Pierre-Antoine Bouit

Dr Kenji Kamada

Dr Patrick Feneyrou

Gérard Berginc

Dr Loïc Toupet

Olivier Maury

Dr Chantal Andraud
email: chantal.andraud@ens-lyon.fr

Dr P.-A Bouit

Two-photon absorption related properties of functionalized BODIPY dyes in the infrared range up to telecommunication wavelengths**

Keywords: BODIPY, Two-photon absorption, Near infrared, Telecommunication

wavelengths, nonlinear transmission Two-photon absorption (TPA) is a resonant third-order nonlinear optical (NLO) process, in which an electron is promoted from its ground state to an excited state by simultaneous absorption of two photons of half-energy in a focused intense light beam like that generated by a laser source. This phenomenon, theoretically predicted in 1931 by Göppert-Mayer, [1] has been experimentally evidenced in 1961 in a TPA-induced fluorescence of Eu 2+ doped CaF2 crystals under ruby laser irradiation. [2] Since then, the intrinsic advantages of TPA excitation i.e. long wavelength and confocal three-dimensionally (3D) resolved absorption, led to numerous applications in the fields of material sciences and biology following the availability of tunable laser sources. In particular the development of fs-Ti-sapphire laser triggers numerous studies in the 700-1000 nm spectral range. For instance TPA was successfully involved in the fabrication of microstructures [3][4] or high density optical memories [5] with a 3D sub-micrometer resolution, in signal processing or for the design of optical limiting devices devoted to the protection of sensors against laser damages. [6][7] In life sciences, the long wavelength TPA excitation is located in the biological transparency spectral range (800-1000 nm), and therefore is able to penetrate more deeply biological tissues. Combined with the above mentioned spatial resolution, TPA found exciting applications for bio-imaging using nonlinear microscopy [8] but also for drug delivery or dynamic phototherapy. [9] All these applications encouraged the scientific community to design a large variety of chromophores with optimised two-photon cross-section (2) in this 700-1000 nm spectral range, e.g. organic dyes, organometallic or coordination complexes, polymers, dendrimers or nanonoparticles and quantum-dots, compiled in a recent exhaustive review. [10] However, the near infrared (NIR) spectral range has been less studied and in particular the telecommunication wavelengths spectral range (1.3-1.55 m) where applications in signal processing (optical power stabilization, pulse suppression, optical limiting) are very attractive. Since 2005, under the impulsion of the groups of Marder-Van Stryland and Osuka-Kim, there is a great interest for the design of chromophores exhibiting strong TPA properties in the NIR. Maximal TPA cross-section of ca. 1500 GM at 1.44 m and 1600 GM at 1.3 m have been reported for dipolar and quadrupolar squarine type chromophores respectively, [11][12] whereas nickel(bisdithiolene) complexes [13] and singlet diradical systems [14] exhibit significant 2 over the entire telecommunication range. On the other hand, numerous fused-, extended-or organized-porphyrin complexes have been reported with very large TPA cross-section ranging from 10 3 up to 10 4 GM. [15][16][17][18][19] Finally we reported the TPA properties of cyanine-type chromophores (2 of 750 GM at 1450 nm) that present enough solubility to carry out additional nonlinear transmittance experiments at telecommunication wavelengths. [20] Herein we report the synthesis, X-rays structures of aza boron-dipyrromethane (BODIPY) NIR dyes functionalized in the  position by donor--conjugated systems and their TPA and nonlinear transmittance properties in the 1.2-1.6 m spectral range.

It is worth noting that BODIPY is a versatile class of chromophores [21][22] largely studied for its exceptionnal luminescence properties and used as biomolecular labels, sensors (pH, NO, Hg 2+ , toxin…) or photonic therapeutic agent. [21][22][23][24][25][26] Great efforts have been recently devoted to shift their spectroscopic properties to the NIR spectral range by (i) introduction of electron donating moieties, [27] (ii) rigidification of the structure [28] or (iii) replacement of the mesocarbon by a nitrogen atom (the "aza"-BODIPY). [29][30][31] But surprisingly, whereas water soluble conventional BODIPY dyes featuring low two-photon cross-section were used for two-photon imaging microscopy, [32][33] these compounds were not fully optimized for TPA applications.

To the best of our knowledge, only multichromophoric dyes containing two or three BODIPY units and exhibiting rather modest 2 values (75 GM at 990 nm) in the visible were described in the literature. [34] The synthesis of the target chromophores B 1-2 (Scheme 1), featuring extended -conjugated skeleton involves as key step a copper free Sonogashira cross-coupling using Pd(PPh3)4 as catalyst between bromo-functionalized aza-dipyrromethene [START_REF] Loudet | Very recently, an iodo analogous of B 0 has been described[END_REF] and various donorphenylenethynyl derivative (D = OHex, D= NHex2) followed by classical complexation with boron trifluoride etherate in basic media (Scheme 1). The intermediate synthon 3 and the corresponding complex B 0 are prepared using the procedure described by the O'Shea's group. [36] Synthetic details and complete characterization are compiled in supporting information. It is important to note that the synthesis of B 0-2 was carried out at a gram scale and B 2 is enough soluble in chlorinated solvent to envisage further nonlinear transmittance experiments. [20] Crystals of B 0 suitable for X-ray diffraction analysis were obtained by slow diffusion of diethylether into a dichloromethane solution. As already observed for other aza-BODIPY dyes, [31,[START_REF] Loudet | Very recently, an iodo analogous of B 0 has been described[END_REF][37][38] the structure (Figure S1 All the linear and nonlinear properties were studied in dichloromethane solution and are summarized in Table 1. Absorption spectrum (Fig. 1) of B 0 is characteristic of aza-BODIPY with an intense sharp transition (1/2 = 1270 cm -1 ) exhibiting a shoulder at higher energy. The extended -conjugated B 1,2 compounds present transitions whose red shift increases with the strength of the donor group ( = 37 and 87 nm for B 1 and B 2 , respectively). It is worth noting that this shift is accompanied by a significant broadening of the absorption band (1/2 = 2850 cm -1 for the amino derivative). Open aperture Z-scan measurements were performed in the 1200-1600 nm spectral range using a fs-optical parametric amplifier pumped by a Ti:sapphire source. At such short pulse duration (130 fs), only TPA process is considered and the propagation equation can be simplified as:

 dI /dz   2 I 2 (1)
where I represents the light intensity, z the propagation axis and 2 the TPA coefficient. As described previously, [20] the 2 coefficient can be determined from the molecular two-photon cross-section 2 by:

  2  h 2 /N (2)
Carrying these fs-Z-scan experiments at various wavelengths allowed the determination of the TPA spectra of B 0,1,2 over the entire 1200-1600 nm spectral range (Fig. 2). As expected for non-centrosymmetric compounds, each TPA spectrum matches the wavelength doubled linear one, indicating that the same transition is one-and two-photon allowed. At these wavelengths, the unfunctionalized B 0 chromophore does not present any TPA response in the contrary of the donor--conjugated substituted systems B 1,2 (Table 1). Whereas B 1 presents a rather small NLO activity, B 2 shows significant values (around 600 GM) over the broad 1300-1450 nm spectral range. Furthermore, at lower wavelength, a significant increase is observed with a maximal two-photon cross-section max 2



of 1070 GM at 1220 nm. This value is close to that of cyanine compounds [20] and in the same order of magnitude than that of dipolar or quadrupolar derivatives. [11,12] In addition to the high solubility, exceptional thermal stability (Td10 about 360 °C) and the large scale availability of B 2 allowed to perform nonlinear transmittance experiments in the telecommunication spectral range. Such experiments were carried out for B 2 at highly concentrated solution (c = 0.1 Mol.L -1 ) using a ns-OPO pumped by a frequency tripled Nd:YAG laser source. [20] Over the 1350-1500 nm range, the nonlinear transmittance curves vs. the laser fluence present characteristic nonlinear transmission properties; this behavior is illustrated at 1350 nm in Figure 3: a good transmission is observed at low energy (T > 0.9), whereas beyond the nonlinear transmission threshold observed for a fluence of 0.7 J.cm -2 , the laser beam intensity is strongly attenuated up to 60% at a fluence of 7 J.cm -2 . Due to the longer pulse duration (7 ns), the simulation of the experimental curves based only on a TPA process failed (blue line, Figure 3) and a higher order effect related to a three-photon absorption, such as a TPA induced excited state absorption (ESA) was taken into account.

Consequently, as already observed in the NIR for cyanine dyes [20] or in the visible, [39][40] the propagation equation ( 1) becomes

 dI /dz   2 I 2   3 I 3 (3)
where 3 is the three-photon absorption coefficient. Numerical resolution of (3), for temporally and spatially gaussian incident pulses and using the TPA coefficient previously measured by fs-Z-scan, allowed the determination of a 3 coefficient of 620 cm 3 .GW -2 at 1350 nm (Table 1). This value corresponds to a two-fold increase compared to that of cyanine dyes measured in the same condition of concentration (230 cm 3 .GW -2 at 1520 nm). [20] This enhancement, which cannot be interpreted only by a stronger TPA efficiency, could be explained by a strong ESA contribution. This is illustrated in Figure S1 in supporting informations by the comparison of the spectral distribution of  and 3 between 1350 and 1500 nm: 3 decreases from 600 to 300 cm 3 .GW -2 , while 2 remains nearly constant. In conclusion, this study clearly underlines the high potentialities of functionalized aza-BODIPY dyes for TPA related applications in the telecommunication spectral range and in particular for nonlinear transmission. The versatile nature of this class of chromophores opens the way for a fine chemical engineering in order to optimize the TPA properties in the NIR by

4 ,Scheme 1 .

 41 Scheme 1. Synthesis of chromophores.

  in supporting informations) shows three very planar central fused rings and four twisted peripherical phenyl groups (twist angle of 12-43°); the boron atom lies in an almost perfect tetrahedral environment and the C-N bond lengths between bridging nitrogen N2 and the two pyrroles C4 and C5 are identical (1.322(3) and 1.326(3) Å, respectively), showing the conjugated nature of the chromophore.

Figure 1 .

 1 Figure 1. Absorption spectra of B 0 (dashed), B 1 (gray) and B 2 (bold).

Figure 2 .

 2 Figure 2. TPA spectra measured by fs Z-Scan method (filled circle) and linear absorption spectra in upper abscissa ( ____ ) for B 1 (up) and B 2 (bottom).

Figure 3 .

 3 Figure 3. Transmittance variations with the laser fluence in a DCM solution of B 2 at a concentration of 106 g.L -1 ; experimental data points (blue circle), theoretical simulation using

The authors thank the Direction Générale de l'Armement for a grant to PAB

additional functionalization of the  position or by rigidification of the lateral -skeleton for example.

