Konstantinos Christodoulopoulos
email: konstantinos.1.christodoulopoulos@nokia-bell-labs.com

Ippokratis Sartzetakis

Polizois Soumplis

Emmanouel (Manos Varvarigos

Machine Learning Assisted Quality of Transmission Estimation and Planning with Reduced Margins

Keywords: overprovisioning, static network planning, end-of-life margins, physical layer impairments, monitoring, cross-layer optimization, incremental multi-period planning, marginless

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Coherent transmission and Elastic Optical Networks (EONs) deployed today promise higher spectral efficiency, increased capacity, and reduced network costs [START_REF] Lopez | Elastic Optical Networks. Architectures, Technologies, and Control[END_REF]. However, optical networks are traditionally planned to be operated statically; planning relies on abundant margins [START_REF] Pointurier | Design of Low-Margin Optical Networks[END_REF][START_REF] Auge | Can we use Flexible Transponders to Reduce Margins?[END_REF] to ensure that all connections/ lightpaths have acceptable Quality of Transmission (QoT) until the end of life (EoL). Lowering the margins and increasing efficiency reduces network costs, motivating various research directions [START_REF] Christodoulopoulos | ORCHESTRA-Optical performance monitoring enabling flexible networking[END_REF]. Ref. [START_REF] Pesic | Faster return of investment in WDM networks when elastic transponders dynamically fit ageing of link margins[END_REF][START_REF] Soumplis | Network Planning with Actual Margins[END_REF] studied the planning of an EON over multiple periods to harvest the evolution of margins over time. Certain connection parameters are adjusted at a given time granularity (e.g., 1-2 years) and new equipment is added, when actually needed, according to current traffic and physical layer conditions. Therefore, instead of overprovisioning to reach EoL, the network is operated with just in time provisioning, increasing the efficiency, postponing or avoiding equipment purchase, and reducing the network costs. An EON can be operated even more dynamically, by adapting the transmission parameters at shorter timeframes. Refs. [START_REF] Christodoulopoulos | Time-varying spectrum allocation policies and blocking analysis in flexible optical networks[END_REF][START_REF] Rafique | Machine Learning for Network Automation: Overview, Architecture, and Applications[END_REF] and other works consider the dynamic spectrum/capacity adaptation according to traffic demands. The efficient management of failures also pertains to dynamic network operation. [START_REF] Sartzetakis | Cross-layer adaptive elastic optical networks[END_REF] studies the dynamic adaptation of the connections' parameters to current physical layer conditions (e.g., in case of QoT degradation). The dynamic operation of a marginless network is experimentally demonstrated in [START_REF] Delezoide | Marginless Operation of Optical Networks[END_REF].

Considering the above, an accurate QoT estimator is the key component for i) reducing the margins during planning/upgrades, and ii) realizing a more dynamic operation of the network. Fig. 1a shows how a QoT estimation tool (or Qtool) is used by an optimization (planning or dynamic) algorithm. The Qtool consists of a physical layer model (PLM), which is analytical or semi-analytical and makes certain assumptions to estimate with certain accuracy the QoT (e.g. SNR, BER) of new or reconfigured connections. The Qtool takes as input the parameters of the established connections (in case of an operating network), and also certain physical layer parameters, such as spans, fibers, amplifiers and node parameters, etc. The values of the physical layer input parameters are not accurately known, due to lack or limited accuracy of measuring equipment, outdated measurements, etc. To cover the model and input parameter inaccuracies, the design margin is used (2dB in SNR as a reference [START_REF] Pointurier | Design of Low-Margin Optical Networks[END_REF][START_REF] Auge | Can we use Flexible Transponders to Reduce Margins?[END_REF]). Moreover, traditional network planning targets for new connections to have acceptable QoT at EOL, e.g. after 10-15 years. Modeling equipment ageing, increase of interference, reparations of fiber cuts, etc. contributes to the system margin (3dB in SNR as a reference [START_REF] Pointurier | Design of Low-Margin Optical Networks[END_REF][START_REF] Auge | Can we use Flexible Transponders to Reduce Margins?[END_REF]). A great deal of recent research effort was directed towards the application of Machine Learning (ML) in various areas of optical networking both at the physical and network layers [START_REF] Musumeci | An Overview on Application of Machine Learning Techniques in Optical Networks[END_REF]. ML has also been applied to improve the accuracy of QoT estimation and reduce the design and system margins (Fig. 1b). Fig. 1. a) QoT estimation tool (Qtool) used by an algorithm to reconfigure or establish lightpaths, its inputs, and design and system margins, b) Machine learning assisted QoT estimation to reduce the margins.

Monitoring

In this paper, which outlines [START_REF] Sartzetakis | Accurate Quality of Transmission Estimation with Machine Learning[END_REF] and [START_REF] Soumplis | Network Planning with Actual Margins[END_REF][START_REF] Soumplis | Multi-period planning with actual physical and traffic conditions[END_REF], we start by presenting two ML assisted QoT estimators. We then evaluate the accuracy of the estimators under various scenarios and observe quite low estimation error with relatively low information (few established lightpaths). We then quantify the cost savings that can be obtained by accurate QoT estimation in a multiperiod planning network scenario. To do so we interface the ML assisted QoT estimator with the incremental algorithm of [START_REF] Soumplis | Network Planning with Actual Margins[END_REF][START_REF] Soumplis | Multi-period planning with actual physical and traffic conditions[END_REF] and use that to (accurately) estimate the QoT of new and existing lightpaths and provision/ reconfigure them with reduced margins.

Quality of Transmission (QoT) Estimation

We assume an EON with Reconfigurable Optical Add/Drop Multiplexers (ROADMs) connected through uncompensated fiber links. Each link consists of a number of fiber spans that terminate at an EDFA that compensates the span loss. We assume that there are no spectrum converters and thus a lightpath is allocated the same spectrum throughout its path. For long connections, regenerators are placed. The set of established lightpaths and their attributes (e.g., modulation format, baudrate) is denoted by P which will also be referred to as the state of the network at a given time. We also assume that we can obtain monitoring information from the coherent receivers. Coherent receivers deployed today are packed with DSP capabilities, so they can function as Optical Performance Monitors (OPMs) [START_REF] Christodoulopoulos | ORCHESTRA-Optical performance monitoring enabling flexible networking[END_REF]. So we assume that an OPM (receiver) monitors the lightpath's SNR with certain accuracy. We use this information to improve the QoT estimation accuracy and reduce the margins of new or reconfigured lightpaths.

