Ejder Baştug

Marios Kountouris
email: marios.kountouris@huawei.com

Mehdi Bennis
email: bennis@ee.oulu.fi

Mérouane Debbah
email: merouane.debbah@centralesupelec.fr

On the Delay of Geographical Caching Methods in Two-Tiered Heterogeneous Networks

Keywords: edge caching, Poisson point process, stochastic geometry, mobile wireless networks, 5G

We consider a hierarchical network that consists of mobile users, a two-tiered cellular network (namely small cells and macro cells) and central routers, each of which follows a Poisson point process (PPP). In this scenario, small cells with limited-capacity backhaul are able to cache content under a given set of randomized caching policies and storage constraints. Moreover, we consider three different content popularity models, namely fixed content popularity, distance-dependent and loaddependent, in order to model the spatio-temporal behavior of users' content request patterns. We derive expressions for the average delay of users assuming perfect knowledge of content popularity distributions and randomized caching policies. Although the trend of the average delay for all three content popularity models is essentially identical, our results show that the overall performance of cached-enabled heterogeneous networks can be substantially improved, especially under the loaddependent content popularity model.

I. INTRODUCTION

Content caching in 5G heterogeneous wireless networks improves the system performance, and is of high importance in limited-backhaul scenarios [START_REF] Baştug | Living on the Edge: The role of proactive caching in 5G wireless networks[END_REF]. Most existing literature for cache-enabled heterogeneous networks using stochastic geometry focuses on the characterization of key performance metrics neglecting the backhaul limitations and the spatiotemporal content popularity profiles [START_REF] Chen | Cooperative caching and transmission design in cluster-centric small cell networks[END_REF]- [START_REF] Yan | User access mode selection in fog computing based radio access networks[END_REF]. In order to capture these aspects, we analyze in this paper the gains of caching in heterogeneous network deployment considering the average delay as a performance metric.

Consider a multi-tier heterogeneous network where base stations in each tier are deployed according to a homogeneous Poisson point process (PPP). More precisely, we model a heterogeneous network which consists of mobile terminals (users), cache-enabled small base stations (SBSs), macro base stations (MBSs) and central routers. In this network setting, a user may experience delays due to downlink transmissions, backhaul and caches. Supposing that SBSs are able to cache contents proactively, we derive expressions for the average delay of typical users when connected to either MBSs or SBSs.

Moreover, in order to capture the spatio-temporal content access patterns of users, we suppose fixed content popularity, distance-dependent and load-dependent content popularities. Assuming that the content popularity distribution is perfectly known at the small base stations, we explore three different caching policies based on content-popularity and randomization.

II. SYSTEM MODEL

Topology: We consider a multi-tier heterogeneous network in the two-dimensional Euclidean plane R 2 , where nodes in each tier k are distributed according to a homogeneous PPP

Φ k = {r (k) i } i∈N of intensity λ k , and r (k) i ∈ R 2 represents
the location of the i-th node at the k-th tier. The above network layout models a multi-tier heterogeneous network that consists of mobile terminals (users), SBSs, MBSs and central routers with densities λ ut > λ sc > λ mc > λ cr , respectively. A typical mobile user is assumed to be located at the Cartesian origin (0, 0) in order to derive the performance metrics of the heterogeneous network.

Signal Model: We shall consider that the MBSs and SBSs are transmitting in the same frequency band and hence interfering with each other. The transmit power is P mc for each MBS and P sc for each SBS, where we assume that P mc > P sc . For notational convenience, let us denote a base station (transmitter) by its position. The received power experienced at a typical user due to a transmitter x is given by P x h x (x), where P x is the transmit power (P mc or P sc ), h x corresponds to the fast fading power coefficient (square of the fading amplitude) of the channel between transmitter x and typical user, and (x) = x -α is the standard power law pathloss function with α > 2. The channel fading power coefficients are independent and identically distributed (i.i.d.) exponential random variables (Rayleigh fading) with

E[h x ] = 1.
Since we assume that the network is interference-limited (i.e., the interference power dominates over the noise power), we simply consider the signal-to-interference ratio (SIR). For a typical user connected to a MBS located at x, the SIR is given as

SIR mc (x) = P mc h x (x) I mm + I sm (1)
where I mm = y∈Φmc\{x} P mc h y (y) is the interference experienced from all MBSs except the serving MBS at x, and I sm = y∈Φsc P sc h y (y) is the aggregate interference experienced from SBSs. For a typical user connected to a SBS located at x, the SIR is given as

SIR sc (x) = P sc h x (x) I ss + I ms (2)
where I ss = y∈Φsc\{x} P sc h y (y) is the interference experienced from all SBSs except the serving SBS, and I ms = y∈Φmc P mc h y (y) is the aggregate interference from MBSs.

The target SIR in our system model is denoted by γ.

Connectivity and Backhaul: Mobile user terminals are associated with the closest base station, either SBS or MBS. As alluded to earlier, each MBS or SBS is also connected to its nearest central router. Each central router has a highrate broadband Internet connection. The wired backhaul is used to provide this broadband connection to MBSs and SBSs via backaul links, such that users' requests can be satisfied. Supposing that a content request is generated by a user, the base station is then in charge of starting immediately its distribution. An illustration of this heterogeneous network under limited-capacity backhaul is illustrated in Fig. 1.

A. Caching Model

When a user has a content request, we assume that the request is drawn from the distribution f pop , which is in decreasing order of content popularities. More formally, the content popularity distribution of a user is a right continuous and monotonically decreasing probability distribution function (PDF), given by [START_REF] Newman | Power laws, Pareto distributions and Zipf's law[END_REF]

f pop (f, η) = (η -1) f -η , f ≥ 1, 0, f < 1, (3)
where f indicates a point in the support of the corresponding content, and η > 1 parametrizes the steepness of the popularity distribution curve.

In fact, higher values of η results in steeper distribution, which in turn means that certain contents are highly popular than the rest of contents in f pop (f, η). Conversely, lower values of η yield a more uniform distribution, which in turns say that almost all contents have similar popularities. The content popularity of a user may be evolving over time and space, influenced by the choice of other users, and can be partially known at the base stations. This is somewhat equivalent to say that the parameter η can take different values depending on the scenario. In our case, each base station perfectly observes the content popularities according to three different models as follows:

-Fixed: The content popularity is identical for all users, with fixed steepness factor of η = η 0 . Therefore, all SBSs observe the same distribution given by f pop (f, η 0 ).

-Distance-dependent: The users have different content popularity distributions, each of them having a distancedependent steepness factor η = r, where r is the (random) distance between a user and its serving SBS. Therefore, we assume that each SBS observes on average a content popularity distribution given by f pop (f, r), where r is the average distance between the SBS and its users. This model is used to mimic the behavior of content popularity based on the distance (i.e., flat distribution in short distances). -Load-dependent: The content popularity of users is loaddependent on average, each of SBS having parameter η = λ ut /λ sc . Therefore, all SBSs observe the content popularity distribution given by f pop (f, λ ut /λ sc ). This model is used to mimic the behavior of content popularity based on the load (i.e., steep distribution in heavy loads).

Note that the choice of such a continuous content distribution is in fact for ease of analysis. When practical issues or analytical tractability are not a priority, Zipf-like discrete power laws can also be considered for modeling [START_REF] Newman | Power laws, Pareto distributions and Zipf's law[END_REF]. Indeed, content access statistics in cache-enabled web proxies [START_REF] Breslau | Web caching and zipf-like distributions: Evidence and implications[END_REF], or more relevantly in base stations [START_REF] Shafiq | Characterizing and modeling internet traffic dynamics of cellular devices[END_REF], [START_REF] Dehghan | On the complexity of optimal routing and content caching in heterogeneous networks[END_REF] are characterized by such discrete power laws (or arguably distributions).

For the (some of) caching policies described below, we shall assume that the content popularity distribution f pop (f, η) is perfectly known at the base stations. Practically, in order to have partial knowledge of f pop (f, η) for the caching policies, statistical estimation methods can be employed either at the base stations in a distributed manner or alternatively at the central routers, by using statistical tools from machine learning (see [START_REF] Paschos | Wireless caching: Technical misconceptions and business barriers[END_REF], [START_REF] Elbamby | Contentaware user clustering and caching in wireless small cell networks[END_REF] for relevant discussions).