Experimental

[CCDC-668638 contains the supplementary crystallographic data for this paper. These data can be obtained free of charge from The Cambridge Crystallographic Data Centre via www.ccdc.cam.ac.uk/data_request/cif.]

[Further details of the crystal structure investigation(s) may be obtained from the Fachinformationszentrum Karlsruhe, 76344 Eggenstein-Leopoldshafen (Germany), on quoting the depository number CSD-...]


Vincenzo Sciancalepore

Marco Di Renzo

Xavier Costa-Perez

STORNS: Stochastic Radio Access Network Slicing

Recently released 5G networks empower the novel Network Slicing concept. Network slicing introduces new business models such as allowing telecom providers to lease a virtualized slice of their infrastructure to tenants such as industry verticals, e.g. automotive, e-health, factories, etc. However, this new paradigm poses a major challenge when applied to Radio Access Networks (RAN): how to achieve revenue maximization while meeting the diverse service level agreements (SLAs) requested by the infrastructure tenants?

In this paper, we propose a new analytical framework, based on stochastic geometry theory, to model realistic RANs that leverage the business opportunities offered by network slicing. We mathematically prove the benefits of slicing radio access networks as compared to non-sliced infrastructures. Based on this, we design a new admission control functional block, STORNS, which takes decisions considering per slice SLA guaranteed average experienced throughput. A radio resource allocation strategy is introduced to optimally allocate transmit power and bandwidth (i.e., a slice of radio access resources) to the users of each infrastructure tenant. Numerical results are illustrated to validate our proposed solution in terms of potential spectral efficiency, and compare it against a non-slicing benchmark.

I. INTRODUCTION

Upcoming service requirements from vertical industries call for a novel design of mobile networks, namely 5G. Keyenablers have been identified as network programambility and virtualization: the former brings the benefits of automation and reactiveness of software modules, allowing to (re)configure mobile networks dynamically while in operation; the latter overcomes the limitations of monolithic network infrastructures by abstracting the concept of "network function" and providing flexibility in composing, placing and managing these functions. In this context, the novel definition of network slicing [START_REF] Alliance | Description of network slicing concept[END_REF] encompasses such new requirements and constitutes an enabler for potential economical benefits. New vertical industries, e.g., automotive, e-health, factories etc., are entering into the telecom market and are disrupting the traditional business models of telecom operators. They are forcing infrastructure providers to open their networks to tenants, a solution that provides incentives for monetizing the availability of isolated and secure (virtualized) network slices [START_REF] Han | A Utility-driven Multi-Queue Admission Control Solution for Network Slicing[END_REF], [START_REF] Pries | Network as a Service -A Demo on 5G Network Slicing[END_REF].

This new disruptive concept has spurred research interest in both academic and industrial communities. Its realization, however, requires the solution of a number of technical challenges that, for the time being, are not completely resolved [START_REF] Foukas | Network Slicing in 5G: Survey and Challenges[END_REF]. In the future, telecom providers envision an increasing demand for end-to-end network slices, which involve heterogeneous service level agreements (SLAs) comprising different key performance indicators (KPIs), such as throughput, latency and reliability [START_REF] Sciancalepore | Mobile Traffic Forecasting for Maximizing 5G Network Slicing Resource Utilization[END_REF]. However, this requires appropriate automated admission control and resource allocation protocols for designing efficient network management systems [START_REF] Salvat | Overbooking network slices through yield-driven end-to-end orchestration[END_REF]. In particular, Fig. 1: Illustration of the Network Slicing Concept the resource management of the radio access network (RAN) is one of the most challenging aspects that needs to be dealt with. In [START_REF] Samdanis | From network sharing to multi-tenancy: The 5g network slice broker[END_REF], a first network slicing brokering solution has been presented for automated network slicing admission control decisions. We build on that architecture to illustrate in Fig. 1 some of the challenges that need to be solved for efficiently slicing the RAN. The available resources of the air interface can be sliced at multiple levels: in frequency, time, and power domains. Slicing at this level of granularity requires to account for the cellular network topology, the other-cell interference and the radio channel conditions experienced by the users of every single tenant [START_REF] Wen | On robustness of network slicing for next generation mobile networks[END_REF]. To solve this never-addressed and challenging issue, we leverage the mathematical tool of stochastic geometry and the theory of point processes [START_REF] Elsawy | Modeling and analysis of cellular networks using stochastic geometry: A tutorial[END_REF]. To the best of our knowledge, we pioneer the design of an automated RAN slicing admission control and resource allocation scheme that provides throughput guarantees in the RAN, where the cellular network topology and the other-cell interference are taken into account.

Our main research contributions can be summarized as follows: i) we derive a new formulation of the network spectral efficiency with the aid of stochastic geometry tools explicitly accounting for the interplay among the transmit power of the cellular base stations (BSs), the available spectrum and the deployment densities of cellular BSs and mobile terminals (MTs) of each tenant, ii) we study the analytical properties of the newly proposed utility function for slicing the RAN and mathematically prove its convexity, and iii) we design a RAN admission control and a novel STOchastic RaN Slicing mechanisms (STORNS). STORNS accounts for slice SLAs in terms of average throughput and assigns, in an automated fashion, time/frequency resources and transmit power levels to each slice of the tenants admitted into the sliced RAN.

II. SYSTEM MODEL

We explicitly account for the topology of cellular networks by using the mathematical tools of stochastic geometry and point processes [START_REF] Andrews | A tractable approach to coverage and rate in cellular networks[END_REF]. Under a stochastic geometry framework, in particular, the locations of BSs and MTs are modeled as points of a point process with some specific spatial properties, due to its tractability.

We consider a RAN with multiple access points (i.e., the BSs), such as long-term evolution (LTE) eNBs, femto-cells, mm-wave access points. Multiple tenants are available in the network. The generic infrastructure tenant, i ∈ I, is willing to pay for managing a "slice" of the resources of the RAN, provided that a certain slice SLA is guaranteed to it, e.g., a minimum average throughput requirement. Users u ∈ U i (i.e., the MTs) belong to a particular tenant i and are assumed to have density λ Ti . We consider a single network operator making available its physical resources to the multiple tenants. In particular, each tenant accesses the RAN resources so that its users share common resources of the (same) air interface with specific privileges. Specifically, the BSs of the cellular network operate in an open access mode for all the users of each tenant. However, each BS serves different tenants in a non-overlapping frequency band and by using part of its available total transmit power. As a consequence, each tenant owns a dedicated part of the spectrum and of the transmit power, which constitute the "isolated" slice of the physical resources requested to the network operator. It is worth noting that the requests of the tenants are not related to the specific spectrum share or transmit power that are eventually assigned to them by the network operator. The tenants are, on the other hand, interested in getting a minimum required spectral efficiency (expressed in bit/sec/m 2 ), which allows them to satisfy the specific service requirements of their own users, regardless of the presence of the other tenants in the network. In this paper, we formulate the minimum spectral efficiency requested by each tenant as a percentage of the spectral efficiency of the network without slicing the RAN, i.e., when tenants do not request any guaranteed service to the network operator. Based on these assumptions, we formulate an optimization problem and identify the optimal transmit power and spectrum to be assigned to each tenant so as to obtain the requested spectral efficiency. The solution of this system-level optimization problem provides insights on the advantages of a sliced network, and sheds light on the feasible set of spectral efficiencies that each tenant can request as a function of the network throughput without applying network slicing. In the sequel, for ease of description, the system model is introduced by considering a two-tenant scenario. The two tenants are denoted by T1 and T2. The generalization to more than two tenants is detailed in Section III-C.

A. Cellular Network Modeling

The BSs are modeled as points of a homogeneous Poisson point process (PPP), denoted by Ψ BS , of density λ BS . The MTs of each tenant are modeled following a different homogeneous PPP, denoted by Ψ Ti , of density λ Ti for i = 1, 2. Ψ BS , Ψ T1 and Ψ T2 are assumed to be independent. The MTs are served by the BS providing the best average received power on the downlink channel. All the other BSs transmitting over the same frequency spectrum act as interfering BSs (i.e., full-frequency reuse is considered). Each BS transmits with constant power. P tot denotes the total power budget of each BS. Each BS transmits in a spectrum of total bandwidth B tot . The percentage of transmit power and bandwidth used by Ti are denoted by P Ti and B Ti for i = 1, 2, respectively, such that P T1 + P T2 ≤ P tot and B T1 + B T2 ≤ B tot . The spectrum bands used by T1 and T2 are non-overlapping and, thus, no inter-tenant interference is available.

Let us consider a generic BS of the network. All the MTs of T1 and T2 served by this BS equally share the available transmit power and bandwidth, i.e., power and spectrum are viewed as continuous resources by the BS's scheduler and, thus, no intra-cell interference is available 1 . As a result, the transmit power spectral density of the BSs of tenant Ti is P Ti /B Ti for i = 1, 2. This implies that a BS is off only if there are no MTs, either from T1 or T2, within its corresponding coverage region. If Ni MTs belong to tenant Ti for i = 1, 2, this implies that each MT uses bandwidth B Ti /N Ti and that the MTs do not interfere with each other. The other-cell interference (among BSs of the same tenant transmitting over the same spectrum) is, on the other hand, taken into account.

Based on this system model, in the next section we formulate the potential spectral efficiency (PSE), i.e., the average network throughput, in bit/sec/m 2 for each tenant of the network, by either using or not network slicing. In the latter case, the tenants equally share the resources of the network operator without any constraints on their minimum service requirements. In this case, in other words, tenants T1 and T2 equally share the transmit power P tot and bandwidth B tot .

B. Potential Spectral Efficiency

For ease of notation, we formulate the PSE for a generic tenant whose MTs constitute a PPP of density λ T and whose BSs allocate transmit power P and bandwidth B. The PSE can be formulated as follows:

PSE (P, B, λ T ) = +∞ n=0 PSE ( P, B, λ T | n + 1) P r {n, λ T } (1)
where PSE ( P, B, λ T | n + 1) is the PSE by conditioning on the number, n + 1, of MTs in a generic cell and P r {n, λ T } is the probability that, given a MT in a cell, there are other n MTs in it.