In particular, we denote by 𝑄 * (𝑃) the vector that contains the SNR values of all established lightpaths p∈P. The mapping 𝑄 * (𝑃) is nonlinear and unknown to us. For the set of established lightpaths P we denote by 𝑌(𝑃) the vector of their monitored SNR values. The monitoring error consists of a systematic and a random error. The systematic error can be reduced through proper calibration, while the random error can be reduced by averaging measurements over time (at a shorter timescale than a natural change of the monitored value would occur). As monitoring errors are small and can be reduced, we ignore them here for simplicity. So, we will assume we monitor the true SNR for the paths (𝑌(𝑃) = 𝑄 * (𝑃) for p ∈P).

We consider the case where a new lightpath 𝑤 ∉ 𝑃 is about to be established, and we want to estimate (i) the QoT of that new lightpath w, and (ii) how QoT of existing lightpaths p∈P will be affected by the establishment of w. Stated formally, we are given the measurements 𝑌(𝑃) = 𝑄 * (𝑃) of QoT metrics at the current network state P. Our objective is to estimate the new QoT metrics 𝑄 * (𝑃 ∪ {𝑤}) after the new lightpath 𝑤 ∉ 𝑃 is established. Note that 𝑄 * (𝑃 ∪ {𝑤}) contains the (new) QoT metrics of existing lightpaths p∈P, which in general will be affected by the establishment of w. Also note that the above problem definition can be extended to include the establishment of a set or the reconfiguration of a single or a set of lightpaths.

Machine Learning Assisted QoT Estimation

In QoT estimation, our goal is to identify a parametric function 𝑄 ̃(𝑟, 𝑃) that approximates well the actual QoT of the connections 𝑄 * (𝑃). Here r is a set of parameters of the model. The parametric function 𝑄 ̃(𝑟, 𝑃) does not have to be a closed form expression; it can also be the output of a simulation. What is important is that (i) 𝑄 ̃(𝑟, 𝑃) approximates 𝑄 * (𝑃) relatively well and that (ii) given the set r, it is relatively easy computationally to obtain 𝑄 ̃(𝑟, 𝑃) for the given state P. The following two subsections describe two approximating architectures, corresponding to different parametric function choices, the first based on a physical layer model (PLM) and the second based on a machine learning (ML) model and features extraction.

Machine Learning Physical Layer Model (ML-PLM)

The first approach considers a physical layer model (PLM) as the approximating architecture. In this case, 𝑄 ̃(𝑟, 𝑃) takes as input r the PLM input parameters, denoted by b, which can include: per span length, fiber attenuation, dispersion and nonlinear coefficients, the noise figure of each EDFA, its gain, etc. So, following the above we denote the PLM model with 𝑄 ̃(𝑏, 𝑃). In this paper, we will use the GN model [START_REF] Poggiolini | The GN Model of Non-Linear Propagation in Uncompensated Coherent Optical Systems[END_REF] as the PLM, but the method is generic and applicable to other PLMs.

The assumption is that the initial parameter values are close to the actual ones but not close enough to provide accurate QoT estimation. They could be based on datasheets or even field measurements that however cannot be taken continuously and with high accuracy, so they would be partially outdated / inaccurate. The objective of the machine-learning (ML)-PLM is to use monitoring information from the established lightpaths 𝑌(𝑃) to train and learn the physical layer parameters and thus improve the accuracy of future QoT estimations. The model also uses the set of lightpaths' parameters included in the network state P, which can be: the route, central frequency, baudrate, modulation format, launch power, etc. The parameters in P are assumed to be perfectly known, as opposed to the parameters in b. Regarding the learning algorithm, if we have closed forms for the partial derivatives ∂𝑄 ̃/∂bj with respect to all physical layer parameters bj of b, we could use gradient decent to obtain better estimates of b as in [START_REF] Seve | Learning Process for Reducing Uncertainties on Network Parameters and Design Margins[END_REF]. Here we assume a generic case where 𝑄 ̃(𝑏, 𝑃) is unknown. Assuming that QoT depends non-linearly on some input parameter, we use a nonlinear method to fit the model.

Machine Learning Model (ML-M)

The second investigated approach uses a Machine Learning Model (ML-M) and features extraction. Features extraction maps the state P into a matrix = (𝑃), called the features matrix. The mapping function f is chosen to summarize in a heuristic way the important characteristics of P with respect to QoT estimation. The main idea is that through the change of variables from P to = (𝑃), the unknown mapping 𝑄 * (𝑃) of the true QoT is approximated well by the function 𝑄 ̃( , ) = 𝑄 ̃( , (𝑃)), where Θ are the ML-M parameters (coefficients). The type of coefficients Θ depends on the particular model (linear regression, neural network-NN, Support Vector Machine-SVM, etc). The ML-M 𝑄 ̃( , ) is trained with an appropriate regression algorithm, using, as in previous section, training data (𝑃, 𝑌(𝑃)).

After implementing and testing several such models, we observed that the choice of features is of utmost importance for the estimation accuracy, at least considering medium size networks (10-20 nodes). So, in the following we present the formulation that achieved the best performance. We follow a link-level feature formulation to account for network heterogeneity. So, in our ML estimation model we consider each lightpath's link attributes separately. Thus, the model distinguishes, for example, two lightpaths that have similar lengths but cross different links and possibly exhibit different QoT due to heterogeneities of link attributes. We do not consider span level features since a lightpath will always cross all spans of a link. We organize the feature matrix X so that each row corresponds to one lightpath p∈P, while the columns represent (link, impairment) features. We defined three sets of link-impairment features corresponding to the major impairment classes affecting the QoT. More specifically, we define A as a |𝑃| × |𝐿| link-level feature matrix designed to account for the ASE noise. Element Apl, corresponding to lightpath p and link l, is set equal to 1 if it contains link l, and is set to zero, otherwise. We also define the link-level feature matrix S to account for Self-Channel Interference (SCI) noise [START_REF] Poggiolini | The GN Model of Non-Linear Propagation in Uncompensated Coherent Optical Systems[END_REF]. Element Spl is set equal to the inverse of the square of the lightpath's baudrate if lightpath p contains link l, and is set to zero, otherwise. Finally, we define a link-level feature matrix W to account for the interference of neighboring lightpaths (cross channel interference-XCI [START_REF] Poggiolini | The GN Model of Non-Linear Propagation in Uncompensated Coherent Optical Systems[END_REF]). The elements of this matrix are derived from an equation that involves the baudrate of the lightpath under consideration, and for each of its neighboring lightpaths the spectrum distance and its baudrate, following Eq. (40) of [START_REF] Poggiolini | The GN Model of Non-Linear Propagation in Uncompensated Coherent Optical Systems[END_REF]. We also consider an additional feature, the bias term (denoted by BT). Bias can account for the monitoring error in Y(P), impairments not modeled, etc. We concatenate all the link-level feature matrices and the bias into one feature matrix = (𝑃):