Given f pop (f, η), the content in the interval [1, f 0 ) is the cacheable content and is called as catalogue, whereas the remaining part [f 0 , ∞] is considered as non-cacheable content (i.e., sensor data, voice streaming and online gaming). An interval [f, f + ∆f ) in the support of f pop (f, η) is dedicated to represent the probability of the f -th content. Each SBS has a storage capacity of S, thus it caches contents according to a given caching policy. Having such a request behavior described above and caching capabilities at the SBSs, we consider the following offline caching policies:

-StdPop [START_REF] Baştug | Cache-enabled small cell networks: Modeling and tradeoffs[END_REF]: The most popular content from the catalogue is stored in the cache of SBSs and requires S p ≥ 0 amount of storage. We additionally assume that the track of content popularity in a SBS requires S 0 amount of storage, defined as a function of the number of contents in the catalogue and the type of algorithm employed for content popularity estimation, thereby it holds that S = S p + S 0 ≥ 0. -UniRand [START_REF] Blaszczyszyn | Optimal geographic caching in cellular networks[END_REF]: The S amount of contents are cached uniformly at random. Note that this policy is not aware of the content catalogue, therefore it does not require any memory to track the content popularity profile. -MixPop: The S p amount of storage is used to cache the most popular content deterministically. The storage overhead is S 0 ≥ 0 and again defined as a function of number of content and the employed algorithm. In addition, we suppose that S u ≥ 0 amount of storage is used to cache content uniformly at random, thus S = S p + S 0 + S u ≥ 0. In fact, if the catalogue size is sufficiently small, the storage overhead in StdPop and MixPop, due to the track of content popularity can be neglected. However, such an overhead may dominate the total storage space when a large catalogue with low-sized chunks is considered. One can also observe that the StdPop and UniRand policies are special cases of the MixPop policy and are given here for the sake of exposition.

The performance of any statistics-aware online cache removal policy (i.e., least-recently used (LRU) and leastfrequently used (LFU)) would be upper bounded by its offline successor that has perfect content statistics; as such an online approach would require iterative estimation of content popularity in a finite time window, yielding to overall performance degradation. Such online policies can also be incorporated to our system model after some specific assumptions (see Independent Reference Model [START_REF] Che | Hierarchical web caching systems: Modeling, design and experimental results[END_REF] for instance).

B. Delay and Quality of Service

Quality-of-service (QoS) is closely related to the delay experienced by users. We consider three different sources of delay which are detailed separately as follows.

Delay in downlink: When MBSs and SBSs have to deliver the contents to their intended mobile users, it is evident that the downlink transmission over the wireless medium incurs a delay mainly due to the interference from concurrent transmissions and channel fading. Consider now a simple retransmission protocol where a packet of requested content is repeatedly transmitted until its successful delivery, up to a pre-defined number of retransmission attempts M . Indeed, inferring whether a packet delivery is successful or not at the base station essentially relies on the signal-to-interferenceplus-noise ratio (SINR) (or SIR in our case) being higher than the predefined threshold γ. If a packet is delivered successfully, we shall assume that the base station (macro or small cell) receives a one-bit acknowledgement message from the mobile user with negligible delay and error. Otherwise, if the delivery fails, the base station receives a one-bit negative acknowledgement message in the same vein. These attempts take T 0 amount of time. An outage event occurs if the packet is not delivered after M attempts. In the remainder, we denote the downlink delay experienced by the typical macro cell users (MUs) and small cell users (SUs) as D dm and D ds respectively.

Delay in backhaul: The delay caused in a wired backhaul link is modeled by an exponentially distributed random variable whose mean being proportional to the product of the average link distance (from typical base station to its nearest central router) and the average number of base stations connected to a single central router. In particular, representing the delay in macro and small cell backhaul links as D bm ∼ Exponential(μ bm ) and D bs ∼ Exponential(μ bs ) respectively, we (in general) suppose that D bs stochastically dominates D bm . 1 This implies that small cell backhaul links are subject to higher delays compared to those of MBSs.

Delay in caches: Serving a user by fetching its content from the local cache is subject to delay as the storage medium is prone to errors, whereas such a delay may also vary depending on the storage type and the underlying mechanisms (i.e., hard disk, solid-state disk (SSD)). In this regard, we model this phenomenon as D ca ∼ Exponential(μ ca ), an exponentially distributed random variable with mean μca being proportional to the storage type. We also assume that the delay of small cell backhaul links stochastically dominates the delay of reading a content from local caches, meaning that the speed of content reads from caches is stochastically higher than the speed of small cell backhaul links.

III. PERFORMANCE ANALYSIS

Considering the aforementioned sources of delay, namely downlink, caching and backhaul, the delay experienced by the typical MUs and SUs are respectively defined as

D m = D dm + D bm , (4)
D s = D ds + 1 {fs∈∆0} D ca + 1 -1 {fs∈∆0} D bs (5)
where f s is the content requested by the typical small cell user and ∆ 0 is the cache of its associated small cell. The indicator function 1 {...} returns 1 if the statement holds, 0 otherwise. Before proceeding to the next step, let us define functions B 1 (T 0 , M, γ, α, P x , P y , λ x , λ y ), B 2 (S p , η) and B 3 (S u , S p , f 0 , η) given on the top of the next page. 1 Given two random variables A and B, we say that A stochastically dominates B if P(A > x) ≥ P(B > x) for all a, or alternatively, F A (x) ≤ F B (x) for cumulative distribution functions F A (x) and F B (x).

B1(T0, M, γ, α, Px, Py, λx, λy) = T0

M -1 i=0 (-1) i M i + 1 1 1 + i ρ(γ, α) + (Px/Py) 2/α (λx/λy)γ 2/α A(α) (6) B2(Sp, η) = 1 -1 + Sp 1-η (7) B3(Su, Sp, f0, η) = Su f0 -Sp 1 -1 + f0 1-η + 1 + Sp 1-η (8)
We now state the following result related to the average delay experienced by the typical MUs.

Theorem 1. The average delay for a typical user connected to its nearest MBS is given by Dm = B 1 (T 0 , M, γ, α, P sc , P mc , λ sc , λ mc ) + 1 2

βλ mc λ -3/2 cs ( 9
)
where B 1 (T 0 , M, γ, α, P sc , P mc , λ sc , λ mc ) is given in [START_REF] Newman | Power laws, Pareto distributions and Zipf's law[END_REF]. The parameter β is a scaling factor, relating to the importance of backhaul delay over the non-backhaul delay.

Proof. See Appendix B.2 in [START_REF] Baştug | Distributed caching methods in small cell networks[END_REF].

In Theorem 1, the function B 1 models the average downlink delay whereas the remaining term in Dm incorporates the average delay caused due to the backhaul. The summation of terms is due to the consideration of independent PPPs. We now turn our attention to SU with and without caching capabilities at the SBSs.

Corollary 1. The average delay for a typical user connected to its nearest small cell (with no caching) is given by

Dm = B1(T0, M, γ, α, Pmc, Psc, λmc, λsc) + 1 2 βλscλ -3/2 cs ( 10
)
where B 1 (T 0 , M, γ, α, P mc , P sc , λ mc , λ sc ) is given in [START_REF] Newman | Power laws, Pareto distributions and Zipf's law[END_REF].

Proof. The result is a direct application of Theorem 1, thus it is immediately proved by following similar steps given in Appendix B.2 in [START_REF] Baştug | Distributed caching methods in small cell networks[END_REF].

Theorem 2. When MixPop caching policy is employed at the SBSs, the average delay for a typical user connected to its nearest small cell under fixed content popularity distribution is given by

D(mix) fix = B1(T0, M, γ, α, Pmc, Psc, λmc, λsc) + 1 2 βλscλ -3/2 cs + μca - 1 2 βλscλ -3/2 cs B2(Sp, η0) + B3(Su, Sp, f0, η0) (11)
where B 2 (S p , η 0 ) and B 3 (S u , S p , f 0 , η 0 ) are given in (7) and (8) respectively.