Let γ I be the reliability threshold for successfully decoding a data packet and γ A be the reliability threshold for detecting the presence of the serving BS during the cell association phase. With the aid of stochastic geometry ( [START_REF] Andrews | A tractable approach to coverage and rate in cellular networks[END_REF] and [START_REF] Renzo | The intensity matching approach: A tractable stochastic geometry approximation to system-level analysis of cellular networks[END_REF]), PSE ( P, B, λ T | n + 1) and P r {n, λ T } can be formulated as follows:

PSE ( P, B, λ T | n + 1) = λ T B n + 1 log 2 (1 + γ I ) × P r SIR (n + 1) ≥ γ I , SNR (n + 1) ≥ γ A (2) P r {n, λ T } = 3.5 3.5 Γ (n + 4.5) (λ T /λ BS ) n Γ (3.5) Γ (n + 1) (3.5 + λ T /λ BS ) n+4.5 (3)
where Γ(•) is the gamma function, SIR (n + 1) and SNR (n + 1) are the signal-to-interference-ratio (SIR) and the average signal-to-noise-ratio (SNR), given the number of MTs, n + 1, in a generic cell, during the information decoding and the cell association phases, respectively. They are defined as follows:

SIR (n + 1) = (P /(n + 1)) h 0 /L 0 k∈ΨBS (P /(n + 1)) h k /L k 1 (L k >L0) (4)
SNR ((n + 1)) = (P /(n + 1))/L 0 N 0 (B/(n + 1)) (5)
where h 0 and h k are the fading power gains of the serving and interfering BSs of a generic MT, respectively, due to the wireless channels, which are assumed to be independent and identically distributed exponential random variables with unit mean, L 0 = κr β 0 and L k = κr β k are the path-losses of serving and interfering BSs, respectively, where κ is the path-loss propagation constant, β > 2 is the path-loss exponent, and r 0 and r k are the distances of serving and interfering BSs, and N 0 is the noise-power spectral density. The indicator function 1 (L k >L0) accounts for the cell association strategy and implies that the path-loss of the serving BS, averaged over the fast fading, is smaller than the path-losses of the interfering BSs. It is worth mentioning that SNR (n + 1) is averaged with respect to the fast fading in order to avoid frequent handovers due to the channel variations, as usual in cellular networks.

By comparing [START_REF] Han | A Utility-driven Multi-Queue Admission Control Solution for Network Slicing[END_REF] with the typical definition of PSE in [START_REF] Andrews | A tractable approach to coverage and rate in cellular networks[END_REF] and [START_REF] Renzo | The intensity matching approach: A tractable stochastic geometry approximation to system-level analysis of cellular networks[END_REF], we note that our definition is more realistic since it accounts for non-zero values of γ A and, thus, for the fact that the MTs cannot detect an arbitrary weak signal. This is a fundamental change of our modeling, which allows us to obtain an expression of the PSE that explicitly depends on the transmit power of the BSs. The PSE is reported in the following proposition.

Proposition 1. The exact mathematical expression of the PSE is given in Eq. (6) below:

PSE (P, B, λ T ) = Blog 2 (1 + γ I ) λ BS L λT λBS 1 + L λT λBS Υ (γ I , β) × 1 -exp -πλ BS τ A P B 2/β 1 + L λ T λ BS Υ (γ I , β) (6)
where τ A = (κγ A N 0 ) -1 and:

L λ T λ BS = 1 -1 + 1 3.5 λ T λ BS -3.5 ≥ 0 (7) Υ (γ I , β) = 2 F 1 - 2 β , 1, 1 - 2 β , -γ I -1 ≥ 0 (8)
with 2 F 1 (•) denoting the Gauss hypergeometric function. Sketch of Proof: Eq. ( 6) is obtained from the definition of PSE in [11, Eq. ( 15)], by computing the coverage probability with the aid of mathematical steps similar to those in [10, Theorem 1]. The difference with respect to [10, Theorem 1] lies in the non-zero value of γ A , which modifies the upperlimit in [10, Eq. ( 2)] from L 0 → ∞ to 1 L0 ≤ N 0 Bγ A /P . The proof follows by taking into account the definition of L 0 , i.e., L 0 = κr β 0 , and by solving the integral in closed-form. If γ A = 0, it is worth nothing that the PSE is independent of the transmit-power, P , of the BSs and the proposed framework simplifies to previously reported formulas in [START_REF] Andrews | A tractable approach to coverage and rate in cellular networks[END_REF] and [START_REF] Renzo | The intensity matching approach: A tractable stochastic geometry approximation to system-level analysis of cellular networks[END_REF]. In addition, the PSE would linearly depend on the transmission bandwidth B. By considering that the BSs have a finite sensitivity for detecting the presence of the BSs, i.e., γ A = 0, on the other hand, we obtain a more accurate mathematical framework where P and B play a fundamental role for system optimization in the context of a multi-tenant cellular network with network slicing capabilities.

III. SYSTEM-LEVEL OPTIMIZATION

Based on the mathematical formulation of the PSE in [START_REF] Salvat | Overbooking network slices through yield-driven end-to-end orchestration[END_REF], the PSE of tenant Ti in the presence of network slicing is PSE Ti = PSE (P Ti , B Ti , λ Ti ) for i = 1, 2. Conversely, the PSE without a network slicing-MTs of T1 and T2 share the available resources of the RAN without any service requirement constraints-is PSE NoSlicing = PSE (P tot , B tot , λ tot = λ T1 + λ T2 ).

Let us now consider a toy scenario. We assume a homogeneous network deployment, e.g., all the BSs have the same bandwidth B tot and maximum transmission power P tot . The tenant SLAs are formulated in terms of average PSE2 , i.e., PSE Ti = α Ti PSE NoSlicing where α Ti ≥ 0, ∀i, which constitute the minimum spectral efficiency requirements of tenant T1 and T2, respectively. The SLAs, in particular, are expressed as a fraction of the spectral efficiency without performing network slicing, i.e., the baseline working operation of current cellular networks.

Let us introduce the short-hand notation:

k (Ti) 1 = log 2 (1 + γ I ) λBSL λ Ti λ BS 1+L λ Ti λ BS Υ(γI ,β)
and

k (Ti) 2 = πλ BS (τ A ) 2/β 1 + L λTi λBS Υ (γ I , β) .
The following optimization problem can be formulated.

Problem Bi-Sharing:

minimize ½ subject to k (T1) 1 B T1 1-e -( P T1 B T1 ) (2/β) k (T1) 2 ≥ α T1 PSE NoSlicing ; k (T2) 1 B T2 1-e -( P T2 B T2 ) (2/β) k (T2) 2 ≥ α T2 PSE NoSlicing ; B T1 + B T2 ≤ B tot ; P T1 + P T2 ≤ P tot ; B T1 , B T2 , P T1 , P T2 ∈ R + ;
where the total throughput without slicing is

PSE NoSlicing = k (tot) 1 B tot 1-e -( P tot B tot ) (2/β) k (tot) 2
. Generally speaking, Problem Bi-Sharing provides the optimal set of values b = {B i } and p = {P i } given the tenants SLAs.

A. The relevance of a sliced RAN

The previous example provides the baseline scenario for our analysis. It unveils important insights when applying the network slicing concept to the RAN of cellular networks.

Lemma 1. The probability that the sum-PSE experienced by all the tenants is greater than the PSE experienced by a monolithic non-sliced network is greater than zero, i.e., Pr{(PSE T1 + PSE T2 ) ≥ PSE NoSlicing } ≥ ǫ, ∀ǫ > 0.

This lemma relies on the convexity property of the multivariable PSE function shown in Eq. ( 6) 3 , showing the potential benefits of appling network slicing to the RAN. In some cases, slicing the RAN may increase the total experienced spectral efficiency, which, in turn, translates into higher operator's revenues. Tighter conditions on the case studies when a sliced network outperforms a monolithic network structure are formulated as follows.

Lemma 2. The sum-PSE of two tenants with network slicing is always greater than the sum of the PSEs of the tenants in a non-sliced network where the tenants equally split the total transmit power and available bandwidth, i.e., PSE T1 + PSE T2 > 2PSE NoSlicing (P tot /2, B tot /2), with B T1 + B T2 = B tot , B T1 = B T2 or P T1 + P T2 = P tot , P T1 = P T2 . If a uniform (equal) distribution of transmit power and bandwidth among the tenants is assumed, i.e., PSE T1 = PSE T2 , then PSE T1 + PSE T2 = 2PSE NoSlicing (P tot /2, B tot /2).

Sketch of Proof:

Assuming the same MT densities, i.e., λ T1 = λ T2 and relying on the convexity property, it yields the following:

PSE NoSlicing ( BT1 2 + BT2 2 ),( PT1 2 + PT2 2 ) ≤ PSET1 2 + PSET2 2 ; PSE NoSlicing Btot 2 , Ptot 2 ≤ PSET1+PSET2 2 .
(9) If B T1 = B T2 and P T1 = P T2 , then B T1 = B tot /2, P T1 = P tot /2, PSE T1 = PSE T2 , and we have:

PSE NoSlicing Btot 2 , Ptot 2 ≤ PSE T1 =PSE T1 Btot 2 , Ptot 2 , (10) where PSE T1 ( Btot 2 , Ptot 2 ) = PSE NoSlicing ( Btot 2 , Ptot 2
). This concludes our proof.

With the aid of this lemma, we can improve the system performance, by designing an admission control scheme for cellular networks that exploits network slicing and opportunistically admits subsets of tenants that maximize the PSE. This is discussed in Section III-C. In particular, the following important proposition holds.

Proposition 2. Problem Bi-Sharing admits a feasible solution even if α

T1 + α T2 ≥ 1.

Sketch of Proof:

The proof is obtained by combining Lemma 1 and Lemma 2. Let us consider two tenants sharing the total available bandwidth and transmit power. From Lemma 1, there is a non-negligible probability that PSE T1 + PSE T2 = α NoSlicing PSE NoSlicing , where α NoSlicing > 1. From Problem Bi-Sharing, we obtain PSE Ti = α Ti PSE NoSlicing , and thus, α T1 + α T2 = α NoSlicing ≥ 1. Therefore, Problem Bi-Sharing admits a feasible solution for α T1 +α T2 ≥ 1 if B T1 + B T2 ≤ B tot and P T1 + P T2 ≤ P tot .

This proposition is a key-finding of this work: telecom operators can slice their radio access resources among the tenants and achieve a sum-throughput higher than that achieved without slicing the RAN, i.e., by sharing the available resources among the tenants without performance guarantees. In the next section, we discuss the solution of Problem Bi-Sharing.

B. Lagrange Decomposition

In order to solve Problem Bi-Sharing, we propose to apply the Lagrange duality theorem [START_REF] Boyd | Convex Optimization[END_REF]. Let us define the Lagrangian L : R m × R n → R, where m = 6 is the number of decision variables and n = 2 is the number of constraints, as follows:

L(µ T1 , µ T2 , B T1 , P T1 , B T2 , P T2 ) = 1 -µ T1 α 1 PSE NoSlicing -K 1 B T1 (1 -e - P T1 B T1 (2/β) k2 ) - µ T2 α 2 PSE NoSlicing -K 1 B T2 (1 -e - P T2 B T2
(2/β) k2 ) .