  | | (3| | 1) P x L X BT A S W + =
Focusing on a lightpath p, its features are designed to represent its noise contributing parameters (considering the major types of impairments, both linear and nonlinear) per link. Assuming xp,j is the j th feature of lightpath p, the noise contribution of the related impairment/link is approximated well with a linear function, i.e. nj(xp,j,Θ)=xp,j . θj, where θj is the related impairment-link coefficient. We assume that the noise of the different types of impairments is additive on a link level and over the path, following the assumptions of the GN model [START_REF] Poggiolini | The GN Model of Non-Linear Propagation in Uncompensated Coherent Optical Systems[END_REF]. In other words, the total noise accumulated over lightpath p is given by ∑ 𝑛 𝑗 (𝑥 𝑝,𝑗 , ) 𝑗 . This assumption makes possible the correlation of the noise contributions at impairment and link levels. Based on these assumptions we estimate the total noise of the lightpaths by 𝑁 ( , 𝛸) = • . From that we obtain the SNR of the lightpaths by

𝑄 ̃( , ) = 𝐵 𝑁 • 𝑙 𝑝 𝐵 𝑟 • 𝑁 ( , 𝛸)
where BN is the noise bandwidth, lp is the launch power, and Br is the baudrate of the lightpaths. Then we use the SNR measurements Y(P) and a linear regression/gradient descent algorithm to learn .

Note that we derived a linear model to approximate 𝑄 ̃. However, we performed a non-linear transformation f of the input P, carefully designed so that the subsequent fitting would be well approximated using a linear model. The above described ML-M concept is quite generic. We can define different features X, and model more complicated features -noise functions nj with other ML models, such as NN or SVM, and train them with appropriate algorithms. Another possibility, under the linear feature -noise functions assumption, is to use a constrained least squares solver, where we can add constraints that exploit certain expected QoT relationships. For example, a link that is 200km longer is highly unlikely to contribute lower noise. The additional constraints help the model to provide better estimations especially in cases with low amount of information (lightpaths).

Machine Learning Assisted QoT Estimation

We evaluated the proposed machine learning QoT estimators through simulations.

We considered the DT topology with 12 nodes and 40 bidirectional links. We assumed 4 traffic loads of 100, 200, 300 and 400 total connections with uniformly chosen source-destinations and random baudrates from the set {32, 43, 56} Gbaud.

Regarding the physical layer, we assumed a span length of 80 km, EDFA noise figure 5 dB and SSMF with mean attenuation coefficient 0.23 dB/km, mean dispersion coefficient 16.7 ps/nm/km, and mean nonlinear coefficient 1.3 1/W/km. We set the launch power at 0 dbm. The actual (unknown) values of the fiber coefficients for each span were drawn from uniform distributions ranging by 0%, 10% or 20% around the above means, thus defining 3 uncertainty scenarios. The GN model was used as the ground truth with these values. For the ML-PLM case, the b vector consisted of the attenuation, dispersion, and nonlinear coefficients, initiated with their mean values. So, the case with 0% variation implies that the ML-PLM estimator has accurate knowledge of the physical layer parameters. The training of ML-PLM was done with nonlinear regression and in particular the Levenberg -Marquardt algorithm. For ML-M, we tried various models and we finally used the features described in Section 3.2 and the constrained least square solver that provided the best results.

For each traffic load and physical layer parameters uncertainty setting we executed 500 iterations with random traffic and span parameters. For each instance we used 85% of the lightpaths for training and 15% for testing. We excluded from the testing set the lightpaths that include links for which we have no QoT information (links that no lightpath from the training set cross). The training goal was to minimize the MSE but the max overestimation is also a very useful metric because it defines the design margin. This has to be used to be on the safe side, so that we never overestimate the QoT and establish a lightpath with unacceptable QoT.

Fig. 3a shows the MSE and Fig. 3b the max overestimation error. For both estimators the MSE decreases as the number of lightpaths increases. The ML-M has consistent performance for all the uncertainty scenarios (0% to 20% variations) since it does not assume any previous knowledge. On the other hand, ML-PLM's performance is affected by the magnitude of the uncertainty, since the ground truth values deviate from the model's initial values. The ML-PLM is more accurate than ML-M in all cases except for the case with high uncertainty (>10%) and few connections (100) where the ML-M achieved lower max overestimation. This is expected since the ML-PLM is a good approximation of the actual physical layer, in the sense that the ground truth function 𝑄 * (𝑃) was taken to be the GN model, which is also the ML-PLM function 𝑄 ̃(𝑏, 𝑃). So, ML-PLM only needs to fit the parameters b that are uncertain and starts from good (mean) values. The ML-PLM's max overestimation was around 0.05 dB for more than 200 lightpaths while ML-M's max overestimation was 0.2 dB. The training time was in the order of seconds for the ML-M/linear regression and minutes for the ML-PLM/nonlinear regression. Once trained, estimations are quite fast (<0.1 sec). So, the estimators can be used even in dynamic (re)configuration cases. We also obtained results for other ML estimators with end-to-end features as opposed to the link-level features. As expected, end-toend features resulted in larger MSE and max overestimation for the high uncertainty cases, since in those cases the network is non-homogeneous and end-to-end features do not cover that. Detailed results are presented in [START_REF] Sartzetakis | Accurate Quality of Transmission Estimation with Machine Learning[END_REF].