In case of distance-dependent content popularity, the average delay is given by

D(mix) dist = B1(T0, M, γ, α, Pmc, Psc, λmc, λsc) + 1 2 βλscλ -3/2 cs + μca - 1 2 βλscλ -3/2 cs B2(Sp, 1 2 √ λsc )+ B3(Su, Sp, f0, 1 2 √ λsc ) . ( 12
)
In case of load-dependent content popularity, the average delay is given by

D(mix) load = B1(T0, M, γ, α, Pmc, Psc, λmc, λsc) + 1 2 βλscλ -3/2 cs + μca - 1 2 βλscλ -3/2 cs B2(Sp, λut λsc )+ B3(Su, Sp, f0, λut λsc ) . (13)
Proof. See Appendix B.3 in [START_REF] Baştug | Distributed caching methods in small cell networks[END_REF].

The functions B 2 and B 3 in Theorem 2 are related to caching popular contents and caching uniformly at random respectively, and captures the cache hit behavior of the MixPop policy. By slightly modifying the steps in the proof of Theorem 2, similar results for StdPop and UniRand caching policies can be readily obtained. Note that the results above are based on the assumption that the typical users are connected to their nearest base stations.

In the above, we have provided the average delay expressions for typical MUs and SUs. The total average network delay, total network cost (including deployment and operational costs), and optimization of these metrics with respect to system design parameters are left for future work.

IV. NUMERICAL RESULTS

In this section, we numerically validate our approximations derived in the previous section. The impact of critical system parameters are discussed as follows.

Impact of MBS density λ mc : The change of average delay with respect to the MBS density is given in Fig. 2a. Therein, as the number of MBSs increases, we observe an increment in average delay. This is mainly due to the backhaul as the delay in backhaul is proportional to the distance and average number of connected MBSs. In this setup, even though the average distance from a MBS to its central central router decreases (thus less delay in the backhaul), the increasing number of base stations contributes more to the average delay, thus yielding such a behaviour. On the other hand, the average delay in SBSs remains static in this setup. However, we note that the average delay experienced by a typical small cell user is reduced by adding caching capabilities at the base stations. For instance, when content popularity is load-dependent and caching policy is MixPop, the average delay is reasonably less than other candidates (including typical users with no caching at SBSs).

Impact of small cell density λ sc : The change of the average delay with respect to the small cell density is depicted in Fig. 2b. Similarly to the previous figure for MBS density, we see that the average delay increases for all kind of small cell

Figure 1 :

 1 Figure 1: An illustration of the considered system model. The snapshots of i) central routers, ii) MBSs, iii) SBSs and iv) mobile user terminals are provided on the right side of figure.

In this work, we have characterized the average delay of MUs and SUs under backhaul constraints and caching capabilities at the small base stations. Several content popularity distributions and caching policies have been considered. The main conclusion from this work is that caching at the small base stations allows for balancing the average access delay to the contents, especially if heterogeneous network densification under limited backhaul is considered.

This research has been supported by the ERC Starting Grant 305123 MORE (Advanced Mathematical Tools for Complex Network Engineering), the projects BESTCOM and 4GinVitro, the Academy of Finland CARMA project and TEKES grant (2364/31/2014).

users. However, in this numerical setup, the rate of increment in delay with no-caching capabilities at the SBSs is higher than the delay experienced by the typical users with cacheenabled SBSs. Compared to the fixed and load-dependent content popularities, the typical user under load-dependent content popularity experiences less delay when the number of SBSs increases. Impact of target SIR γ: In our setup, yet another important design parameter is the target SIR. In this regard, the average delay variation with respect to the target SIR is illustrated in Fig. 2c. As observed in the figure, the average delay increases by imposing higher target SIR values. This change is only visible in low values of target SIR, whereas the variation of delay in higher values of target SIR is negligible. This might stem from the fact that the downlink delay is not a dominating factor in our scenario compared to the backhaul delay. A typical user connected to the small cell with no caching capabilities experiences the highest delay, whereas the minimum delay is achieved by using MixPop policy under load-dependent content popularity. The delay of a typical MU remains between a SU with no-caching and caching capabilities at the base stations.

Impact of storage size S: Yet another crucial design parameter in our setup is the storage size. The impact of storage size on the average delay is shown in Fig. 2d. Indeed, as observed from the figure, dramatical decrease in delay is observed by increasing the storage size of small base stations. Similarly to previous observations, the most sensitive content popularity for the average delay is the load-dependent content popularity.

V. CONCLUSIONS


Bruno Vallespir
email: bruno.vallespir@ims-bordeaux.fr

Yves Ducq
email: yves.ducq@ims-bordeaux.fr

ENTERPRISE MODELLING: FROM EARLY LANGUAGES TO MODELS TRANSFORMATION

Keywords: Enterprise modelling, modelling framework, meta-modelling, models transformation, model-driven. 1

During the last thirty years, enterprise modelling has been recognised as an efficient tool to externalise the knowledge of companies in order to understand their operations, to analyse their running and to design new systems from several points of view: functions, processes, decisions, resources, information technology. This paper aims at describing the long evolution of enterprise modelling techniques as well as one of the future challenges of these techniques: the transformation of enterprise models. So, in a first part, the paper describes the evolution of enterprise modelling techniques from the divergence era to the convergence period. In a second time, the paper focuses on the recent advances in the use of enterprise models through model driven approaches, interoperability problem solving and simulation, all these advances having the same characteristic to use the transformation of enterprise models.

INTRODUCTION

Formalizing the organization of companies is a necessity to obtain a better efficiency and allow them to evolve in relation with the market and the socio-technical environment. During the last thirty years, enterprise modelling thus stood out as a powerful mean in the service of the industrial organizations with a focus on performances improvement. A large number of tools and methods emerged at first, in particular in the United States and in Europe, highlighting scientific concepts today recognized at the international level with an asserted industrial application purpose. This development led, according to the diverse needs for every study, to a significant number of tools and specific methods sharing however numerous concepts or aiming at identical application domains. In the face of this diversity, a second phase appeared who leads to a link, even to an integration of the existing approaches, rather than to an emergence of new ones. From then on, the problem was to define how to make modelling languages to communicate. Although based on common concepts stemming from more general theories as the systems theory, the graph theory or the information theory, the languages are nevertheless interpretable of different manners when they are concretely implemented. In summary, the development of enterprise modelling went at first through an era of brainstorming, followed by an era of convergence.

In this article, first of all we shall remind quickly the foundations of enterprise modelling and the main currents having led to the development of modelling languages. We shall then address the approaches the role of which is to move closer, to put in coherence and to integrate these concepts.

In parallel with the process of convergence which has been presented, other developments appeared around enterprise modelling and are presented for some of them. The purpose of these developments is to ensure the alignment between models at the business level and at the technical level through model driven approaches, the modelling in situation of interoperability and the combine use of enterprise modelling and simulation. All these parts have an essential common point that is the use of model transformation. The purpose of this paper is to present the frames where models transformation take place, and some technical aspects of models transformation including an application case.

MODELS AND LANGUAGES: THE INITIAL ERA OF BRAINSTORMING AND DIVERGENCE

An enterprise is a complex reality that it is necessary to model for, on one hand, making it understandable and, on the other hand, allowing actors having a well defined project in mind to reason, decide and act. In other words, a model of a company has for objective to formalize all or a part of the company with the aim of understanding or of explaining an existing situation or to realize and then validate a designed project [START_REF] Braesch | La modélisation systémique en entreprise[END_REF].

The actions based on enterprise modelling concern the design, the evaluation and the reengineering of enterprises. Vernadat (1999a) considers that enterprise modelling is a discipline that consists of describing the organization and the operational processes of a company either with the aim of simulating these processes to compare diverse scenarios, or with the aim of analysing and restructuring them to improve the performance of the company. So, it emerges that there is no really a life cycle of enterprise modelling because it intervenes during all the phases of the life of the company: since its creation until its destruction and during its evolution due to the possible re-use of some of its components.