(11) We can derive the Lagrange dual function g(µ T1 , µ T2 ) = inf BT1,BT2,PT1,PT2

L(µ T1 , µ T2 , B T1 , P T1 , B T2 , P T2 ) that satisfies the constraints B T1 + B T2 ≤ B tot and P T1 + P T2 ≤ P tot .

The unconstrained dual problem can be formulated as follows.

Problem Bi-Sharing(DUAL):

maximize g(µ T1 , µ T2 ) subject to µ T1 , µ T2 ≥ 0.
This problem can be solved by using the iterative subgradient update method to optimize the Lagrange multipliers µ T1 and µ T2 :

µ (k+1) T1 = µ (k) T1 + ζ (k) ∂L(µT1,µT2,BT1,PT1,BT2,PT2) (k) ∂µT1 + µ (k+1) T2 = µ (k) T2 + ζ (k) ∂L(µT1,µT2,BT1,PT1,BT2,PT2) (k) ∂µT2 + (12)
where ζ (k) is defined as the step-size and can be chosen as follows ( [START_REF] Shi | Step-size estimation for unconstrained optimization methods[END_REF]):

ζ (k) = ν (k) ∂L(µT1,µT2,BT1,PT1,BT2,PT2) (k) ∂µ 2 (13)
where

ν (k) = µ (k) -µ (k-1)
2 with • 2 denoting the norm-2 operation whereas ζ (k) must be greater than zero. The iterative process stops when the convergence is reached and the optimal Lagrange multipliers µ If two tenants are considered, the Lagrange multipliers can be obtained without using this recursive approach. This can be done as follows. Let us generalize the integrity constraint as B T1 + B T2 = B tot and P T1 + P T2 = P tot . This allows us to derive L(µ T1 , µ T2 , B T1 , P T1 ), where B T2 and P T2 can be obtained from the equalities B T1 + B T2 = B tot and P T1 + P T2 = P tot . Therefore, we can rewrite Eq. ( 11) and obtain the optimal µ ( * ) T1 and µ ( * ) T2 for given B T1 and P T1 . The following is obtained:

µ ( * ) T1 = α -1 T1 µ ( * ) T2 = α -1 T2 . (14)
Based on this finding, Problem Bi-Sharing can be reduced to an unconstrained feasibility problem as follows.

Problem Bi-Sharing(UNCONSTRAINED):

minimize 1-2PSE NoSlicing - k (T1) 1 αT1 B T1 1 -e - P T1 B T1 (2/β) k (T1) 2 - k (T2) 1 αT2 (B tot -B T1 ) 1 -e - P tot -P T1 B tot -B T1 (2/β) k (T2) 2 subject to 0 ≤ B T1 ≤ B tot ; 0 ≤ P T1 ≤ P tot .
Based on this result, Problem Bi-Sharing can be solved with the aid of conventional numerical methods.

In the next section, we provide a generalized formulation of the problem for a multi-tenant system where i α i ≥ 1.

C. Generalized Problem Formulation

Let us consider a set of tenants i ∈ I each of them requesting a network slice. The objective it to efficiently split the resources of the RAN among them. This encompasses the allocation of adequate transmit powers and spectrum bandwidths to each tenant, i.e., p = {P Ti } and b = {B Ti }. Based on Problem Bi-Sharing, we can formulate a general optimization problem as follows.

Problem MultiTenant-Optimizer:

minimize ½ subject to PSE(b, p, λ) ≥ α PSE NoSlicing b 1 ≤ B tot ; p 1 ≤ P tot .
where • 1 is the norm-1 operator, λ = {λ Ti } is the set of user densities of the tenants and α is the vector of SLA requirements of the tenants, which are formulated in terms of percentages of the PSE without performing network slicing, i.e., PSE NoSlicing . We can formulate the dual problem of Problem MultiTenant-Optimizer and calculate the optimal Lagrange multipliers µ = {µ Ti } by using the iterative function as follows:

µ (k+1) Ti = µ (k) Ti + ζ (k) ∂L(µ, b, p) ∂µ Ti + . ( 15
)
This leads to a problem similar to Problem MultiTenant-Optimizer (UNCONSTRAINED) studied in Section III-B.

If the solution of the optimization problem consists of small values of transmit power or bandwidth, i.e., P Ti ≈ 0 or B Ti ≈ 0, the system discards the request of the slice that originates from tenant i: this is the essence of the proposed admission control protocol. To speed up the admission control phase, we propose to pre-filter the requests of RAN slices beforehand.

In particular, we propose to process all the slices that need less than half of the non-sliced spectral efficiency, i.e., the Algorithm 1 Stochastic RAN Slicer (STORNS) 1) Initialise set µ and γ.

2) Initialise sets b ← 0, p ← 0 and value k ← 0.

3) Solve Problem MultiTenant-Optimizer (DUAL)

(with INPUT b (k) and p (k) ) and get µ (k) . 4) Calculate µ (k+1) based on Eq. ( 15). 5) Update ζ (k+1) based on Eq. ( 13).

6) Solve

Problem MultiTenant-Optimizer (UNCONSTRAINED) (with INPUT µ (k+1) ) and get b (k+1) and p (k+1) . 7) If (µ (k+1) = µ (k) ), then increase k = k + 1 and Go to step (3). 8) Mark µ ( * ) = µ (k+1) as the optimal Lagrange multipliers. 9) Solve Problem MultiTenant-Optimizer (UNCONSTRAINED) (with INPUT µ ( * ) ) and get the optimal solution of b ( * ) and p ( * ) . slices with α Ti <= 0.5 are processed for optimal resource allocation while the others are discarded. This approach is motivated by the finding in Lemma 2. This leads the sliced RAN to have a sum-throughput that is in general close and may be higher than its non-sliced counterpart, as shown in Sec. IV-C. More precisely, our proposed admission control scheme works as follows: if, after the first assignment round, the sum-throughput of the admitted tenants is below the nonsliced PSE, then other slice requests from other tenants are considered in ascending order until the best sum-throughput is found. This is the essence of STORNS.

In Alg. 1, we provide the pseudo-code description of STORNS, which yields the set bandwidths and transmit powers for each admitted slice. Steps 3 -7 are repeated until the Lagrange multipliers reach convergence. The speed and accuracy of the proposed algorithm are determined by the step-size ζ (k) , defined in Eq. ( 13). An empirical analysis of the convergence of the algorithm is provided in Sec. IV-D.

IV. PERFORMANCE EVALUATION

We have carried out an extensive simulation study to i) validate the analytical framework based on stochastic geometry theory, ii) show the near-optimality of the proposed lowcomplex algorithm (STORNS) and iii) prove the finding that slicing the RAN may provide a higher sum-PSE compared to its non-sliced counterpart. We have implemented frameworks and algorithms by using commercial mathematical tools, such as MATLAB and MATHEMATICA. We consider different random instances of cellular network deployments based on the PPP model, as explained in Sec. II-A. Unless otherwise stated, the simulations parameters are those reported in Table I, summarizing the Urban Micro-cell model (UMi) which is in agreement with IMT ITU-R specifications [START_REF]Guidelines for evaluation of radio interface technologies for IMT-Advanced[END_REF]. The pathloss exponent is chosen based on the empirical evaluations performed in [START_REF] Sun | Propagation path loss models for 5G urban micro-and macro-cellular scenarios[END_REF].

A. Stochastic Framework Validation

In Fig. 2, the PSE in Eq. ( 6) is validated against Monte Carlo simulations. The simulations are obtained by considering several instances for the locations of BSs and MTs, which follow two independent PPPs. For each network realization, the potential throughput of each MT is computed based on its definition in Eq. [START_REF] Han | A Utility-driven Multi-Queue Admission Control Solution for Network Slicing[END_REF]. It is worth mentioning that, to make the validation sound, none of the mathematical equations in Sec. II are used. In particular, the number of users per cell is directly obtained from Monte Carlo simulations. The network spectral efficiency is obtained by summing the potential throughput of all the MTs and normalizing it to the area of the network. Fig. 2 shows a good agreement between analysis and simulations. The small inaccuracies for a large ratio of the density of MTs and BSs is due to the limited number of network realizations that can be simulated in a reasonable amount of time. We note, in particular, that PSE increases as a function of the ratio of densities of MTs and BSs, but saturates as this ratio gets large. This is due to the fact that, as the number of MTs increases, all the BSs are activated and the bandwidth allocated to each MT decreases at the same rate as the number of MTs per unit area. Mathematically speaking, the L(•) function in Eq. ( 6) tends to one as the ratio λ T /λ BS increases towards infinity.

B. Optimality of STORNS

In this section, we validate our algorithm, STORNS, against a benchmark optimal algorithm that is obtained by using a brute-force optimization method that is denoted by OPT. In particular, OPT is obtained by means of an exhaustive greedy search algorithm that explores all possible solutions of Problem MultiTenant-Optimizer. Due to complexity issues, we are able to employ this method for up to 6 tenants.

In Fig. 3, we show numerical results by setting λ T = 100λ BS and by considering different thresholds γ I and γ A . The objective is to compare STORNS and OPT as the number of tenants requesting a RAN slice increases. We assume that each additional tenant asks for a fraction α Ti of PSE NoSlicing that is drawn from a normal distribution with mean µ distr and variance σ distr , as defined in Table I.

The larger the number of slice requests, the higher the PSE. We observe that STORNS exhibits near-optimal performance and the gap with respect to OPT is around 6%, 8% and 12% for γ I = γ A = -5dB,0dB and 5dB, respectively.

C. The RAN slicing benefits

In Fig. 4, we illustrate the performance offered by STORNS as a function of the number of tenants admitted into the cellular network. More precisely, we provide numerical evidence that STORNS is capable of appropriately admitting tenants and allocating their slices (i.e., bandwidth and power) in a way ]

OPT i = A =5dB STORNS i = A =5dB OPT i = A =0dB STORNS i = A =0dB OPT i = A =-5dB STORNS i = =-5dB
Fig. 3: Optimality study of STORNS that the PSE of the sliced RAN is higher than its monolithic network counterpart that does not exploit network slicing. We assume a network slices demand up to 32 network slice requests. In addition, we assume that each tenant requests, on average, a network slice that provides a PSE that is 10% of the achievable PSE without using slicing. This implies that the non-sliced network would be able to admit, on average, up to ten tenants (non-shaded region in the figure) 4 . By using STORNS, we can accommodate a larger number of tenants and achieve a sum-PSE that is higher than the non-sliced sum-PSE (shaded region in the figure). This is possible by admitting the "best" network slice requests among the 32 available and by optimally allocating the transmit power and bandwidth to each of them. STORNS allows telecom operators to achieve up to 120% of the throughput of a monolithic non-sliced cellular network. This motivates telecom operators to use network slicing not only as a means for accommodating the specific request of vertical industries, but also as a powerful means for enhancing the overall network performance and, in turn, for increasing their revenues by simply sharing their physical infrastructure among multiple tenants.