Planning with Reduced Margins -Algorithm

In Section 3 we presented two ML QoT estimators that were shown in Section 4 to achieve good accuracy and a low design margin. To quantify the benefits of an accurate QoT estimator we use/interface that to an incremental planning algorithm [START_REF] Meusburger | Optimizing the Migration of Channels With Higher Bitrates[END_REF][START_REF] Eira | Optimal Multi-Period Provisioning of Fixed and Flex-Rate Modular Line Interfaces in DWDM Networks[END_REF] and in particular to the incremental planning algorithm of [START_REF] Soumplis | Network Planning with Actual Margins[END_REF][START_REF] Soumplis | Multi-period planning with actual physical and traffic conditions[END_REF]. This incremental planning algorithm is involved before a planned network upgrade or even at the initial period (greenfield). So, the algorithm starts from a given network state P, with a set of deployed equipment, e.g. transponders/regenerators, established lightpaths with known transmission parameters, routes and spectrum, or even from an empty network. The algorithm takes as input a new set of demands and the available equipment at that time, e.g. available transponders, their capabilities and costs.

To serve the demands, it interfaces with a QoT Estimator (see Fig. 1a) to account for the physical layer behavior.

If the network is planned with high margins, which is the traditional approach, then the QoT Estimator uses a high design margin and EOL system margin (to target acceptable QoT after e.g. 10-15 years). So, the efficiency is low, more transponders/regenerators are placed, but lightpaths are ensured to be uninterrupted, to have acceptable QoT, until EOL. Instead, if we leverage an accurate QoT estimator we can use an appropriate algorithm to plan the network with reduced margins. In the first period, we will serve the new demands with a high design margin (no feedback/no establish connections to monitor, train and refine the QoT estimator), and with a reduced system margin to reach the next upgrade period (several months -few years, but lower than EOL). Then at an intermediate period, we train the QoT estimator, and we reduce the design margin. So, at each period the incremental planning algorithm performs two tasks: i) checks the remaining margins of previously established connections and reconfigures/adds transponders or regenerators to restore those that run out of margins (will have unacceptable QoT performance before the next or a targeted period), ii) serves the new demands by placing transponders/regenerators. In both cases it chooses the configuration of the transponders/regenerators, and decides the routes, and spectrum allocation, by interacting with the QoT estimator to check the physical layer performance.

In more detail, the algorithm starts in phase 1 by freeing resources for removed connections (not demanded anymore). This equipment can be repurposed, reused at its current position (for free) or moved (with a penalty) to serve some other demand. Then the algorithm in phase 2 fixes the established connections that are running out of margins. The algorithm decides how to reconfigure and/or add new transponders/ regenerators so that they have enough margins to reach the next or certain periods ahead. The algorithm's objective is to minimize the added cost, and as a secondary objective (controlled through a weight) to avoid extensive reconfigurations. For example, when adding a regenerator, the used path is broken into two segments and the same spectrum is allocated at both segments, avoiding any reconfiguration to other lightpaths. Then the algorithm in phase 3 serves the new demands by adding adequate transponders/ regenerators and choosing the routes and spectrum. Since, adding new lightpaths (phases 2 and 3) increases interference, the algorithm rechecks with the QoT estimator the QoT all lightpaths. The algorithm repeats phase 2 considering all previous and new lightpaths. Since the algorithm in phase 2 might add new lightpaths to restore some problematic, interference can increase again, and the algorithm repeats phase 2 until QoT is acceptable for all lightpaths.

The incremental algorithm is a heuristic. It processes the previously established lightpaths and serves new demands one by one, in some particular order. Since the performance depends on that order, we can use simulated annealing or some other meta-heuristic to search among orderings and find better solutions. Note that the execution time is not a major concern, since calculations are supposed to be offline, to decide the changes for the next upgrade. In any case planning is NP-hard. Jointly planning and accounting for the physical layer (also called impairment-aware or cross-layer) requires the integration of the planning algorithm and the Qtool. Some optimal formulations for DWDM [START_REF] Christodoulopoulos | Offline Routing and Wavelength Assignment in Transparent WDM Networks[END_REF] and EON [START_REF] Yan | Joint Assignment of Power, Routing, and Spectrum in Static Flexible-Grid Networks[END_REF] have appeared, where the resource allocation formulation is extended to include (simplified) impairment/ Qtool constraints. Such simplifications would require a higher design margin. In this paper we consider the Qtool as an external optimization module; such separation could result in some loss of optimality. Nevertheless, the Qtool can be more complicated and accurate, and we can use techniques such as ML to reduce and regulate the related margins, which in the end could yield better performance than a joint algorithm.

Planning with Reduced Margins -Case Study

To quantify the benefits of the developed accurate QoT estimators and the incremental planning algorithm with low margins we dimensioned a network over multiple periods and calculated the capital expenditure (CAPEX) at each period. We compare that with planning with EOL margins. The network topology we studied was the 12 node DT network topology as in Section 4. We planned the network over 11 periods (initial/brownfield and 10 incremental/ greenfield periods); one period would roughly correspond to one year. The initial traffic (period τ0) consisted of 200 connections with uniformly chosen source-destination pairs and uniformly demanded traffic between 100 to 200 Gbps. So, the initial traffic was 30 Tbps and it was increased by 20% each period by creating new demands in the same way. We assumed two types of elastic transponders (ET): i) 32 Gbaud, modulating with DP-QPSK, or DP-8QAM, or DP-16QAM, supporting capacities of 100, 150, and 200 Gbps, respectively, and ii) 64 Gbaud, modulating from DP-QPSK to DP-32QAM, supporting capacities of 200 up to 500 Gbps, respectively. The first ET of 32 Gbaud was assumed to be available at the initial period τ0 with price equal to 1 cost unit (CU) and the second ET of 64 Gbaud was assumed available at period τ5 with price again 1 CU (at that period). The ET prices were assumed to fall by 10% per period (so when the second ET is introduced the price of the first was 0.59 CU).

We again used the GN model to model the physical layer. In the first period τ0 we initialized the model with heterogeneous span parameters with 10% uncertainty, similar to Section 4. We executed 10 problem instances with different traffic and span parameters and averaged the results. To model the ageing of the network we considered the increase of fiber attenuation (e.g. due to cuts), ageing of ETs, EDFAs, and nodes (OXC). The interference was modeled according to the network load. Table 1 shows the increase of the model parameters per period. Note that the increase was uniform for all spans, but since we started with heterogeneous and uncertain conditions, they remained heterogeneous and unknown for all subsequent periods.