A model is always built on the basis of a language that could be informal (natural language, for example), semi-formal (language essentially graphic for example) or formal (mathematical language). Most of the time, the models based on an informal language are used to describe an existing situation whereas the models based on a formal language allow the check of the properties required in a given project (Chapurlat et al. 1999, Kamsu-Foguem and[START_REF] Kamsu-Foguem | Requirements modelling and formal analysis using graph operations[END_REF]. [START_REF] Kosanke | CIMOSA: enterprise engineering and integration[END_REF] state that a model is a representation of an abstraction of a part of real world, expressed in a language of representation. So, a model of company is always associated with a purpose and it must, according to needs, be capable of taking the structural, functional and behavioural aspects into account. Besides these aspects, it must also be capable of understanding the particular point of view of an actor. Finally, a specific approach is often associated with every modelling language. This approach clarifies the various stages necessary for the building and for the exploitation of the model (life cycle). [START_REF] Doumeingts | Methodologies for designing CIM systems: A survey[END_REF] consider that a model is a representation of knowledge to be communicated without ambiguity. [START_REF] Neaga | An enterprise modeling and integration framework based on knowledge discovery and data mining[END_REF] go towards the same direction by linking enterprise modelling and knowledge management.

The supplied models have also to perform a role concerning the human actors of the project. Indeed, they have to play a role of support in:

 the communication inside the project, in space (exchange between actors of different skills and belonging to structures or departments with different purposes) and in time (archiving and documentation of the project); and

 the decision of design or reengineering of the system.

These two requirements result in one of the major difficulties of enterprise modelling. On one hand, it has to propose languages leading to models allowing the valuation of the system and from which a calculation approach can be implemented (evaluation). On the other hand, the obtained models must be easily understandable by the actors of the company and allow to gather the various advices and expertise, while taking the behavioural variability of the system into account.

These various observations explain why the works led within the framework of enterprise modelling gave rise to numerous languages and modelling tools. So, it is very difficult to supply an exhaustive list, especially as most of the disciplines having to deal with artificial complex systems made their contribution. We shall quote in particular the engineering of information systems, the automatic control and the industrial engineering. In the case of information systems, it must be noticed that this domain was always source of innovation in enterprise modelling whether it is through already old methods such as MERISE, more recent such as PROCESS UNITED or works such as those led by the OMG (Object Management Group) or the WfMC (Workflow Management Coalition), which have an incidence on enterprise modelling, in particular within the framework of the integration of industrial applications.

Within the framework of industrial engineering, the first developments were led in the United States in the 70s and produced in particular SADT, SSAD, IDEF0, Data Flow Diagram [START_REF] Doumeingts | GIM, GRAI Integrated Methodology -A Methodology for Designing CIM Systems[END_REF][START_REF] Doumeingts | Methodologies for designing CIM systems: A survey[END_REF]. Concerning Europe, the programs of the European Commission widely allowed developing and spreading these tools, this since the first Framework Program of Research and Development (Esprit program, the 80s). Since then, a large number of languages of enterprise modelling appeared in North America and in Europe. We can quote for example MERISE [START_REF] Tardieu | La méthode MERISE, Principes et outils[END_REF], GRAI [START_REF] Chen | GRAI integrated methodology and its mapping onto generic enterprise reference architecture and methodology[END_REF], [START_REF] Vallespir | Modélisation d'entreprise et architecture de conduite des systèmes de production[END_REF]Doumeingts 2007a, 2007b), NIAM [START_REF] Abrial | Data Semantics in Data Base management[END_REF], M* (Di [START_REF] Di Leva | Information system analysis and conceptual database design in production environments with M *[END_REF], CIMOSA (AMICE 1993), [START_REF] Kosanke | CIMOSA: enterprise engineering and integration[END_REF], [START_REF] Vernadat | The CIMOSA Languages[END_REF], [START_REF] Zelm | CIMOSA: a primer on key concepts, purpose and business value[END_REF][START_REF] Zelm | Enterprise modelling -constructs comparison (state of the art)[END_REF], OMT, IEM/MO2GO [START_REF] Mertins | MO2GO: User Oriented Enterprise Models for Organisational and IT solutions[END_REF], IDEFx [START_REF] Menzel | The IDEF Family of Languages[END_REF], METIS and AKM [START_REF] Lillehagen | Active Knowledge Models and Enterprise Knowledge Management, in Enterprise Inter-and intra-organizational integration[END_REF] or ARIS Toolset [START_REF] Scheer | ARIS architecture and reference models for business process management[END_REF][START_REF] Scheer | Business process engineering: reference models for industrial enterprises[END_REF], 2000[START_REF] Scheer | ARIS -Architecture of Integrated Information Systems[END_REF].

At the end of the 80s, fewer new languages appear as if, due to being each other inspired, the variety of the used concepts reached a relative stability. The problem changed then. Based on the observation of their complementarity, it appears a need to move closer, to assemble even to integrate these languages to obtain more powerful and coherent methodological sets. It is the beginning of the era of convergence.

THE ERA OF CONVERGENCE

Several stages punctuate this convergence. It begins with the development of multi-languages methods, continues by the analysis of the methodological components and the standardization and next by the semantic fusion of the languages [START_REF] Vallespir | L'intégration en modélisation d'entreprise : les chemins d'UEML[END_REF][START_REF] Vallespir | Modélisation d'entreprise et architecture de conduite des systèmes de production[END_REF].

First stage of convergence: the interoperability of languages supported by modelling frameworks

The modelling languages allow to instantiate generic concepts within the framework of the modelling of the studied system. These formalisms are enough important in number to allow to cover all the requirements of the domain by taking into account:  The diverse objectives of the modelling (analysis of the performances of the system, upgrade of an existing system, design of a new system),  Considered points of view (physical system, control system),  Levels of analysis or questioning (functional or semantic level, level of components supporting these functions), and  Levels of detail (analysis of the system as a whole or focus on a precise point).

So, considering this, when a methodological approach aims at a wide cover of modelling, it uses several languages. The problem of the consistence of the various models is then put. A solution is to propose a modelling framework which characterizes the relative positioning of models, the overlapping areas and the mechanisms to go from one to another.

The first approach having put clearly this concept is certainly MERISE which, by proposing three levels of abstraction (conceptual, organizational / logical and realizational / physical) and two views (data and processes), leads to a modelling framework with six domains of modelling. Each of these domains uses a language (for the most abstract levels) or several in the case of the lowest abstract levels.

Other methods followed the same way. They propose different views in kind and in number, different levels but the underlying concept of modelling framework remains the same. Among these approaches, let us quote in particular GIM [START_REF] Doumeingts | GIM, GRAI Integrated Methodology -A Methodology for Designing CIM Systems[END_REF][START_REF] Chen | GRAI integrated methodology and its mapping onto generic enterprise reference architecture and methodology[END_REF], CIMOSA [START_REF] Amice | CIMOSA: Open Architecture for CIM[END_REF] and the Zachman framework [START_REF] Zachman | The Concise Definition of The Zachman Framework[END_REF].

Figure 1. The GIM modelling framework

The GIM modelling framework is composed of two axes (Figure 1):

 An abstraction levels axis based on a multi-strata approach and allowing the modelling of the system according to several semantic levels considering the required depth of analysis (conceptual level: no consideration about organization or techniques, structural level: consideration of the options of organization, realizational level: consideration of the technical options); and  A views axis which allows to take into account several points of view stemming from a systematic decomposition (for the upper part -user oriented): decisional view, informational view or physical view to which a functional view is added, allowing to present a simple model of the studied system and the relations with its environment.

The well-known CIMOSA cube follows the same type of reasoning with a three-dimension framework (Figure 2):

 The Generation dimension corresponds to the same concept as the GIM views and is composed of Function, Information, Resources and Organization;  The Derivation dimension is linked to the life cycle of the system with Requirement definition, Design specification and Implementation description and is closed to the GIM abstraction levels; and  The Instantiation dimension goes from the language to the specific model with Generic, Partial and Particular.

The CIMOSA cube has evolved within the ISO 19439 standard (ISO, 2006). Unlike GIM and CIMOSA, The Zachman framework does not propose any modelling languages. So, it does not really participate to the convergence and the interoperability of modelling languages.

Figure 3. The Zachman framework

Second stage of convergence: the macroscopic integration (GERAM)

GERAM (Generalised Enterprise Reference Architecture Methodology) was developed by a task force on the architectures for the enterprise integration (IFAC/IFIP Task Force) (GERAM 1999) based mainly on CIMOSA, GRAI/GIM and PERA [START_REF] Williams | SLMToolBox: enterprise service process modelling and simulation by coupling DEVS and services workflow[END_REF].

GERAM proposes an organization of the main concepts constituting the existing approaches. The final result is an articulation of a set of components (languages, structured approach, reference models, etc. -Figure 3) allowing to cover domains related to enterprise engineering and integration. Most of these components are based on works already widely recognized by enterprise modelling.