In Table II

D. Algorithm complexity

We study the complexity of our algorithm against that of the exhaustive greedy search. The main parameter for STORNS is the number of rounds to converge and to compute the optimal Lagrange multipliers, as explained in Sec. III-C. In Fig. 5, we show with a solid green line the number of rounds (k) that are needed to converge while increasing the number of tenants requesting a RAN slice. The behavior of the curve unveils that the complexity of our algorithm does not exponentially increases with the number of constraints (i.e., the number of tenants) of the optimization problem (Problem MultiTenant-Optimizer) but it converges to a stable number of iterations. On the right y-axis of Fig. 5, we compare the computational time for solving the optimization problem and compare STORNS against OPT. We evince that STORNS is capable of achieving near-optimal performance with a limited complexity compared to greedy approaches.

V. CONCLUSIONS

We have analyzed the benefits of applying network slicing to radio access networks. In particular, we have considered network slice requests with diverse service level agreements (SLAs) in terms of required average throughput per tenant. To analytically formulate the problem, we have capitalized on stochastic geometry theory, which allowed us to consider cellular network topologies in a tractable yet sufficiently realistic manner. We have introduced a new mathematical formulation for network slicing throughput and have defined an optimization problem to design an admission control, STORNS, that identifies the best tenants to be admitted into the network along with their spectrum and transmit power allocation such that the overall system throughput is maximized. Our results have shown through mathematical proofs, numerical and simulation results that networks where network slicing is applied can achieve a higher network throughput than non-sliced ones. Finally, we have provided quantitative results of reduced computational complexity of STORNS as compared to brute-force optimization methods. Our work puts forth network slicing as a suitable approach for optimizing the radio resource utilization of future sliced cellular networks.

Fig. 2 :

 2 Fig. 2: Framework validation. Solid lines: Eq. (6). Markers: Monte Carlo simulations.

Fig. 5 :

 5 Fig. 5: Computational Analysis

TABLE I :

 I Simulation parameters (ITU UMi[START_REF]Guidelines for evaluation of radio interface technologies for IMT-Advanced[END_REF])

	System Parameters	Values
	Inter-site distance (ISD)	200 m
	Base station density (λBS )	(π ISD 2 ) -1
	Mobile terminal density (λT )	100 λBS
	Decoding threshold (γI )	0 dB
	Detecting threshold (γA)	0 dB
	Carrier frequency (fc)	2.1 • 10 9 Hz
	Transmission wavelength (λ) Path-loss constant (κ) Noise power spectral density (N0)	3•10 8 fc ( 4π λ ) 2 m -174 dBm/Hz
	(τA)	(κ γA N0) -1
	Path-loss exponent (β) [15]	3.5
	Bandwidth (Btot)	20 • 10 6 Hz
	Transmit power (Ptot)	43 dBm
	Slice generation process (mean µdistr)	10
	Slice generation process (variance σ 2 distr ) Simulation instances (STORNS)	5 1000

  , we evaluate the gain provided by RAN slicing by Fig.4: Potential benefits of network slicing using STORNS. The gain is defined as PSE max /PSE NoSlicing , where PSE max = i PSE Ti , ∀i admitted. We evaluate different user densities λ T and threshold parameters γ I , γ A . When the average number of users increases, we note that network slicing provides additional performance gains (about 19%). TABLE II: RAN Slicing Gain

	γ I = γ A = 5dB	λ T λ BS 7.17% = 50	λ T λ BS 13.2% = 200	λ T λ BS 18.1% = 500
	γ I = γ A = 0dB	6.63%	16.6%	18.8%
	γ I = γ A = -5dB	4.52%	14.8%	17.69%

Considering continuous resources (e.g., bandwidth) makes our analysis tractable and easy to explain. However, this assumption can be relaxed by accounting for discrete resources (e.g., physical resource blocks (PRBs))[START_REF] Renzo | The intensity matching approach: A tractable stochastic geometry approximation to system-level analysis of cellular networks[END_REF].

The PSE is interpreted as the average network throughput experienced by the users of the tenant. This is a reasonable assumption when considering tenant SLAs in terms of cell throughput.

Note that the Hessian condition for that function is not fully satisfied. Therefore, as shown in[START_REF] Boyd | Convex Optimization[END_REF], it is needed to check that g(tx 1 +(1-t)x 2 , ty 1 + (1t)y 2 ) ≤ tg(x 1 , y 1 ) + (1t)g(x 2 , y 2 ).

It is worth nothing that the overall PSE of 10 admitted network slice requests that corresponds to 100% of PSE NoSlicing is slightly above the dashed line shown in the figure. This is due to the randomness of the network slice generation process. In fact, 10% is only the mean value (see TableI).


Amaury Vignolles
email: amaury.vignolles@laas.frelodie.chanthery@laas.frpauline.ribot@laas.fr

Elodie Chanthery

Pauline Ribot

An overview on diagnosability and prognosability for system monitoring

The complexification of systems has brought the emergence of a new field of study: system health monitoring. This field is deemed necessary because it improves system availability and it avoids unnecessary maintenance costs. System health monitoring is performed through diagnosis and prognosis methods. Diagnosis consists in detecting and identifying faults that may lead to system failures. Prognosis is related to the prediction of the system Remaining Useful Life (RUL) that corresponds to the remaining time until the system failure. This paper aims at giving an overview on the properties related to diagnosis and prognosis on different types of systems. We will focus on the diagnosability and prognosability properties. This paper will first briefly present the different types of systems of interest for the system health monitoring community. We will consider Discrete Event Systems (DES), Continuous Systems (CS), Hybrid Systems (HS) or Heterogeneous Systems (HtS). The rest of this paper will present the definitions given in the literature for the concepts of diagnosability and prognosability. The similarities and differences in these definitions for the different types of systems will be highlighted. Some metrics associated with the prognosability property will also be discussed.

INTRODUCTION

Nowadays, systems are more and more complex and a more complex system is subject to more faults and failures reasons. To detect and identify these faults, a diagnosis reasoning is required. Diagnosis algorithms [START_REF] Isermann | Supervision, fault-detection and faultdiagnosis methods-an introduction[END_REF] were developed to monitor systems and know if a fault occurred (detection) and which fault (or faults) occurred (isolation). Re-Amaury Vignolles et al. This is an open-access article distributed under the terms of the Creative Commons Attribution 3.0 United States License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. cently, industrials want not only to be able to know if a fault, and which one, occurred, but also to be able to predict the remaining time during which the system will be able to fulfill its purpose, or in other words, the time until the system encounters a failure, also known as the Remaining Useful Life (RUL) of a system. Prognosis is often related to this temporal prediction of the RUL. To understand well the difference between faults and failures, the definitions of faults [START_REF] Isermann | Supervision, fault-detection and faultdiagnosis methods-an introduction[END_REF] and failures [START_REF] Villemeur | Dependability of industrial systems[END_REF]) must be reminded.

Definition 1 (Failure)

A failure state is a state in which the system is not able to fulfill its purpose anymore.

Definition 2 (Faults) Faults represent a non-acceptable deviation from a characteristic property or a parameter of the system.

Faults might lead to failures, but not necessarily, as in some cases, the system might still be able to fulfill its purpose. In this paper, we will survey the notion of diagnosability and prognosability in different types of systems. Diagnosability is a notion that has been well studied for a long time, while prognosability is a more recent notion. Intuitively, we can define diagnosability and prognosability as follows, based on the etymology of the words themselves.

Definition 3 (Diagnosability-intuition) Diagnosability represents the ability for a system to be diagnosed. In other terms, diagnosability is the ability for a system and its monitoring capacities to exhibit different symptoms for each fault situation.

Definition 4 (Prognosability-intuition) Prognosability represents the ability for a system to be prognosed. In other terms, it represents the ability to predict a failure.

In this paper, proper definitions given in the literature are to be found for different kinds of systems. Four different kinds of systems will be considered: discreteevent systems (DES), continuous systems (CS), hybrid systems (HS) and finally heterogeneous systems (HtS). A DES is defined as a system that will only encounter, and produce, discrete data. If a DES encounters continuous data, it will be abstracted to generate discrete events. Among the most widespread formalisms for DES are automata, state machines or Petri nets, interested readers can find more details in [START_REF] Cassandras | Introduction to discrete event systems[END_REF]. A CS will have continuous dynamics. In general, a CS is described by a dynamic equation C of the form:

C = x k+1 = f (x k , u k ) + v(x k , u k ) y k = h(x k , u k ) + w(x k , u k ) , (1)
where x k ∈ R nx is the continuous state vector of n state variables at time k, u k ∈ R nu is the vector of n u continuous input variables at time k, f is the noise-free continuous evolution function, v is a noise function, y k ∈ R ny is the vector of n y continuous output variables at time k, h is the noise-free output function and w is the noise function associated with observation.

A HS will always be subject to continuous and discrete data.

The most known formalisms for hybrid systems are hybrid automata [START_REF] Cassandras | Introduction to discrete event systems[END_REF], hybrid Petri nets [START_REF] Alla | Continuous and hybrid Petri nets[END_REF] or hybrid bond graphs [START_REF] Mosterman | A theory of discontinuities in physical system models[END_REF].

Finally, we consider that a HtS is a system that could be described as different sub-systems, which are either DES, CS or HS. It means that a HtS will sometimes be affected by discrete data only, sometimes by continuous data only, and sometimes both.

For each of these type of systems, Σ f will denote the set of faults that may occur on the system. For each one of these formalisms, one specific definition of diagnosability or prognosability can be proposed. The goal here is to see where these definitions converge or diverge and if it is possible to merge some interesting concepts for the specific HtS.

This paper is organized as follows. Section 2 will survey the diagnosability notion as defined by the literature for each of the previously presented category of systems. Based on Section 2 layout, Section 3 will survey the prognosability notion and introduce prognosability's metrics. Finally, Section 4 will conclude, criticize and highlight some interesting points on which advancements could be made.

DIAGNOSABILITY

This section exposes how diagnosability is defined in the literature. Table 1 shows the reviewed articles and the studied types of systems. The following subsections give more details about the definitions given by the articles.