When planning the network with EOL margins, the Qtool uses a system margin based on the parameters of Table 1 assuming 10 periods (~3 dB), and full network load (each link with 60x32 Gbaud connections). The design margin was set to 2 dB (1 dB for the model inaccuracy and 1 dB for the input parameters inaccuracy). When planning the network with reduced margins, we used the ML-M estimator (Section 3.2). The system margin was based on the parameters of Table 1 assuming 2 periods (~0.6 dB). Then we assumed that at each period we monitor and obtained the SNR values, Y(P), for the established connections (the monitored parameters were calculated by the GN model, with the random created initial parameters and ageing according to the period and Table I -unknown to the used QoT estimator). We used the monitored SNR values to train the ML-M and obtain the coefficients for that period, and also the max overestimation error. The design margin was set equal to 1 dB for the model inaccuracy, plus the maximum between 0.2 dB (the design margin of the extensive simulations in Section 4) and the max overestimation in the training for that period. Note that we chose the system margin for the next 2 periods and a design margin greater than 1.2 dB to be conservative. In total when we plan the network with reduced margins we harvest ~3dB when compared to EOL planning. Fig. 4a presents the total cost of the deployed elastic transponders (ET) per period for the two provisioning approaches. As expected, reducing the margins yields lower costs. Reducing the system margins postpones the purchase of ET and we obtain savings from the 10% depreciation. Reducing the design margin avoids the purchase of equipment. Fig. 4a also presents the relative savings, found to be about 20% at the end of the examined periods. Note that the savings would reduce at later periods if we only assumed a single type of ET for all periods. The introduction of the higher rate transponder boosts the savings after period τ5. Fig. 4b shows the design margin used and the actual maximum estimation error. In all simulations (10 instances x 11 periods) we never observed a QoT problem, all lightpaths had adequate QoT when reaching the next period. The closest we got to margins limit was at period τ5, where the new ET (thus a new baudrate) was introduced, and some parameters were not learned accurately enough with only the first ET (32Gbaud). Fig. 4: a) Total cost (in cost units) and savings of elastic transponders (ET) per period when planning with accurate QoT estimation/reduced margins and with high margins, b) Used and needed design margin.

A factor not included in the above evaluation is the time value of money; money saved at intermediate periods can be invested (or loans can be avoided) resulting in extra savings. Additional savings can be also obtained by power optimization [START_REF] Soumplis | Network Planning with Actual Margins[END_REF].

Conclusions

Estimating the Quality of Transmission (QoT) is typically performed before establishing new or re-configuring existing connections using a physical layer model (PLM). Traditional planning uses high margins to account for inaccuracies and to target connections with acceptable QoT after several years (EOL). Reducing the margins increases network efficiency but requires accurate QoT estimation. We presented two machine learning (ML) assisted QoT Estimators that leverage monitoring information of existing connections to understand the actual physical conditions and achieve good estimation accuracy. We then quantified the benefits of planning/ upgrading a network with accurate QoT estimation. We interfaced the developed accurate ML QoT estimator with an incremental planning algorithm that handles the reduction of margins. We compared the multi-period planning of the network with accurate QoT estimation/ reduced margins to planning with EOL margins, and observed savings that reach 20% at the end of the examined periods.

Fig. 2 .

 2 Fig. 2. Flowcharts of a) machine learning physical layer model (ML-PLM) and b) machine learning model (ML-M) QoT estimators.

Fig. 3 a

 3 Fig. 3 a) MSE and b) maximum overestimation for 0%, 10% and 20% uncertainty.

Acknowledgment I.

  Sartzetakis was supported by IKY Greek State PhD scholarship co-funded by the European Social Fund and the Greek State.

  : 1dB + max (0.2dB + training max overest) needed design margin: 1dB + max error (after establishing)

Table 1 .

 1 Parameters to model the network evolution over time.

	Physical layer parameters evolution	Increase per period
	Ageing	Transponder margin (dB)	0.05
		Attenuation (dB/km)	0.0015
		EDFA noise figure (dB)	0.1
		OXC loss (dB)	0.3
		Interference	According to load


Yan Cui
email: yan_cui@student.uml.edu

Vinod M Vokkarane
email: vinod_vokkarane@uml.edu

Analytical Modeling of Survivable Anycast Communication in Optical Networks

Keywords: Survivability, Analytical Model, WDM Networks, Anycast

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

In recent years, with rapid growth in demands on network connections, the optical network, acting as the foundation of network connectivity, have been becoming more and more indispensable and important to our daily life. Hence, failures in optical networks may cause catastrophic disasters. Those network failures happen on both network nodes and network links are common due to the human error, such as fiber cuts during construction accidents [START_REF] Modiano | Survivable lightpath routing: A new approach to the design of WDM-based networks[END_REF] and natural disasters. The arrival of Hurricane Sandy in New York and New Jersey in October 2012 resulted in the failure of three hundred Verizon facilities along the eastern seaboard [START_REF] Kwasinsk | Hurricane sandy effects on communication systems[END_REF]. Other unavoidable natural disasters, such as the catastrophic destruction brought about by 2011 Tohoku earthquake and tsunami in Japan [START_REF] Adachi | The restoration of telecom power damages by the Great East Japan Earthquake[END_REF], can occur, often without warning. Therefore, the ability of recovery of network connectivity, which is called network survivability [START_REF] Ellison | Survivable network systems: An emerging discipline[END_REF], is of significance in the design of modern networks. Regarding traffic engineering strategies, it can be beneficial to use different transmission techniques with respect to the characteristics of realized demands. Nowadays, the conventional transmission paradigm is unicast, i.e., data transmits from a source node to a destination node. However, the unicast paradigm cannot accommodate the novel distributed network applications, e.g., content distribution and Data Center (DC).

In the distributed network applications, available network services are provided by more than one network service providers (e.g., the DCs). In order to support these services, anycast could be applied which refers to the transmission of data from a source node to any one member in the candidate destination set [5]. The anycast client can establish a connection with any of the available DCs and selects the target DC based on different network performance criteria (e.g., DC response time, distance to DC, DC load, etc.). As a consequence, anycast is capable of improving network performance, remarkably reduce the network load and may also provide protection against the selected the target DC failure [START_REF] Goścień | Survivable multipath routing of anycast and unicast traffic in elastic optical networks[END_REF].

The designs of survivable networks are often modeled as the linear programming problem with graph theory, wherein, the nodes represent the network components (such as computers, routers, etc.), and edges represent the communication links between the components. Therefore, the survivable network design problem can be modeled as a problem of finding a subgraph satisfying certain connectivity constraints, or augmenting a given network to certain connectivity requirements [START_REF] Davis | Failure-Aware Protection for Many-to-Many Routing in Content Centric Networks[END_REF]. In particular, the input is an undirected graph (or digraphs) with weights on the edges or nodes and prescribed demands on connectivity between nodes in the graph with the objective being the computation a subgraph of minimum weight that satisfies the connectivity demands.