For example, GERA (Generic Enterprise Reference Architecture) supplies an analysis and modelling frame the structure of which was widely inspired by CIMOSA and PERA. Besides, EEM (Enterprise Engineering Methodology) makes a reference to modelling methodologies such as GIM or PERA. The latter use themselves languages sometimes supported by software tools gathered in EET / EML (Enterprise Engineering Tools / Languages). IDEF, IDEF3, CIMOSA and the GRAI grid on one hand and ARIS Toolset, FirstSTEP, MOGO (IEM) (Mertins and Jochem, 1998) on the other hand appear among the languages and the tools which may be selected.

The concepts are formalized in the form of ontologies, of meta-models or of glossaries in GEMC (Generic Enterprise Modelling Concepts). The models are, if possible, included in PEMs (Partial Enterprise Models). Finally, EMOs (Enterprise Modules) and EMs (Enterprise Models) contribute to the representation of all or part of the company and are used to guide the implementation of the operational systems of the company (EOS -Enterprise Operating Systems) and to estimate the current and future performances as well.

In conclusion, GERAM, being inspired by existing approaches, proposes an organized set of relevant concepts of enterprise modelling and integration as well as their relations (Vernadat, 1999a), (Figure 4). So, it is possible to determine the positioning and the potentialities of any approach with regard to this set. However, it constitutes only a theoretical framework of metamodelling, non-operational thus and not dealing with the integration of the languages themselves.

Figure 4. GERAM -Set of enterprise engineering and integration components

The fact remains that reference tables common and accepted by all are essential to succeed in federating the modelling approaches. This is why the standardization works constitute an important stage of the reflection.

Third stage of convergence: the standardization

To increase the quality of systems and propose work reference tables, numerous organizations as CEN (European Standardization Committee), ISO (International Organization for Standardization), ISA (Instrument Society of America), IEEE (Institute of Electrical and Electronics Engineers), OMG (Object Management Group) and the OAG (Open Application Group) defined a set of standards based widely on the concept of "industrial consensus" [START_REF] Shorter | Enterprise Engineering and Integration -The Standardization Scene[END_REF].

These bodies and working groups for standardization [START_REF] Chen | Standardisation on Enterprise Modelling and Integration: Achievements, on-going Works and Future Perspectives[END_REF]) developed frameworks and basic constructs [START_REF] Panetto | Unified Modelling Language for meta-modelling: towards Constructs definition[END_REF], Chen et al. 2002a[START_REF] Chen | Enterprise-control system integration -an international standard[END_REF]) now reused and applied in related domains (ISA 95 for example). So, the concepts necessary for the modelling of the activity, the functions, the organization or the resources of a company, but also the rules of construction and manipulation of these concepts, are now established and approved by a more and more wide industrial, scientific and academic community. At the international level, the main actor in the field of enterprise modelling and integration is the technical committee ISO TC184 SC5/WG1 and CEN TC310/WG1. Figure 5 gathers the main CEN and ISO standards of the domain. It is generally recognized that there are too many heterogeneous modelling languages available on the market. So, it is difficult for the users to understand them really and to choose the most adequate. Furthermore, every language has its own syntax, textual or graphic. Finally, even if they possess a defined syntax, most of the languages do not possess a clear semantics. However, as we have already mentioned, the concepts on which these various languages are based are similar or differ only in detail.

The initiatives around UEML (Unified Enterprise Modelling Language) [START_REF] Petit | Formal Semantics of Enterprise Models[END_REF][START_REF] Geram | GERAM: Generalised Enterprise Reference Architecture and Methodology[END_REF]) have turned to a consensus between the various actors and the various standardization bodies working in the field of enterprise engineering and integration. The result would be the development of a high-level pivot language allowing several modellers to exchange information, data and knowledge contained and already taken into account within their own models established by means of languages, of basic constructs and different methods (CIMOSA, GRAI, etc.). However, it is not obvious to get a really unified language that answers to all the problems of modelling that could occur. So, it is not possible to ignore the approach which would consist in making only communicate the computing tools rather than to integrate the languages.

This approach is based on the principles of integration of software tools by the data that they exchange. The current enterprise modelling languages would not then be questioned. This integration can basically correspond to the exchange of neutral files (said low integration as proposed in the past to describe the architectures of the control systems (PTA 1992)), with all possible losses or bad understandings of the information. For example, the PRIMA II ESPRIT project (EP 20775) allowed, by focusing only on CIMOSA, to define neutral formats of exchange by using the XML standard [START_REF] Salvato | Presentation and Exchange of Business Models With CIMOSA-XML[END_REF]. The integration of tools can also be based on a semantic identity of the data and information processed by tools (said strong integration as proposed to model and exchange data products [START_REF] Bouaza | La norme STEP[END_REF]). The main problem is then to define these semantic links. This objective can be reached only with metamodelling and the definition of constructs.

The development of a unified enterprise modelling language seems thus inescapable. The expected result has not for objective to be a new language that must replace those existing and at present used but rather to be capable of interpreting the latter. UEML could be an "Esperanto" (Vernadat 1999b[START_REF] Vernadat | UEML: Towards a Unified Enterprise Modelling Language[END_REF][START_REF] Vernadat | UEML: Towards a Unified Enterprise Modelling Language[END_REF] in the field of enterprise modelling and engineering. It does not establish the ultimate enterprise modelling language replacing all the precedents but a standard meta-model (with its associated ontology) widely accepted by the users as by the developers of software tools. In other words, UEML does not aim at being integrated into the analyst's toolbox. Then, it is not constrained by operational usability criteria [START_REF] Vallespir | Problems and Research orientation for UEML: A point of view[END_REF]. Thus, UEML would be a compatible language with the most used operational languages.

In this sense, UEML can become a general conceptual base. The existing modelling languages become then operational interfaces (a projection of UEML in a specific semantic plane, Figure 6, Chen et al. 2002b). It means implicitly that UEML covers more than the individual modelling languages do. UEML so appears as the union of the existing languages. The bottom-up approach begins with the analysis of the existing enterprise modelling languages. The approach can be structured into four stages:  Choice of the existing enterprise modelling languages (state of the art and then choice of languages according to criteria established as usability, recognition, etc.),  Decomposition of the chosen formalisms into elementary concepts,  Union of the elementary concepts, and  Fusion and establishment of a unique syntax.

The advantage of the bottom-up approach is to be fast and to avoid rediscovering what already exists. The inconvenience is that nothing insures that the chosen languages are representative.

The top-down approach is based on a conceptual analysis. It can be also structured in four stages:

 Precise definition of the needs and domain,  Choice of paradigm (for example: systems theory),  Definition of elementary concepts, and  Definition of a syntax around these concepts.

Seman c plane corresponding to a given opera onal language UEML projec on into a given opera onal language

UEML model

The advantage of the top-down approach is the theoretical coherence. The problems are the time of development and the fact that existing formalisms risk certainly to be redefined.

An interesting compromise between these two approaches consists in using a hybrid approach that seems better adapted to the development of UEML. It is composed of the following stages:  Precise definition of functional needs that UEML has to match,  Choice of the existing languages in coherence with the required features and, at the same time, choice of one paradigm to ensure the theoretical coherence,  Definition of a set of elementary concepts obtained by decomposition of the chosen languages,  Union of the elementary concepts and elimination of the redundancy, and  Definition of the syntax around the elementary concepts.

It is admitted that the development of UEML would contribute to a better interoperability and a capacity of communication between modelling actors in heterogeneous environment. This is the main objective. However, this development presents interests that exceed this operational objective because it allows:

 A clear definition of the common semantics of the languages and a better demarcation of the domain of enterprise modelling and engineering;  A better definition of the scientific corpus of enterprise modelling and engineering and thus an increase of their visibility within the framework of the scientific community;  A vocabulary accepted and used by standardization bodies at every level (national, European and international) working in the domain.

This observation is generalizable to all the points developed in this chapter about convergence.

The modelling frameworks allowed to establish modelling domains in connection with the underlying modelling paradigms (systems theory for example); GERAM allowed to define the big methodological components and to answer consequently the question "what is an enterprise modelling method?"; the standardization participated in the organization of the concepts and the stabilization of the terminology and UEML is the opportunity to work and to organize semantics of the domain. At the end, this era of convergence is the era of the conceptual and semantic structuring of the domain.