Diagnosability in DES

Diagnosability in DES is quite well defined in the literature. One of the first definition was given in [START_REF] Sampath | Diagnos-ability of discrete-event systems[END_REF] for systems modeled by automata.

Definition 5 (Diagnosability by [START_REF] Sampath | Diagnos-ability of discrete-event systems[END_REF] Diagnosability is the ability for a DES to provide a particular set of observations after the occurrence of a fault f ∈ Σ f belonging to a fault's partition.

A fault's partition is a set Ω f ⊆ Σ f containing one or several faults, represented by unobservable discrete events. For example, the fault's partition Ω 1 = {f 1 ; f 2 } contains the faults f 1 and f 2 , whereas Ω 2 = {f 3 } contains only the fault f 3 . So, in this case, the two faults f 1 and f 2 are not distinguished as they are represented in the same fault partition. The set of observations must allow the monitor (also called the diagnoser) to detect the occurrence of a particular fault's partition. However, this definition of diagnosability does not imply that the fault which occurred will be precisely known. Moreover, the previous definition is quite strict and might not be applicable to some real systems. For example, if you consider the case of a door, it is impossible to know if the door is broken and stuck closed (event f sc ) without trying to open the door. Based on this idea, a relaxed definition has been proposed by the same authors in [START_REF] Sampath | Diagnos-ability of discrete-event systems[END_REF], named the I-diagnosability.

Definition 6 (I-diagnosability) I-diagnosability is the ability for a DES to provide a particular set of observations, allowing the monitor to detect the occurrence of a fault f ∈ Σ f and to identify the fault's partition not after the occurrence of the said fault, but after the occurrence of an indicator event related to the fault.

Considering the previous example of the door, the event open door is considered as an indicator event related to the fault f sc . Definition 5 is used as a base to define most of the notions related to diagnosability and is sometimes used as it is like in [START_REF] Yoo | Polynomial-time verification of diagnosability of partially observed discreteevent systems[END_REF], along with Definition 6. It was also applied to Petri Nets in [START_REF] Basile | Decentralized k-diagnosability of Petri nets[END_REF] and [START_REF] Madalinski | Diagnosability verification with Petri net unfoldings[END_REF]. A new definition, k-diagnosability, is given in [START_REF] Basile | Decentralized k-diagnosability of Petri nets[END_REF].

Definition 7 (k-diagnosability) k-diagnosability is the ability to be k-diagnosable. To be kdiagnosable, a fault event f ∈ Σ f must be diagnosable after the firing of at most k transitions after its occurrence. If all the faults f ∈ Σ f are k-diagnosable, the system is kdiagnosable.

In [START_REF] Zanella | Asymmetric diagnosability analysis of discrete-event systems[END_REF], || //l -diagnosability is presented to take into account temporal uncertainty into the diagnosability def-System Type Articles DES [START_REF] Sampath | Diagnos-ability of discrete-event systems[END_REF] and [START_REF] Ye | Manifestability verification of discrete event systems[END_REF] [Automata] [START_REF] Zanella | Asymmetric diagnosability analysis of discrete-event systems[END_REF] [Twin plant : finite automata] [START_REF] Basile | Decentralized k-diagnosability of Petri nets[END_REF] [Labelled Petri Nets] [START_REF] Yoo | Polynomial-time verification of diagnosability of partially observed discreteevent systems[END_REF] [Finite-state automaton] [START_REF] Madalinski | Diagnosability verification with Petri net unfoldings[END_REF] [Labelled Petri Nets] CS [START_REF] Staroswiecki | Analytical redundancy relations for fault detection and isolation in algebraic dynamic systems[END_REF] [algebraic dynamic systems] [START_REF] Zuniga | Flotation process fault diagnosis via structural analysis[END_REF]) [START_REF] Cordier | Comparing diagnosability in continuous and discreteevent systems[END_REF]) HS [START_REF] Zaatiti | Counterexample-guided abstraction-refinement for hybrid systems diagnosability analysis[END_REF] [Partially observed hybrid automata] [START_REF] Bayoudh | Diagnosability analysis of hybrid systems cast in a discrete-event framework[END_REF]) [START_REF] Fourlas | Diagnosability of hybrid systems[END_REF] [Hybrid I/O Automata] HtS [START_REF] Hsieh | Strongly diagnosable product networks under the comparison diagnosis model[END_REF] [distributed] [START_REF] Hua | A data driven method for quantitative fault diagnosability evaluation[END_REF] Table 1. Reviewed articles on diagnosability classified by system types inition for systems modeled by automata. This temporal uncertainty is up to a certain level, denoted l. For example, let us consider four events a, b, c, d, such that events a and d are temporally unrelated as illustrated by Figure 1. In this case, when a and d both occur, it is not possible to know if the sequence is a.d or d.a. To represent this temporal uncertainty, the authors propose to write a//d. Thus, an example of observations for the system represented in Figure 1 is b.a//d.c.

In this example, as there are two temporally unrelated events, we consider l = 2.

Figure 1. Temporaly unrelated events, from [START_REF] Zanella | Asymmetric diagnosability analysis of discrete-event systems[END_REF] Definition 8 (|| //l -diagnosability) || //l -diagnosability represents the ability for a system to be || //l -diagnosable. For a faulty behavior f ∈ Σ f to be || //ldiagnosable, it should produce a set of observations that cannot be mistaken, even with temporal uncertainty on the events up to a level l, with a set of observations produced by a nominal behavior or a different fault. If all the faults f ∈ Σ f are || //l -diagnosable, the system is || //l -diagnosable.

In [START_REF] Ye | Manifestability verification of discrete event systems[END_REF], a weaker definition is introduced, and called manifestability. This notion is based upon the idea that diagnosability might be expensive in terms of sensors requirement and is defined for systems modeled by automata.

Definition 9 (Manifestability)

The notion of manifestability represents the possibility for a fault occurrence to be manifestable, i.e. to be recognized through observations. A manifestable fault f ∈ Σ f will have effects on the observation given by the system. If all the faults f ∈ Σ f are manifestable, the system is said manifestable.

Based on this notion, if a fault is not manifestable, then, whatever the event following its occurrence, it is not possible to know for sure whether it occurred or not. A non-manifestable fault is therefore not diagnosable.

Diagnosability in CS

Diagnosability in CS is usually composed of detectability and isolability and is based upon what is called Analytical Redundancy Relations (ARRs) [START_REF] Staroswiecki | Analytical redundancy relations for fault detection and isolation in algebraic dynamic systems[END_REF], [START_REF] Pérez | Fault-driven structural diagnosis approach in a distributed context[END_REF], which are analytical relations based upon the set obs of measured variables.

Definition 10 (ARRs)

A dynamic constraint C obs (ū k , ȳk , ¯ k ) = 0 is an ARR for Equation 1 if for all (u k ,y k ) consistent with Equation 1, the dynamic constraint is statistically satisfied, with u k being the vector of input variables, y k the vector of output variables and k the vector of noise variables, and ūk (resp. ȳk ; ¯ k ) stands for the vector u k (resp. y k ; k ) and its derivatives up to some unspecified order.

Each ARR gives rise to a consistency indicator, called a residual.

Definition 11 (Residual)

The residual r is time-dependent and is given by a boolean value by the following application:

r(t) = 0 if C obs is satisfied by (u(s), y(s)), s ≤ t 1 otherwise (2)
When the residual is equal to 0 it means that the observations are consistent with the model, so no fault has occurred. If the residual is 1, it means that a fault is detected.

We can split the residuals in two parts: a deterministic (depending on ū(t) and ȳ(t)) and a stochastic part (depending on ¯ (t)).

The two following concepts related to diagnosability are defined in [START_REF] Staroswiecki | Analytical redundancy relations for fault detection and isolation in algebraic dynamic systems[END_REF].

Definition 12 (Detectability) Detectability is the ability for a fault to be detectable. A fault f ∈ Σ f is said to be detectable if its residual evaluation differs from zero. If all the faults f ∈ Σ f are detectable, the system is detectable.

Definition 13 (D-detectability) D-detectability is the ability to be D-detectable. A fault f ∈ Σ f is said to be D-detectable if the deterministic part of its residual evaluation form differs from zero. If all the faults f ∈ Σ f are D-detectable, the system is D-detectable.

D-detectability is a weaker concept than detectability as a non-D-detectable fault might influence the stochastic part of their residuals and, thus, could be detected. After detecting a fault, it has to be isolated, for this purpose, a fault signature has to be defined.

Definition 14 (Fault signature) A fault signature is a function Sig associating a set of observables (residuals) to each fault.

The following definition of isolability is given in [START_REF] Zuniga | Flotation process fault diagnosis via structural analysis[END_REF].

Definition 15 (Isolability) Isolability is the ability to be isolable. Two faults f i and f j ∈ Σ f are isolable iff their fault signatures differ. If all pairs of faults are isolable, the system is said isolable.

The following definition of diagnosability for CS is given in [START_REF] Cordier | Comparing diagnosability in continuous and discreteevent systems[END_REF].

Definition 16 (Diagnosability by [START_REF] Cordier | Comparing diagnosability in continuous and discreteevent systems[END_REF] Diagnosability is the ability to be diagnosable. Two faults f i and f j ∈ Σ f are diagnosable iff the junction of their signatures is empty. Formally, it can be written as:

Sig(f i ) ∩ Sig(f j ) = ∅.
If all pairs of faults are diagnosable, the system is said diagnosable.

Diagnosability in HS

In [START_REF] Fourlas | Diagnosability of hybrid systems[END_REF], a first definition of a diagnosable fault is given for HS, on Hybrid I/O Automata. Here, a fault is modeled as an event whose occurrence impacts either the continuous variables, the discrete state or both.

Definition 17 (Diagnosability by [START_REF] Fourlas | Diagnosability of hybrid systems[END_REF] Diagnosability is the ability to be diagnosable. A fault f ∈ Σ f is said diagnosable in HS if its occurrence can be detected after t ∈ N changes of trajectories (changes of continuous variables values) and actions (events) occurred on the system.

If all the faults f ∈ Σ f of the system are diagnosable, so is the system.

The following notions around the concept of diagnosability, for multimode systems, are defined in [START_REF] Bayoudh | Diagnosability analysis of hybrid systems cast in a discrete-event framework[END_REF]. The first one is mode diagnosability and the second one is fault diagnosability, the former being stronger than the latter. From the ARRs of the systems, we can compute mode signatures, from which fault signatures are determined as the set of the systems modes in which the fault is present.

Definition 18 (Mode)

A mode q i corresponds to a working state of the system and is associated with a set of continuous dynamics C i (see Equation 1) representing the evolution of the continuous variables.