There are mainly a drawback in the existing works on design of survivable networks, 1) authors are mainly focused on the design of optimal network survivability algorithms without considering the negative effects caused by new designed survivability algorithms. For instance, by selecting a subgraph for a pair of source and destination, there will be less available network resources for other traffics. This might cause severe network congestion when the amount of traffics is large.

In this work, we provide the theoretical analysis demonstrating the network-wide block rate with considering the survivability algorithms for anycast traffics. We assume two protection policies for anycast traffic networks. One is called survivable routing policy (SRP); The other is called survivable routing with relocation policy (SRRP). In the first scheme, two link-disjointed path are allocated to each pair of traffic source and destination. For efficient resource provision, in the second scheme, the Anycast connection is composed of two link disjointed routing paths between an anycast client and two same content DCs, wherein one path is used as the backup path just in case the primary path that carries the data fails.

The remainder of this paper is organized as follows: Section II introduces the network model and assumptions, and we describe the proposed blocking probability analytical model in Section III. Numerical evaluation and model verification are discussed in Section IV, and Section V concludes this paper.

Network Model and Assumptions

We consider a stochastic connection request arrival process and model connection arrivals in the network as a Poisson process. We also assume that the holding time of connection requests is exponentially distributed. In the analysis, the total offered load of the network is uniformly distributed between different anycast client.

We adopt the first-fit wavelength assignment (FF-WA) policy. In this scheme, all wavelengths are indexed and lowest indexed available wavelength is assigned before a higher indexed wavelength. We also assume that the resource provisioning and allocation for the dynamic connection request starts as soon as the request arrives into the network. The connection requests are holding-time-aware, each providing an exact duration.

Moreover, we assume that full wavelength conversion between all input and output links at all intermediate optical cross-connects is available. This provide the ability of fully making use of available wavelength spectrum for every link in a path.

Survivable Routing Policy (SRP)

We denote a anycast connection request as ( , ). Where is anycast connection request node, and = ( , , … , ) is a destination set and all candidate destinations are numbered. We assume that a source node is randomly chosen from a node set and a candidate destination set is randomly chosen from the same node set except the source node for each incoming connection request.

For each source and candidate destination pair, we use Dijkstra's algorithm to find the optimal link-disjointed primary path and backup path. We denote the primary path as ( , ), and the backup path as ( , ), wherein is the index of candidate destination. For example, Fig. 1 shows for the anycast client located at node 1, the destination set includes two destinations: node 2 and node 5. For the first destination node 2 , the primary path generated by Dijkstra's algorithm is ( , ) and the related link-disjointed backup path is ( , ). For the second destination node 5, the primary path and backup path are , and ( , ).

Anycast service requires selection of a destination for the candidate destinations. We use a first-fit destination selection (FF-DS) policy. In the FF-DS, all candidate destinations are numbered. Each destination will be checked one by one to verify if there are any available wavelengths for the incoming request. If at least one wavelength is available on each link along the primary and backup paths at a destination, the reservation is successful. Otherwise, the next destination will be checked. If there are no available wavelength on any link along all destinations, the connection request will be blocked. As an example a anycast client at node 1, the first destination node 2 will be checked first, followed by the second destination node 5. If there are no available wavelengths on any link all the two destinations, the anycast request will be blocked.

Survivable Routing with Relocation Policy (SRRP)

We denote an connection request as ( , D), where is the anycast connection request node, = ( , ) and is the selected target DC node, and is backup data center node with same content. We assume that each node may be a source node or a data center node for the incoming connection request. We assume there are DC nodes provide same content as backup choice. The backup data center node is generated randomly among of the same DC nodes.

For each source -target DC-backup DC pair, we use Dijkstra's algorithm to find the optimal link-disjointed primary path and backup path. We denote the primary path as ( , ), and the backup path as ( , ). For example, Fig. 2 shows for the anycast client located at node 1, the primary path generated by Dijkstra's algorithm is ( , ) and the related link-disjointed backup path is ( , ).

In order to cope with link failures, a connection needs to be allocated with a primary path and a backup path. The primary path ( , ) is the working path used to transfer its data from the source to the destination . The data is rerouted through the backup path in case of a failure on the primary path. We adopt dedicated path protection scheme to define the blocking problem for incoming connection request. When a connection request arrival at the network, the request tries to get resource allocation on the primary path and backup path, if currently there are available resource on each link traversed by primary path or backup path, we call this connection request is allocated successfully. Otherwise, the connection request will be blocked.

Analytical Blocking Model

There are two parts in this section. In the first part, we give the computation process of link arrival rates to calculate the link blocking probability. We present our theoretical model on how to calculate the average network blocking probability in the second part.

Computation of Link Arrival Rate Survivable Routing Policy (SRP)

Link arrival rate when |D|= 1 This is called unicast traffic, when only one destination is included in destination set for incoming request. Since each node may be a source node or a candidate destination node, we can find that the total number of combinations of source-candidate destination pairs is • ( -1), if the network has nodes. Since the total offered load to the network is uniformly distributed among source-candidate destination set pairs, we can derive the arrival rate between a source and a destination as

, = , = !(!" ) (1)
We obtain the arrival rate # for link $ by combining the contributions of requests from all primary and backup paths that traverse such a link. Hence,

# = ∑ , , |#'( , ) *( #∈ ( , , )
.

(2)

Link arrival rate when |D|>=2

Each source-candidate destination set pair includes a source node and two destination nodes, which means the size of the route set is 2. As each node in an anycast network may be a source node or a destination node, if a network graph includes nodes, the number of combinations of source-candidate destination set pairs is ( -1)( -2).

Since the total offered load to the network is uniformly distributed among sourcecandidate destination set pairs, we can derive the arrival rate of a route set between a source and candidate destination set as

, = !•(!" )(!" ) (3)
The request to the destination set will arrive at the first destination and attempt resource allocation, so the arrival rate of the first destination is the same as the arrival rate of the route set. However, if the request coming to first destination set is blocked, the request then arrives to the second destination and tries to receive resource allocation. So the contributed arrival rate to the second route is from the requests which are already blocked on the first destination.