In conclusion of this part, meta-modelling and models transformation explicitly arose within UEML initiatives and stay a hard issue. Even if UEML did not match the intended results because of a methodological lack at the level of meta-modelling, this domain remains a scientific challenge because, the models transformation is at the base of several current developments on enterprise modelling. The next chapter will present some of them.

CURRENT DEVELOPMENTS BASED ON MODELS TRANSFORMATION

Three approaches based on model transformation are described hereafter. They are:  model driven approaches which aim to ensure the progressive modelling of the system from business to execution,  the works on enterprise interoperability, and  the combined use of enterprise modelling and simulation aiming at verifying the performance of the future system corresponding to the model.

Model driven approaches

Nowadays, a major topic frequently encountered in companies is the gap between the strategic vision in the process running and the technical activities and tools that are really implemented to do so. This alignment of the technical aspects with the strategic ones opposes clearly the business view to the technical view.

To combine both views and to separate the concerns, several approaches have been developed for around fifteen years under the label of model driven approaches. The approaches described below use enterprise models and then require model transformation to ensure the coherence between modelling level from the business view to the simulation.

The Model Driven Architecture (MDA) is the most illustrative even if some approaches were developed before with the same spirit of modelling levels. However, MDA can come in adaptations to specific situations.

4.1.1. Model Driven Architecture MDA (OMG 2005) was defined and adopted by the Object Management Group (OMG) in 2001. It is designed to promote the use of models and their transformations to consider and implement different systems. It is based on an architecture defining four levels, which goes from general considerations to specific ones:  CIM Level (Computation Independent Model) is describing the system and its environment. It contains function, decision and process models independent from implementation.  PIM Level (Platform Independent Model) models the system sub-set that will be implemented.  PSM Level (Platform Specific Model) focuses on parts of the development platform.  Coding Level consists in coding Enterprise Software Application.

To complete this description, a Platform Description Model used for the transformation between PIM level and PSM level is added to these four kinds of models corresponding to four abstraction levels.

However, the idea of MDA to propose a multi-level architecture is not new. As presented previously in this paper, it was also at the origin of CIMOSA with the requirement / design / implementation levels using model derivation and model transformation. This spirit was also proposed in MERISE with conceptual / organisational-logical / realizational-physical levels and in GIM with conceptual / structural / realizational levels.

MDA is at the moment significantly used for IT service implementation. In major cases, the final stage of the MDA or other MD Development method is the generation of a Service Oriented Architecture SOA. An execution workflow language can be used like BPEL or simulation components. Obviously, MDA is an approach that must privilege enterprise modelling. Nevertheless the development is mainly IT-directed and the models are already built with this IT goal without placing always the users point of view and requirements at a central position. Moreover, in MDA, no real modelling languages are promoted at each level, leading the practitioner to develop his own architecture of models.

The Model Driven Service Engineering Architecture (MDSEA)

MDSEA is inspired from MDA. This methodology was developed in the frame of the FP7-MSEE project (Manufacturing Service Ecosystem) (MSEE 2012). One objective of MSEE was to facilitate the transition of companies from a traditional business of product manufacturing to a challenged business of Product and Service manufacturing. MDSEA provides an integrated methodology dealing with modelling languages at various abstraction levels to support Service models and Service System design and implementation. To better define and implement the service system supporting the service life cycle, it is again necessary to separate the preoccupations, from the user/business point of view to the technical point of view. The business point of view is more focused on the definition of the service product and service system, in particular from the business process, the decision and the information system modelling points of view, while the technical point of view is more focused on the progressive implementation of the service -product and service -system taking progressively the technical constraints into account. The MDSEA defines a framework for service system modelling based on three abstraction levels: BSM, TIM and TSM as well as the dedicated modelling languages at each level. Figure 7 presents the MDSEA.

Figure 7. MDSEA: Architecture for interoperability and alignment of service system Figure 7 shows a declination of MDSEA with two collaborative enterprises. Even if each level is detailed hereafter, the figure shows that the interest of such an architecture is on one hand to develop and implement at the same time a service -product and on the other hand a dedicated service -system coherent with business service models, represented by enterprise models. So, the consideration of BSM models for both entities at the same time ensures the interoperability at the model level that will then be ensured at the service system level, which is required to allow an efficient running of the service system network. It can be seen that from TIM to TSM level the methodology separates three kinds of elements. The resources to achieve the service are categorized into IT, Human and Physical Means in order to tackle the different consideration coming from the different natures of the resources at the implementation step of the service system. Then after implementing the cloud of services, it must be kept separated on one side the service -product that is proposed to the environment of the enterprise and on the other side the system to generate the services (the way to generate) that can be specific to the enterprise.

CONTINUM

Business Service Model (BSM). BSM specifies the system, at the global level, describing the service running inside a single enterprise or inside a set of enterprises as well as the links between these enterprises. These models can include business models to specify the value creation. The models at the BSM level must be independent of the future technologies and skills that will be used for the various resources. The languages used at this level must be simple to be used by non experts in modelling but also powerful to describe business aspects. The languages must represent the various points of view of service processes, service process control for the design and operation of the service system. At this level, MDSEA proposes to use GRAI Grid for decision modelling and Extended Actigram* (a process oriented language, Bazoun 2015) for business process modelling instead of BPMN in the sense that the language will not be used only for IT.

Technology Independent Model (TIM).

TIM delivers models at a second level of abstraction independent from the technology used to implement the system. It gives detailed specifications of the structure and functionality of the service system which do not include technological details. More concretely, it focuses on the operational details while hiding specific details of any particular technology in order to stay independent from any technology, used for the implementation. At TIM level, the detailed specification of the components of a service -system (as skills for a human resources or function of a machine or a function of a software) are elaborated with respect to IT, Organization/Human and Physical means involved within the production of the service. This is important to mention that in comparison to MDA or SOMA (Service Oriented Modelling and Architecture), the objective of MDSEA is not only IT oriented and then this requires enabling the representation of human and technical resources from the BSM level. At TIM level, the representations must add information in comparison to BSM models. Of course, the intention is to obtain models at TIM level using transformation mechanisms of BSM model. For instance, transformation mechanisms have been developed between extended actigrams at the BSM level and BPMN 2.0 at TIM level. At TIM level, it can be necessary to add detailed information. Also horizontal transformation mechanisms have been developed between BPMN 2.0 and DEVS still at TIM level (Zacharewicz et al. 2016).

Technology Specific Model (TSM).

TSM enhances the specifications of the TIM model with details that specify how the implementation of the system uses a particular type of technology (such as, for example IT applications, machine technology or a specific person with dedicated skill defined at TIM). At TSM level, the models must provide sufficient details to allow developing or buying suitable software applications, hardware components, recruiting human operators / managers or establishing internal training plans, buying and realizing machine devices, for supporting and delivering services in interaction with customers. For instance for IT applications, a TSM model enhances a TIM model with technological details and implementation constructs that are available in a specific implementation platform, including middleware, operating systems and programming languages (e.g. Java, C++, EJB, CORBA, XML, Web Services, etc.). Based on the technical specifications given at TSM level, the next step consists in the realization and the implementation of the designed service -system in terms of IT components (applications and services), physical means (machine or device components or material handling) and human resources and organization ensuring human related tasks/operations.

Enterprise interoperability: framework, maturity model and model driven approach

The current economic situation, favouring the networking of the economic players for the benefit of a collective project, leads enterprise modelling to be interested in the networks of technico-economic activities the size of which overtakes the statutory borders of the companies involved. In this sense, enterprise modelling plays a major role in the analysis of the complex transactions networks appearing within the theme of the extended enterprise.

Furthermore, as the evolution of the economic environment in which companies have to evolve leads to envisage permanently new organizations (decentralization, transformation of big size structures into networks of communicating companies, etc.), the proposed approaches must be multi-purpose to adapt itself to these various contexts and especially not be dedicated to a type of specific organizations. This situation leads to focus more on enterprise interoperability rather than enterprise integration. That is why interoperability raised up as an important challenge and a scientific domain since the beginning of this millennium.