Definition 19 (Mode signature)

The signature of a mode q i is the vector obtained by the concatenation of the vectors of boolean residuals associated to all the different modes q j of the system considered with the observations consistent with the considered mode q i . It characterizes the behaviour of this particular mode with regard to all the other modes.

Definition 20 (Mode diagnosability in HS) Two modes q i and q j are said diagnosable if their signatures are different (Sig(q i ) = Sig(q j )). If all pairs of modes are diagnosable, the system is said to be mode diagnosable.

Definition 21 (Fault signature by [START_REF] Bayoudh | Diagnosability analysis of hybrid systems cast in a discrete-event framework[END_REF] The signature of a fault f ∈ Σ f is given by the set of signatures of the system modes in which the fault is present. Assuming that the model does not account for actions that repair the faults, the signature of a fault f i is equal to the set of signatures of the modes that are reachable from a transition labeled with the fault event f i .

Definition 22 (Fault diagnosability)

A fault f ∈ Σ f is considered diagnosable if its occurrence can be detected after a given time window thanks to the observation of the discrete events and the measurements of the continuous variables available on the system. Two faults f i and

f j ∈ Σ f are diagnosable from the continuous observa- tions if Sig(f i ) ∩ Sig(f j ) = ∅.
As the notion of mode diagnosability is stronger than the notion of fault diagnosability, a mode diagnosable system is fault diagnosable.

The following property is introduced and proved in [START_REF] Bayoudh | Diagnosability analysis of hybrid systems cast in a discrete-event framework[END_REF].

Property 1 (System diagnosability) For a hybrid system to be diagnosable, either its underlying DES or multimode system must be diagnosable.

The diagnosability in Partially Observable Hybrid Automaton (POHA), in which some variables are not observed, is defined in [START_REF] Zaatiti | Counterexample-guided abstraction-refinement for hybrid systems diagnosability analysis[END_REF] as follows.

Definition 23 (∆-diagnosability) ∆-diagnosability represents the ability to be ∆-diagnosable.

A fault event f ∈ Σ f is said ∆-diagnosable in a POHA iff it is possible to detect its occurrence at least ∆ time units after it occurred. If all faults f ∈ Σ f are ∆-diagnosable, then the system is ∆-diagnosable.

Here, ∆ represents the minimum required time (real time) before detecting and identifying the fault.

Diagnosability in HtS

Only a few articles study HtS, and even fewer the specific point of diagnosability in HtS. The first of the two following articles we found in the literature, [START_REF] Hsieh | Strongly diagnosable product networks under the comparison diagnosis model[END_REF], is based on a multiprocessors system. It is comprised of multiple communicating processors called nodes. These nodes can be of different natures: DES, CS or HS and can be faulty.

The authors have defined what they called t-diagnosability and strong t-diagnosability.

Definition 24 (t-diagnosability) t-diagnosability consists in the fact that, as long as the number of faulty components does not exceed t, they can be detected without a replacement (a faulty node is replaced by a non-faulty one).

Definition 25 (Strong t-diagnosability) Strong t-diagnosability means that the system is t-diagnosable and can achieve (t + 1)-diagnosability.

A data driven method for quantitative fault diagnosability evaluation is proposed in [START_REF] Hua | A data driven method for quantitative fault diagnosability evaluation[END_REF]. As this method is data-based, we consider that it can be applied on complex systems such as HtS. The definition of distinguability is introduced.

Definition 26 (Distinguability) Distinguability represents the difference between two modes (both faulty or one faulty and one normal) and is calculated with fuzzy numbers. It is represented by a number between 0 and 1.

The closer distinguability is from 0, the harder it is to distinguish the two modes. The faults leading to the said modes are not isolable (therefore, not diagnosable).

Synthesis on diagnosability

Diagnosability is a quite-well defined notion for the different types of systems (DES, CS, HS). Following the needs and the specifications of systems, advanced notions of diagnosability have emerged, to consider uncertainty or to give more precise outputs.

Table 2 classifies the studied articles regarding the type of information provided by the diagnosability analysis. While some of the outputs are just boolean (diagnosable or not), some tend to be more precise and parameter dependent, giving for example a number of transitions representing the maximum delay until the detection of the fault, or a time window representing the minimum time from which the fault's occurrence can be detected. Another precision on the result of the diagnosability analysis is to represent the uncertainty on the diagnosability analysis. This uncertainty could be up to how much uncertainty on the time occurrence of the observations the diagnosability of the fault is sure. Another one could be up to represent the upper limit on the number of faulty components in our distributed system for it to be diagnosable. Finally, another type of result is a real number, between 0 and 1, representing how 'difficult' it is to distinguish the fault from another.

Different ways to consider uncertainty of the system are represented in Table 3.

• The temporal uncertainty, representing the idea that when events are temporally unrelated, it is not possible to know which event occurred first, • The noise appearing on the sensors, potentially not giving the exact output. This has to be somehow taken into account to have a precise analysis, • The reliability of the testing nodes in distributed systems, represented with uncertainty on the output. If a testing node is faulty, its output is totally unreliable.

Table 4 shows how the time is represented and taken into account in the diagnosability evaluation. As a matter of fact, some represent the time through a number of transitions or events (usually the case for DES), while some others represent it with continuous time. Some others represent the time with a number of actions or changes occurring on the system. Finally, the time could also not be taken into account and the process could be done online.

PROGNOSABILITY

In contrast to diagnosability, prognosability is not so well defined in the literature, as we will notice in this section. Prognosability is a property relative to the prediction of the future state of the system, related to the prognosis idea. For the PHM community, prognosis aims at predicting the system future states and its RUL or End of Life (EoL) from diagnosis and future inputs available from a mission scenario for example.

Prognosability in DES

In DES, prognosability has been studied for a very long time, even before the idea of diagnosability appeared. In discrete event models, the idea of time does noes appear as only sequence of events are considered. Prognosability in DES was

Results of the diagnosability analysis Articles Diagnosable or not [START_REF] Sampath | Diagnos-ability of discrete-event systems[END_REF][START_REF] Ye | Manifestability verification of discrete event systems[END_REF]) [START_REF] Bayoudh | Diagnosability analysis of hybrid systems cast in a discrete-event framework[END_REF]) [START_REF] Staroswiecki | Analytical redundancy relations for fault detection and isolation in algebraic dynamic systems[END_REF]) [START_REF] Yoo | Polynomial-time verification of diagnosability of partially observed discreteevent systems[END_REF]) [START_REF] Madalinski | Diagnosability verification with Petri net unfoldings[END_REF]) [START_REF] Fourlas | Diagnosability of hybrid systems[END_REF] Parameter Dependent [START_REF] Zanella | Asymmetric diagnosability analysis of discrete-event systems[END_REF] [Dependent on temporal uncertainty window] [START_REF] Basile | Decentralized k-diagnosability of Petri nets[END_REF] [Dependent on the number of transition] [START_REF] Zaatiti | Counterexample-guided abstraction-refinement for hybrid systems diagnosability analysis[END_REF] [Dependent on a given time] [START_REF] Hsieh | Strongly diagnosable product networks under the comparison diagnosis model[END_REF] [Dependent on number of detectable faulty components] Real number [START_REF] Hua | A data driven method for quantitative fault diagnosability evaluation[END_REF] Table 2. Reviewed articles on diagnosability classified by the type of result given by the diagnosability analysis Uncertainty Articles Temporal uncertainty [START_REF] Zanella | Asymmetric diagnosability analysis of discrete-event systems[END_REF]) Noise [START_REF] Bayoudh | Diagnosability analysis of hybrid systems cast in a discrete-event framework[END_REF]) [START_REF] Staroswiecki | Analytical redundancy relations for fault detection and isolation in algebraic dynamic systems[END_REF]) Node reliability [START_REF] Hsieh | Strongly diagnosable product networks under the comparison diagnosis model[END_REF] Table 3. Reviewed articles on diagnosability classified by the type of appearing uncertainty indeed studied under different names such as predictability or trajectory prediction.

The very first work on predictability is [START_REF] Cao | The predictability of discrete event systems[END_REF], who works on automata, and introduces one of the first definition of predictability, allowing the monitor to predict the occurrence of a system event or set of events. It is based on the projection operation, defined as follows.

Definition 27 (Projection)

The projection P L of an event e on a language L is defined as P L (ε) = ε, with ε being the empty word and

P L (e) = e if e ∈ L ε if e / ∈ L (3)
Definition 28 (Predictability by [START_REF] Cao | The predictability of discrete event systems[END_REF] Predictability is the ability to be predictable. Considering two automatons H and G, it is said that a finite sequence of events t, defined as a string, belonging to the language of H denoted L(H) is predictable from a string s in L(G) if it matches the projection of s on H. This definition is extended to a language, which is said predictable if all the strings contained in the said language are predictable.

A string is defined as a finite sequence of events, but an event by itself could be considered as a string of length 1.

The following additional definition for systems modeled by automata is introduced in [START_REF] Genc | Predictability of event occurrences in partially-observed discrete-event systems[END_REF].

Definition 29 (Predictability by [START_REF] Genc | Predictability of event occurrences in partially-observed discrete-event systems[END_REF] Predictability is the ability to be predictable. Let us consider any string s ending with the event e to be predicted. Such an event is said to be predictable if we can find, for all existing s, a prefix t of s, not containing e at first but which will contain the desired event after a sufficiently long continuation. If all events of a system are predictable, the system is said predictable.

In this work, predictability is also proved to be a stronger property than diagnosability (see Definition 5), as both require to identify and isolate the desired event, but predictability must do it preemptively. An algorithm to evaluate the predictability property applicable on distributed systems is proposed in [START_REF] Ye | A predictability algorithm for distributed discrete event systems[END_REF], based on automata as well.

Definition 29 is extended to define k-predictability on timed automata in [START_REF] Cassez | Predictability of event occurrences in timed systems[END_REF] as follows.

Definition 30 (k-predictability) k-predictability represents the ability to be k-predictable. Let f ∈ Σ f be a fault. f is considered as k-predictable if it is predictable k event occurrences before its occurrence.

A system is k-predictable if all the faults f ∈ Σ f are kpredictable.

In the case k = 0, the definition of 0-predictability matches Definition 29 .