In summary, we can derive the arrival rate of each source-destination pair by combining the contributions of requests that arrive at the destination set as In above equation, the first term is the sum of the arrival rate of source-candidate destination set pairs in which the destination is the first destination in a destination set. The second term is the sum of the arrival rate of destination sets in which the is the second destination. as shown in Fig. 3. Considering that the blocking probability / 0 that one source destination pair is blocked has already been computed in unicast network, we can compute the arrival rate of a source-candidate destination pair using the above equation. We can use Eq. ( 2) to calculate the link arrival rate after having computed the route arrival rate. Survivable Routing with Relocation Policy (SRRP)

, = ∑ , , + ∑ , / 0 , . (4)
We denote the total mean arrival rate to the network as λ, the arrival rate for each primary-backup paths set as , ) , , , the arrival rate of primary path between source and destination as , ) and the arrival rate of backup path as , , . We will derive the link $ arrival rate # based on the connection request traversed by link $.

Since each node may be an anycast client node or a target DC node and the backup DC node is fixed based on the target DC node, we can find that the total number of combinations of client-DC pairs is • ( -1), if the network has nodes. Since the total offered load to the network is uniformly distributed among client-DC pairs, we can derive the arrival rate between a client node and DC node as , ) , , = , ) = , , = !(!" ) .

(5)

The connection request will go to primary path and backup path and try to get resource allocation. For a route (source-destination pair) in a network generated by route algorithm, as shown in Fig. 4, the route may be as a primary path or as backup path to do the resource allocation. We denote a route in a network as ( , ) and the cumulated arrival rate going through this route as , . We can get the arrival rate of a route (source-destination pair) generated by routing algorithm as

, = , ) + , , (6)
We obtain the arrival rate # for link $ by combining the contributions of requests from all routes ( , ) that traverse such a link. Hence, # = ∑ , , |#'(( , ) .

(7)

Network-wide Blocking Model

The average generalized network blocking model is obtained in three steps. First, we provide a link blocking model based on Erlang-B model. We then present the primary and backup path set blocking computation. After having computed primary and backup path set blocking, we can calculate the average network blocking probability.

Link Blocking Analysis

If there is no available wavelength for incoming request on a link, the request is called blocked on this link. We can model a link as a queuing system. We consider the number of wavelengths for each link to be equal and denoted as 3 and the average holding time of request is τ. We can calculate the blocking probability of link $, denoted as B # , which is equal to the Erlang loss formula [9],

5 # = 5(3, # 6) = 78 9 :; < =! ∑ 78 9 :; ? ?! < ?@A . ( 8
)
In the above equation, # represents the arrival rate of link $.

Primary -backup Path Set Blocking Computation

Under SRP

• For Anycast| | = 1 One destination node means only one pair of primary and backup paths. We can assume that the wavelength allocation on a path is independent between the links which are traversed by this path. With wavelength converters, different wavelength can be assigned on different links along a path. For survivability, there is at least one available wavelength on each link traversed by primary path and backup path for incoming request to get resource allocation. Considering link disjoint constrain, the probability that a connection request gets blocked which means at least one link along a primary path or backup path has no available wavelength when the connection request arrives to the network is equal to 1 minus the probability that the connection is not blocked in any of the corresponding link $ along the primary path ( , )and backup path ( , ), Hence, we can derive the probability that a primary-backup set is blocked as

• For Anycast| | = 2 If the candidate destination set size is two, there are one primary backup path for each destinations. We know there are no common link between primary and backup paths going to same destination. However common links probably happen between paths belongs to different destinations. We classify the links traversed by the all paths of this anycast traffic as three sets: set FG includes common links traversed by the first and second destinations; set G includes links going to the first destination only; set G includes links going to the second destination only. If any link belongs to FG has no available wavelength, the incoming request will be blocked, because the traffic cannot go the first or second destinations. We define this blocking probability as

5 = 1 -∏ (1 -5 # ) #∈HI . ( 10
)
If all links belongs to FG has at least one available wavelength, but any link belongs to set G and any link belongs to set G has no available wavelength. The incoming traffic is still blocked. We define this blocking probability as

5 = 1 -∏ (1 -5 # ) #∈I J 1 -∏ (1 -5 # ) #∈I 0 ∏ (1 -5 # ) #∈HI . (11)
In the above, the two cases are mutually exclusive, so we can sum the probability that each case happens to affect the primary-backup path set blocking probability:

5 , = 5 + 5 . ( 12
)
Under SRRP

We can assume that the wavelength allocation on a route is independent between the links which are traversed by this route. With wavelength converters, different wavelength can be assigned on different links along a route. For survivability, there is at least one available wavelength on each link traversed by primary path and backup path for incoming request to get resource allocation. Considering link disjoint constrain, the probability that a connection request gets blocked which means at least one link along a primary path or backup path has no available wavelength when the connection request arrives to the network is equal to 1 minus the probability that the connection is not blocked in any of the corresponding link $ along the primary path ( , )and backup path ( , ), Hence, we can derive the probability that a primarybackup set is blocked as

5 , ) , , = 1 -∏ (1 -5 # ) #:#∈ , ) *( #∈ ( , , )
.

(13)

Network-wide Blocking probability

We can calculate the network wide blocking probability after having computed every individual primary-backup path set blocking probability, which is simply defined as

K = ∑ L,M ) ,M , •N•O L,M ) ,M , L,M ) ,M , ∑ L,M ) ,M , •N L,M ) ,M , . (14)

Numerical Results and Analysis

In this section, we assess the analytical blocking model proposed in Section III and compare its performance with simulation results on the 14-node National Science Foundation network (NSFnet) shown in Fig. 5.

In the simulations, we assume a Poisson arrival process with an average arrival rate of and an exponential distribution for request holding time with average holding time τ equal to 1. The total offered load is uniformly distributed among source-candidate destination set pairs. For a given simulation set, we change the arrival rate in order to generate the desired offered load (P = 6). The resource provisioning and allocation for a connection request starts as soon as the call arrives into the network. We use the first-fit wavelength assignment and fixed route policy (Dijkstra's shortest path) to determine the link-disjointed primary and backup path for one source-destination pair or one source-target DC-backup DC pair. We use first-fit destination selection (FF-DS) policy for anycast requests to do the simulation. We assume 8 wavelengths on each link to do the simulation. The simulation results were averaged over 30 seeds of 10 R connection demands each.

