Reno does not have two components in the congestion window. The squares and triangles in Figure 4 correspond to simulation results. For both versions of TCP there is a good match between the probabilistic model and the simulations, obtained using ns-2 with a Linux implementation of CTCP [START_REF] Andrew | Compound TCP Linux module[END_REF]. In the simulations there is a single TCP connection with no cross traffic going through a bottleneck of 100 Mb/s with propagation delay of 26.4 ms and where each packet of 1500 B is dropped with probability p. As the difference between different simulation runs is very small (less than 1%) we did not plot errorbars.

In Figure 4 w max = 370 MSS, the bandwidth-delay-product is 220 MSS and the buffer size is 150 MSS. While in Figure 3 w max = 1600 MSS this explains why in Figure 4 the throughput is constant for small drop probabilities (most of the packets are dropped when w = w max ) a similar behavior takes place for larger values of w max but for drop probabilities smaller than those included in Figure 3. In each case the dotted lines represent the corresponding distribution obtained from ns-2 simulations.

Figure 5 corresponds to the case where, most of the time, the window is smaller than 41 so that CTCP behaves similarly to Reno. This is confirmed by the fact that w d0 = 0 most of the time. Figure 6 corresponds to the case where most of the packet are dropped when the window is quickly growing as w n+1 = w n +αw k n . In this case the distribution of w c0 is greater than the distribution of w d0 , so that, on average, w c0 < w d0 .

Figure 7 corresponds to the case where most of the packets are dropped during the constant window phase as confirmed by the jump in the distribution of w 0 when w 0 = 215 MSS which correspond to θ/2 (as θ = 430 MSS in this case). The smaller jumps in the distribution for the ns-2 simulations are caused by the oscillations of the window during the constant window phase. These oscillations are not taken into account by the probabilistic model.

Figure 8 corresponds to the case where a significant fraction of packets is dropped during the Reno phase (almost 50%). Again the jump at w 0 = 215 MSS represents the packets dropped during the constant window phase (with the simulations having smaller jumps caused by the oscillations of the window). The jump at w = 300 MSS correspond to the case when packets are dropped due to a buffer overflow (in this case w max = 600 MSS and the buffer size is 200 MSS for the simulations). In this case the distribution of w c0 is smaller than the distribution of w d0 , the opposite of what happens in the previous two cases.

Figure 9 shows the steady state distribution of Y n for different values of p with θ = 430 MSS and w max = 2000 MSS. Again the dotted lines represent the same distribution for the corresponding ns-2 simulations. As expected, with increasing drop probabilities, each distribution is strictly greater than all the previous ones. In Figures 8 and9 while the probabilistic models have a sharp jump for w = 215 MSS and w = 430 MSS the simulations (dotted lines) have smaller jumps. This is caused the oscillations of the sending window during phase 2 in the simulations. As mentioned in section 2 this is consistent with the CTCP algorithm but it is not taken into account by the probabilistic model. While it is, at least in principle, possible to incorporate this aspect into the model, we prefer using a simpler model with a constant value during phase 2 given that the differences between this simplified model and the simulations are not significant (especially as far as the throughput is concerned). Figure 12 shows the average value and the standard of Y n for CTCP.

Figures 11 and10 show the variance and the coefficient of variations (CoV) for CTCP and Reno. In both cases the difference between the deterministic and probabilistic model is more pronounced than in the case of the average window (response function). This can be explained by the fact that the average window depends only the first moment of Y n while the variance and the CoV depend on the second moment as well. For the CoV, in particular, the difference between the two models is significant. For small values of p the CoV of CTCP is smaller than Reno but for larger values of p the opposite is true indicating that for p > 10 -4 CTCP might not be the best solution. Note that the height and the location of the peak of the CoV for CTCP are a function of θ.

Figures 13 and14 show the distribution of w 0 and Y n for TCP Reno under the same conditions : µτ = 400 MSS and buffer size 200 MSS. These distributions have been computed using a Markov chain similar to the one presented in section 3.1 but with only one state variable, given that Reno uses only one component for its sending window.

Conclusions and Future Work

In this paper we have presented a Markovian model of CTCP under random losses. This kind of model is a first attempt to roughly assess the impact of varying network conditions on a CTCP connection. The network is seen as a black box randomly dropping packets, due to buffer overflows. This model could also be used to describe the impact of transmission in some "challenging environments" (e.g. wireless networks) as expected from new TCP versions [START_REF] Floyd | Specifying New Congestion Control Algorithms[END_REF].

In this first analysis, we have assumed that the loss arrivals follow a simple Bernoulli process. We have computed the distribution of the sending window on loss events with a Markovian model, and then the average throughput. Using Palm Calculus we have computed the steady state distribution of the window. Its value has a direct influence on the buffer occupancy and on the jitter experienced by all the flows sharing the same bottleneck link. This analysis can be extended in many ways. The Bernoulli loss process could be replaced with a more bursty and realistic process. In this case, multiple losses could take place during the same round-trip time, and the recovery time could be longer. We could also consider time-outs in the case of high loss rates, as in [START_REF] Padhye | Modeling tcp reno performance : a simple model and its empirical validation[END_REF]. A similar analytical study also be applied to the other TCP versions, currently under standardization, comparing their efficiency and their robustness.

Fig. 1 -

 1 Fig. 1 -The evolution of w and w c in CTCP

Fig. 2 -Fig. 3 -

 23 Fig. 2 -The deterministic response function for θ = 100, 250, 1000

Fig. 4 -

 4 Fig. 4 -Average Throughput (µτ = 400 MSS, w max = 600 MSS)

Figures 5 , 6 ,

 56 7 and 8 show the distributions for w 0 , w c0 and w d0 when θ = 430 MSS for p = 3 • 10 -3 , 3 • 10 -4 , 4 • 10 -5 and 5 • 10 -6 respectively. Corresponding to the four different regions of the response function in Figure 2.

Fig. 5 -Fig. 6 -

 56 Fig. 5 -Distribution of w 0 ,w c0 and w d0 (µτ = 400 MSS, w max = 600 MSS, , p = 3 • 10 -3 )

Fig. 7 -Fig. 8 -

 78 Fig. 7 -Distribution of w 0 ,w c0 and w d0 (µτ = 400 MSS, w max = 600 MSS, p = 4 • 10 -5 )

p = 3 • 10 - 3 p = 3 • 10 - 4 p = 4 • 10 - 5 p = 5 • 10 - 6 Fig. 9 -Fig. 10 -

 3103310441055106910 Fig. 9 -Steady state distribution of Y n (µτ 400 MSS, w max = 600 MSS)

Fig. 11 -Fig. 12 -p = 3 • 10 - 3 p = 3 • 10 - 4 p = 4 • 10 - 5 p = 5 • 10 - 6 Fig. 13 -p = 3 • 10 - 3 p = 3 • 10 - 4 p = 4 • 10 - 5 p = 5 • 10 - 6 Fig. 14 -

 1112310331044105510613310331044105510614 Fig. 11 -Variance of Y n for CTCP and Reno (µτ = 220 MSS, w max = 2000 MSS)

RR n°6736

Blanc & al.