Stochastic DES are DES in which each event e has a probability p(e) to occur. The following definition is given in [START_REF] Chen | Stochastic failure prognosability of discrete event systems[END_REF]:

Definition 31 (S m -prognosability) S m -Prognosability means that for any threshold value ρ > 0 ∈ R and error bound τ > 0 ∈ R, there exists a reaction bound k ≥ m, such that if a fault f ∈ Σ f cannot be pre- [START_REF] Bayoudh | Diagnosability analysis of hybrid systems cast in a discrete-event framework[END_REF] [For the HS underlying DES] Time [START_REF] Zaatiti | Counterexample-guided abstraction-refinement for hybrid systems diagnosability analysis[END_REF]) [START_REF] Bayoudh | Diagnosability analysis of hybrid systems cast in a discrete-event framework[END_REF]) [START_REF] Staroswiecki | Analytical redundancy relations for fault detection and isolation in algebraic dynamic systems[END_REF] Can be done online [START_REF] Hua | A data driven method for quantitative fault diagnosability evaluation[END_REF] Table 4. Reviewed articles on diagnosability classified by the way to express time during the diagnosability analysis dicted k steps (i.e. k events) in advance with confidence level ρ (we cannot be ρ% sure that the fault is going to happen), the probability of its occurrence is smaller than τ .

Definition 29 is also used in [START_REF] Nouioua | Predictability in probabilistic discrete event systems[END_REF] and applied on probabilistic DES. Based upon the probabilities associated to the occurrence of each event, they refine the definition and propose a probabilistic predictability for the system. As a matter of fact, they compute, for each fault f ∈ Σ f , a probability p ko (f ) to be in a trace in which the fault could not be predicted but occurred, a probability p ok (f ) to be in a trace where the fault could be predicted and occurred and a probability p nf (f ) to be in a trace in which the fault will not occur.

Definition 29 is extended to labelled Petri Nets in [START_REF] Yin | Verification of prognosability for labeled Petri nets[END_REF].

The notion of predictability given in Definition 29 is extended to decentralized systems in [START_REF] Kumar | Decentralized prognosis of failures in discrete event systems[END_REF] and [START_REF] Wu | Verification of coprognosability in decentralized fault prognosis of labeled Petri nets[END_REF], specifically Petri Nets for the latter. The notion of coprognosability is also introduced. The decentralized systems are monitored by local prognosers, each equipped with its own event observation sensors, observing a behavior (sequence of events) executed by a system and issuing their local prognostic decisions. These local prognostic decisions are gathered at a central decision unit and used to issue a global prognostic decision.

Definition 32 (Coprognosability)

Coprognosability is the ability to be coprognosable. If a fault f ∈ Σ f is prognosable by a local prognoser, it is said coprognosable. If all faults f ∈ Σ f are coprognosable, the system is coprognosable.

This article is one of the first to use the word "prognosability" on DES to refer to the predictability property.

Prognosability in CS

The community working on CS is quite active and consequent. However, prognosability in CS is not that studied and only a few works give a formal definition of this notion.

(M. J. Daigle & Sankararaman, 2013) work on prognostic on CS. They define the system with the set of equations:

x k+1 = f (k, x k , θ k , u k , v k ) y k = h(k, x k , θ k , u k , n k ) ( 4
)
where k is the discrete time variable, x k ∈ R nx is the state vector at time k, θ k ∈ R n θ is the unknown parameter vector at time k, u k ∈ R nu is the input vector at time k, v k ∈ R nv is the process noise vector at time k, n k ∈ R nn is the measurement noise vector at time k and h is the output equation. θ k is used to capture the model parameters whose values are unknown and time-varying stochastically.

In this work, their purpose is not to compute the date of occurrence of a fault but to compute the RUL of the system. They use an estimation process, computing the joint stateparameter estimate based on the observations up to the considered time and a prediction process, which computes a probability distribution of the EoL for each time. This probability distribution is computed with the joint state-parameter distribution and the system model, along with the distributions of possible parameters, inputs and process noise trajectories.

No formal definition of prognosability is given in the article, but some metrics, defined later in this paper, are used to check the performances of the result. Many works on CS are actually based on these two estimation and prediction processes, among them are [START_REF] Sankararaman | Analytical algorithms to quantify the uncertainty in remaining useful life prediction[END_REF] or [START_REF] Acuña | Theoretical advances in lebesgue-samplingbased prognostic algorithms[END_REF], which differs through the use of Lebesgue sampling, compared to the more resource consuming Riemman sampling, and thus use probability distribution for the occurrence of faults or particular events.

Prognosability in HS

On HS, the work of (M. [START_REF] Daigle | Modelbased prognostics of hybrid systems[END_REF] can be considered. Although a definition of prognosability is not given, we can deduce one from the article, as follows. A HS is considered as a set of components, each described by a set of discrete modes, with a set of constraints describing the continuous dynamics of the component in each mode.

Definition 33 (Prognosability based on [START_REF] Daigle | Modelbased prognostics of hybrid systems[END_REF] Prognosability is the ability to be prognosable. An event e is said prognosable on a hybrid system if, given the system model, the initial state s 0 , a time horizon k, the future parameters trajectory, the future input trajectory and the future process noise trajectory, it is possible to find one particular trace of events and variables evolution leading to the event e. If all the events e of the system are prognosable, the system is prognosable.

Prognosability in HtS

We have only found a few articles on HtS, as this type of system is not really studied. No definition of prognosability is given in [START_REF] Bartram | Bayesian reliability prognosis for systems with heterogeneous information[END_REF] but the article is interesting as it apply a prognostic algorithm. The occurrence of a fault is represented with Bayes Networks and probabilities, and is updated through the occurrence of events on the system's components.

Even if most works do not directly consider prognosability, a set of metrics have to be defined to evaluate the prognostic results.

Metrics for Prognosability

When evaluating prognostic performance, commons metrics are used:

• The precision, representing the spread of the results,

• The accuracy, representing the bias,

• The Mean Squarred Error (MSE),

• The Mean Absolute Percentage Error Some more advanced metrics are defined in [START_REF] Saxena | Metrics for offline evaluation of prognostic performance[END_REF]. These metrics are computed one after the other as illustrated in Figure 2. The prognostic horizon represents the ability for the pronostic algorithm Φ to predict within a specified error margin (represented by a parameter α ∈ R) the EoL and how much time before it happens. Then, the second metric, based on the accuracy, is the Cumulative Relative Accuracy (CRA), representing the ability for Φ to improve its estimation of the EoL as more information become available. Finally, the third metric introduced is convergence, showing how fast Φ converges. The α -λ-performance property, representing the ability for Φ to predict within a specified error margin (α) the EoL at any time λ.

Three metrics to choose a feature interesting to monitor to prognose a fault f ∈ Σ f are introduced in (Coble & Hines, Each of these metrics ranges from 0 to 1, and are based on data obtained through Run-to-Failures (RtF) of the system. A feature is a variable of the system, and is sampled during the RtF.

The first metric, monotonicity, characterizes the trend of the feature. It is given by the average absolute difference of the fraction of the derivatives of the data. When monotonicity is close to 1 it represents a constant trend of the feature all along the life of the system (it only decreases or increases).

Prognosability gives a measure of the variance in the critical failure value of a feature of the system. The prognosability metric is obtained by computing the deviation of the feature at the time where the fault occurs. The closer the prognosability metric is to 1, the better the chosen feature is to prognose the studied fault. Finally, the last metric, trendability, shows if chosen features have the same underlying shape and can be described by the same functional form (i.e. the evolution of these different parameters are similar through the life of the system).

Synthesis on prognosability

As we have seen, prognosability is a notion which is not as well defined as diagnosability in the literature. The definitions of prognosability differ, be it on the object of prediction (what is prognosed) or the type of results given by the analysis.

For diagnosability property, the occurrence of a particular event (usually a fault) is what is looked for. For the prognosability property, works look to predict the occurrence of an event or of a set of events. The prognosis result is sometimes the prediction of the occurrence of a fault event or the temporal prediction of the RUL. These different types of prediction are gathered in Table 5. The results of the analysis also differs in most of the definitions, as can be seen in Table 6. Some definitions of the prognosability property give a prediction on how long, in terms of number of events, of steps, or time units before the occurrence of the fault we can know it is going to occurs. For some definitions, the idea of time is not present, as the results of the prognosability analysis just give the knowledge that the fault is prognosable or not. Finally, some works give a timed probability information: at time k, a fault f ∈ Σ f has p(f ) probability to occur. Some other works give a non-timed probability information: there is probability p 1 (f ) for a fault f ∈ Σ f to occur and be predicted, p 2 (f ) for the fault to occur without being predicted or p 3 (f ) not to occur.

CONCLUSION

Through this review, it was shown that diagnosability is a notion quite well defined. This is not the case for prognosability, for which the definition and the ways to use it are quite different depending on the studied system. It can be explained by the fact that the PHM community is quite recent and first works were guided by applications. For discrete event systems, that will only produce and encounter discrete data, both definition of diagnosability and prognosability are well defined.

For continuous systems, described by a set of continuous dynamic equations, diagnosability is well defined, whereas prognosability is not. For hybrid systems, who will always encounter both continuous and discrete data, diagnosability is well defined whereas prognosability is not, although the emergence of works on this type of system is really recent. Finally, for heterogeneous systems, that we consider as a system that will sometimes be affected by discrete data only, sometimes by continuous data only and sometimes by both, neither diagnosability nor prognosability are well defined, and only a few works consider this type of system. For our future works, we would like to consider an heterogeneous system under uncertainty and have some ideas about which definitions of diagnosability and prognosability will be used as a base for this kind of system. In the case where data are available for this kind of system, the prognosability metrics proposed by [START_REF] Coble | Identifying optimal prognostic parameters from data: a genetic algorithms approach[END_REF] are interesting to choose which of the feature are best to prognose a fault f . The others metrics could be useful to evaluate the performance of an health monitoring algorithm implemented on this system.

Figure 2 .

 2 Figure 2. Sequence of Saxena's four metrics

Object of prediction

Articles Fault [START_REF] Bartram | Bayesian reliability prognosis for systems with heterogeneous information[END_REF]) [START_REF] Cassez | Predictability of event occurrences in timed systems[END_REF]) [START_REF] Wu | Verification of coprognosability in decentralized fault prognosis of labeled Petri nets[END_REF]) [START_REF] Kumar | Decentralized prognosis of failures in discrete event systems[END_REF]) [START_REF] Chen | Stochastic failure prognosability of discrete event systems[END_REF]) [START_REF] Ye | A predictability algorithm for distributed discrete event systems[END_REF] (Yin, 2017) Any system event (M. [START_REF] Daigle | Modelbased prognostics of hybrid systems[END_REF]) [START_REF] Genc | Predictability of event occurrences in partially-observed discrete-event systems[END_REF] Any system event or set of event [START_REF] Cao | The predictability of discrete event systems[END_REF]) Reach of a particular state [START_REF] Nouioua | Predictability in probabilistic discrete event systems[END_REF]) Remaining Useful Life (M. J. Daigle & Sankararaman, 2013)