There are many initiatives to develop solutions for enabling enterprise interoperability. This ranges from conceptual approaches focusing on frameworks, models and methodologies to practical experimentation with tools dealing with IT interoperability platforms, semantic interoperability, utility tools and models interoperability etc. A quite complete state-of-the-art on those approaches and initiatives have been developed under the three main European projects addressing enterprise interoperability: IDEAS (IDEAS 2002), ATHENA [START_REF] Athena | Advanced Technologies for Interoperability of Heterogeneous Enterprise Networks and their Applications[END_REF]INTEROP NoE (INTEROP 2003). In this section, focus is given to frameworks, maturity models and model driven approach.

Enterprise Interoperability Framework

Concerning the Enterprise Interoperability Framework development (state-of-the-art on existing works on interoperability frameworks reported by [START_REF] Chen | Architectures for Enterprise Integration and Interoperability: Past, Present and Future[END_REF][START_REF] Chen | Framework for Enterprise Interoperability and Maturity Model[END_REF], initial works such as the IDEAS Interoperability framework (IDEAS 2002), ATHENA interoperability framework (ATHENA 2003), European Interoperability Framework (EIF 2004) provide interesting structuration of various issues and concerns on enterprise interoperability. However they do not address explicitly interoperability problems, and do not allow structuring interoperability solutions in relation with the problems. Framework for Enterprise Interoperability (FEI), initially drafted under the frame of INTEROP NoE [START_REF] Interop | Enterprise Interoperability -Framework and knowledge corpus[END_REF] and then accepted as an international standard, is built on the existing ones.

The Framework for Enterprise Interoperability (FEI) has been specified in CEN/ISO 11354-1 (CEN/ISO 2009, Figure 8). It aims at structuring basic enterprise interoperability concepts and issues. The framework has three basic dimensions: Interoperability concerns that defines the content of interoperation that may take place at various levels of the enterprise (data, service, process, business), Interoperability barriers that identifies various obstacles to interoperability in three categories (conceptual, technological, organisational, as in GIM, Figure 1) and Interoperability approaches that represents the different ways to remove barriers (integrated, unified, and federated). Concerning the development of enterprise interoperability maturity model, there exist several approaches [START_REF] Guedria | Interoperability Maturity Models -Survey and Comparison[END_REF][START_REF] Guedria | A Maturity Model for Enterprise Interoperability[END_REF]. One can mention some earlier works such as LISI (Levels of Information Systems Interoperability) (C4ISR 1998), [START_REF] Kasunic | Measuring systems interoperability: challenges and opportunities[END_REF] The Maturity Model for Enterprise Interoperability (MMEI, Chen 2013) is based on the standard framework (FEI) and inspired from and influenced by some existing maturity models as mentioned in previous paragraph. MMEI has been specified in CEN/ISO 11354-2 (CEN/ISO 2011). It aims at enabling an assessment of an enterprise capability to interoperate with another enterprise. MMEI is built on the FEI described above and covers the whole problem space of the FEI (four interoperability concerns and three kinds of interoperability barriers). Five levels of interoperability maturity are defined as shown in Table 1.

Each level identifies a certain degree of capability to interoperate with another enterprise. The table also gives a high level view of MMEI and shows the main focus for each combination of each maturity level and for each interoperability barrier category. These interoperability barriers are already defined in the framework (FEI) by its interoperability barrier dimension (conceptual, technological, organisational). The Model Driven Interoperability (MDI) methodology was elaborated in the framework of the Task Group 2 (TG2) of INTEROP-NoE [START_REF] Bourey | Deliverable DTG2.3 from the INTEROP project[END_REF]) which proposed an approach, inspired from OMG MDA (see above). The goal is to tackle the interoperability problem at each abstraction level defined in MDA and to use model transformation techniques to link vertically the different levels of abstraction or horizontally to ensure interoperability of models at each level (Figure 9). The main goal of this methodology is to allow a complete follow-up from expressing requirements to solution coding and also a greater flexibility thanks to the transformations automation. Yet, MDI was focused on pure IT systems in conjunction with SOA.

An originality of this MDI framework is also to decompose the CIM level into two levels:  The TCIM (or TOP CIM) level aims at describing the enterprise from a business perspective using business models and classical enterprise modelling languages, and  The BCIM (or BOTTOM CIM) level aims at extracting from the TCIM models the part of the processes only dedicated to IT system.

The interest of this decomposition is to focus the transformation between CIM and PIM only on the IT part of the company, MDI being focused on this part. However, the main problem is to determine real criteria to define exactly the items that must be kept in the BCIM models.

Model transformation

As mentioned previously (chapter 4. After defining the meta-models and the description languages, a mapping between meta-models is built. This mapping can be characterized by a table defining constructs matching. Later on, the mapping rules described in the table will be implemented using Atlas Transformation Language (ATL).

The Eclipse "BPMN modeler" plugin is used to visualize and validate BPMN target models. It requires a specific xml format with graphical elements definition. To address this issue, we rely on XSLT transformation sheet which will transform the BPMN models (XMI format) resulting from the ATL transformation into xml format that conforms to the BPMN modeler requirements.

Mapping of concepts

The mapping of concepts proposed for the transformation creates correspondences and links between concepts and their relations from EA* to BPMN language. It is a translation of constructs and their relations from one meta-model to another one. As a result, deep analysis and understanding of the EA* and BPMN meta-models, represent the main issue to start the translation and drawing of the links. The mapping is accompanied with conditions to manage the creation of relations between concepts. The model transformation was developed in this case using ATL.

Case Study

The concept of electronic marketplace (e-marketplace) has been based on grouping new and/or used products coming from several sellers on a unique Internet platform and under the same ecommerce catalog. The result is a trading platform as the e-market itself does not sell nor buy directly physical products or services traded on the platform. On one hand e-marketplaces and supplier directories are B2B Internet platforms. On the other hand, e-marketplaces offer a portal for B2C interface. Several enterprises took this opportunity to extend their offer. In France, major e-selling platforms have initiated e-commerce configurations since 2010 and are moving progressively to integrate this service offer. This selling concept fits the service orientation. It proposes two service interfaces, one dedicated to sellers and the other one to customers. Dedicated services can be provided. On the B2B side, the broker can choose or remove new merchants for his platform based on existing performances on the platform as visibility, customer satisfaction, low cost delivery, etc. A solution of export product catalogue is embarked on each marketplace for a win/win strategy between merchant and broker. The merchant can reuse its catalogue and be part of several marketplaces.

In this context, the case study presents one of the processes run within an e-marketplace collaborative network. It details the process of purchasing products using a marketplace website. The marketplace is maintained by a broker agent that offers services to customers who choose, configure, and buy their products online. On the other hand, sellers are targeting customers and selling their products via the broker. As a result a collaborative network, formed of the broker website, sellers, and delivery companies, offers a service to online customers. The goal of this section is to demonstrate the support provided by the tool described previously. The business model is assisted in its transformation for generating the service platform that will be implemented.

Figure 11 is a transformation example of a private sale e-marketplace purchase process model. It formalizes the business considerations captured with EA* language, then, in order to prepare the definition of the electronic platform, it transforms the model to BPMN 2.0 diagram. In detail, in this model the customer log in with his user account, browses available brands, chooses a brand, browses brand available product, and configures his product (colour, size, etc.). When the customer terminates the configuration, the broker agent verifies product's availability and delivery details from the seller company. These details will be transferred to the customer, who will decide to validate his basket or not. Then he can either pay for his products or go back to choose other products. The figure shows that the tool has been able to identify the partners and to isolate the service they solicit or generate. In the BPMN model, the lanes represent on the upper side the B2C link and service required. On the lower side the collaborative network within a B2B relation where a competition is done between sellers to provide better proposition to customer demand (eg. about the delivery time and price). The fact that the electronic service is not ready at this step needs to be taken into account.

Simulation

The question regarding models and then implementation of systems and solutions is related to the time required to validate the approaches and to obtain an ROI on the real system. This time could be too long considering the project duration. For this reason, traditional enterprise specifications based on conceptual modelling such as enterprise modelling benefits more and more from simulation techniques in helping to extend the models and overpassing design flaws, by adding dynamic dimension. It helps learning system behaviour, providing training and correcting potential problems before a final implementation of the real system. It is becoming a standard practice for building complex systems.

Simulation is still mostly used for running models of enterprise processes or anticipating decision level forecast. These categories of simulation are different by nature targeting different audience. They are not used in combination or coupled at execution time. Nevertheless, the current demand within the context of industry 4.0 reports that coupling of real systems and virtual parts (with IoT, virtual reality, CPS) pushes enterprises using more embedded, interoperable and loosely coupled simulations and organizational models to anticipate the correct behaviour of the whole system. To make these benefits happen, modelling and simulation have to be aligned in the enterprise systems development lifecycle in a model driven approach (vertical alignment) and interoperable with other simulations (horizontal alignment).

For instance, efforts have been done in proposing simulation as an integrated step in the MDSEA cycle (Zacharewicz et al. 2016). In this approach, the model transformation concerns the transformation of BPMN 2.0 models at the TIM or TSM level into DEVS Models directly usable for simulation. The originality of this transformation coupled with MDSEA is to reuse the existing models instead of developing new simulation models from scratch and then to verify the hypothesis of BSM models in terms of performance improvement.

At the technical level (such as MDSEA TSM), in the current context of open-source software development, service-oriented architectures (SOAs), HPC and Model Driven Approaches, modelling and distributed (cloud) simulation will assume even more significance in assessing business process management effectiveness, alignment between business process models and configuring future enterprise information system services [START_REF] Wang | Use of Fuzzy Clustering for Discrete Event Simulation Model Construction[END_REF]. Alignment for interoperability between the client and vendor business strategies in a Supply Chain ecosystem remains critical too. These considerations are identified by MDI.

Despite these efforts, modelling and simulation are still mostly used as offline tools in isolation from each other. Also the decision level is not connected gathering at run time with process information or either the operative level is also not fully able to feed with live inputs and to receive live outputs from simulation. In the future, the current advances can already promise that coupling between real and virtual at run time will be enhanced thanks distributed simulation and IoT. Then the most important challenge concerns the integration of the human in the simulation loop. The human can be "augmented" thanks to the results of simulation but also viewed as an actor of the simulation process to be trained. No doubt that the model driven approaches will facilitate to close the loop from the simulated part to the real world.

CONCLUSION AND FUTURE WORK

During the last thirty years, enterprise modelling has been recognised as an efficient tool to externalise the knowledge of the company in order to understand its operations, to analyse its running and to design new systems from several points of view: functions, processes, decisions, resources, information technology. This paper has shown that enterprise modelling methods evolved all along this period, based on theoretical concerns but also on practitioner matters. After a phase of divergence with many developments in parallel and then attempts of reconciliation (from modelling frameworks towards UEML), this is now the phase of usage when enterprise modelling is less and less used as a stand-alone approach but rather as a component of larger engineering methods (such as MDA and enterprise engineering). Several model driven approaches have been defined and experimented from business modelling through simulation in order to get an efficient usage of these methods. Then the coupling of enterprise modelling with simulation is a very promising approach. In this way, because the system is not only considered during its design phase but also during its exploitation, enterprise modelling does not stay as an approach usable only during the design time but moves closer to runtime. This trend is fully consistent with the industry 4.0 objectives. In this sense, enterprise models move to be used during the total enterprise life cycle. This trend provides a very exciting scientific challenge for the next future.

In this paper, we tried to show that many current challenges address the models transformation:  Translation of a model at a given abstraction level into a model at a lower level within the framework of systems design (vertical transformation, MDA),  Translation of static models in simulated models,  Translation of models from a language to another one (horizontal transformation),  Establishment of the coherence of models related to different semantics (interoperability).

So, we consider that the main challenge today concerns the models transformation and, more precisely, the development of methods of transformation that allow a fluid and fast process with verifiable results.

The models transformation is based on a meta-modelling activity. So, behind the problem of the models transformation, and in a more theoretical perspective, the challenge concerns the development of meta-modelling methods which would allow to meta-model a language in a systematic way. It is to be noted that these methods do not exist today and that the metamodelling activity is a work burdensome, specific to each situation and with an uncertain result. This domain constitutes a real domain of research for the future years.

Figure 2 .

 2 Figure 2. The CIMOSA cube (left) and its evolution within the ISO 19439 standard (right) The Zachman framework (Hay 1997) has two dimensions (Figure 3):  The first one is based on the same concept than the GIM View dimension or the CIMOSA Generation dimension and is composed of Data (answering the "What?" question), Function (How), Network (Where), People (Who), Time (When) and Motivation (Why); and  The second dimension is closed to the GIM Abstraction levels dimension or the CIMOSA Derivation dimension with Objectives / Scope, Model of the Business, Model of the Information System, Technology Model, Detailed Representation and Function System.

Figure 5 .

 5 Figure 5. Main CEN and ISO standards of the enterprise modelling domain 3.4. Fourth stage of convergence: the microscopic integration of languages (UEML)

Figure 6 .

 6 Figure 6. UEML position with regard to the operational languages (projection) Several approaches are possible for the development of UEML: bottom-up, top-down or hybrid (Chen et al. 2002b).

Figure 8 .

 8 Figure 8. The Framework for Enterprise Interoperability 4.2.2. Maturity Model for Enterprise Interoperability

  , OIM (Organizational Interoperability Model)[START_REF] Clark | Organisational interoperability maturity model for C2[END_REF], OIAM (Organisation Interoperability Agility Model)(Kingston et al. 2004), LCIM (Levels of Conceptual Interoperability Model)[START_REF] Tolk | The levels of conceptual interoperability model[END_REF], EIMM (Enterprise Interoperability Maturity Model)[START_REF] Athena | Advanced Technologies for Interoperability of Heterogeneous Enterprise Networks and their Applications[END_REF]. Existing maturity models focus, in most cases, on one simple facet of interoperability (data, technology, conceptual, enterprise modelling, etc.). They are complementary rather than contradictory. Consequently, it is necessary to structure the different approaches into one single interoperability maturity model to avoid redundancy and ensure consistency.

Figure 9 .

 9 Figure 9. The Model Driven Interoperability framework

  Figure 10. Transformation architecture of EA* into BPMN The first step consists in the identification of the source and target meta-models (respectively Extended Actigram Star and BPMN2.0 meta-models), and the languages used for the model representation. The ecore meta-model is used as a meta-meta-model which defines the structures of the Extended Actigram star and BPMN meta-models. Both EA* and BPMN2.0 meta-models are defined in ecore. In addition XML Meta-data Interchange (XMI) is used to save source and target models.

Figure 11 .

 11 Figure 11. Example of model transformation from EA* to BPMN[START_REF] Bazoun | Contribution to a methodology for service systems modeling and engineering through a model driven approach: architecture, transformation and model simulation[END_REF]

Interoperability at the service system level Service Product Service System Service Product Service System

		Enterprise A Decision Business Extended Actigram * GRAI Grid		Business Extended Actigram *	Decision Enterprise B GRAI Grid
		Business Services Models (BSM) Processes Processes	Interoperability at	Business Services Models (BSM) Processes Processes
						the model level
			Organisation Human Domain Technology Independent Models (TIM) Physical means Domain IT Domain BPMN 2.0		Organisation Human Domain Technology Independent Models (TIM) Physical means Domain IT Domain BPMN 2.0	CONTINUM
			Technology Specific Models(TSM)		Technology Specific Models(TSM)
	Generation of "components"	BPMN 2.0		Generation of "components"	BPMN 2.0
	( IT_ Organisation/	Human_Physical	means		( IT_ Organisation/	Human_Physical	means
				Services in virtual enterprises		Services in virtual enterprises
				(IT Applications, Processes, Products	,		(IT Applications, Processes, Products	,
				Services, Organisation/Human	, Physical		Services, Organisation/Human	, Physical
				Means(machine, robots), etc…)			Means(machine, robots), etc…)

Table 1 .

 1 Maturity Model for Enterprise Interoperability[START_REF] Chen | Framework for Enterprise Interoperability and Maturity Model[END_REF]

	Maturity Levels/ Barriers	Conceptual	Technological	Organizational
	Level 4 -Adaptive	Accommodated	Reconfigurable	Agile
	Level 3 -Organized	Mapped	Open-architecture	Trained
	Level 2 -Aligned	Adhered	Arranged	Flexible
	Level 1 -Defined	Modelled	Connectable	Specified
	Level 0 -Unprepared	Incomplete	Inaccessible	Inexplicit
	4.2.3. Model Driven Interoperability