Reduced load [START_REF] Cui | Analytical blocking model for anycast RWA in optical WDM networks[END_REF] is a scientific method for the purpose of making theoretical model more close to reality. It is widely adopted in theoretical model of optical networks. To make our work valuable, we all involve reduced load in our analytical models. Fig. 6 shows the average network blocking probability performance as a function of different offered load for unicast traffic (| | = 1) in survivability NSFnet. We can observe that the analytical results accurately match the simulation results. As expected, the blocking probability increases as the offered load increases. If the offered load is fixed, the blocking rate for unicast traffic with survivability is much higher than that unicast traffic without survivability algorithm. This is because two linkdisjointed paths are provided for each source-destination pair in survivability networks, which needs much more network resource to allocate one single traffic, However the network resource provided are same. Fig. 7 shows the average network blocking probability as a function of different offered load under SRP and under SRRP in NSFnet. We can observe that the analytical results accurately match the simulation results. It is worth noting that comparing the results between | | = 1 and | | = 2 under SRP for the NSFnet with same offered load, and same number of wavelengths on each link, the blocking probability on the latter is lower. This is because increasing the number of candidate destinations means increasing the size of the route set for anycast, providing more opportunities for a request to succeed in resource allocation based on FF-DS policy. To gain more insight in the efficiency of survivable routing with relocation scheme, We can observe that performance under survivable routing with relocation scheme are much better than that of under survivable routing for anycast (| | = 2) paradigm, with same offered load.

Conclusion

We presented analytical models to compute network-wide blocking probability for anycast traffic paradigm in survivability optical WDM networks. Results demonstrate that our model provides good accuracy compared to simulation results. The models can be considered a useful design tool for anycast survivability optical WDM networks. For future work, we will work on analytical models for many-to-many traffic paradigms in survivability networks [START_REF] Davis | Failure-Aware Protection for Many-to-Many Routing in Content Centric Networks[END_REF].

Fig . 1 .

 1 Fig .1. Anycast connection with primary and backup paths for each DC.

Fig . 2 .

 2 Fig .2. Anycast connection with primary and backup paths.

  p r im a r y c o n n e c t io n 1 3

Fig . 3 .

 3 Fig .3. A source destination pair (1,2): (a) node 2 as first destination (b) node 2 as second one

Fig . 4 .

 4 Fig .4. A path ( , ) generated by routing algorithm: (a) as a primary path (b) as backup path for different connection request.

Fig . 5 .

 5 Fig .5. NSF network.

Fig . 6 .

 6 Fig .6. Network blocking comparison between uncast paradigm | | = 1 without survivability and unicast paradigm with survivability.

Fig . 7 .

 7 Fig .7. Network blocking probability under SRP and SRRP


Anuj Agrawal

Student Member, IEEE, OSA Vimal Bhatia
email: vbhatia@iiti.ac.in

Senior Member, IEEE Shashi Prakash
email: sprakash@ietdavv.edu.in

Topology and Failure Modeling for Optical Network Resilience Analysis Against Earthquakes

des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Earthquakes have been a major cause of optical fiber cable (OFC) destruction in the past. For instance, the 2008 China Sichuan earthquake of magnitude 8.0 destroyed around 30, 000 km of OFCs [START_REF] Tran | Enhancing physical network robustness against earthquake disasters with additional links[END_REF]. In a telecom-cloud infrastructure (TCI) supported by optical network, various nodes interconnect datacenters (DCs), multiprotocol label switching (MPLS) areas, and cable landing stations (CLS) through OFCs [START_REF] Velasco | Provisioning, Recovery, and In-Operation Planning in Elastic Optical Networks[END_REF][START_REF]Ciena webinar: Diving into the Data Center Interconnect (DCI) Seascape[END_REF]. A rapid increase in the number of DC and OFC deployment has been observed in the recent past, and many more DCs and OFCs are planned to be deployed in future to enable beyond 5G telecommunication, and cloud services [START_REF]Ciena webinar: 5G will need fiber and lots of it[END_REF]. Moreover, to realize the maximum benefits of wireless network densification, backhaul optical networks also need to be densified [START_REF]Ciena webinar: 5G will need fiber and lots of it[END_REF][START_REF] Bhushan | Network densification: the dominant theme for wireless evolution into 5G[END_REF]. Thus, with a view to make the future TCI resilient against earthquakes, the optical network components (nodes/links) should be physically located in the low seismic-risk regions.

To design robust future optical networks, a method for comprehensive and long-term analysis of various network-resilience schemes is required to assess optical networks against a large number of possible earthquakes. Thus, in this work, we propose a stochastic failure and optical network topology (SFONT) model that can be used to estimate and analyze the impact of various network-resilience schemes on optical networks.

Since large-scale natural disasters such as earthquakes affect a large geographical region, all the fiber cables lying in that region may fail simultaneously. Such a large-scale failure caused due to a common disaster occurrence is referred to as geographically correlated multi-link failure (GCMF). In the studies [START_REF] Tran | Enhancing physical network robustness against earthquake disasters with additional links[END_REF][START_REF] Tran | Geographical route design of physical networks using earthquake risk information[END_REF][START_REF] Msongaleli | Disaster-aware submarine fiber-optic cable deployment for mesh networks[END_REF][START_REF] Saito | Spatial design of physical network robust against earthquakes[END_REF][START_REF] Cao | Survivable topology design of submarine networks[END_REF] that address the problem of robust physical network design, most of the works propose physical fiber route modification as a solution to achieve network robustness. However, in Fig. 1, we show that GCMFs are highly dependent on the physical location of nodes.

In Fig. 1, an example mesh-connected 6-node optical network is shown. Here, two earthquakes causing disaster in a circular area of radius r from their epicenters are shown. It can be observed that the earthquake that occur in the vicinity of node is vulnerable to a GCMF affecting three fiber links. However, the earthquake that occur far away from nodes is vulnerable to single-link failure that can be survived using the conventional dedicated path protection (DPP), or shared path protection (SPP) methods. Thus, for resilient network design against GCMFs, the locations of nodes are of utmost concern. For simulations, we consider GCMFs caused due to earthquakes having epicenters in the vicinity of nodes.

In this work, we study an optical network densification (OND) scenario with a view to design the future dense optical networks resilient against earthquakes. We propose a seismic-risk aware optical network densification (SRA-OND) scheme for robust physical network topology design that finds the low seismic-risk regions to locate the new nodes. We evaluate the proposed SFONT model by analyzing various OND stages using the proposed SRA-OND as well as a seismic-risk unaware (SRU)-OND approach (where nodes are randomly located without considering seismic risk). To verify the capability of the proposed SFONT model to emulate real-world network topologies and failure scenarios, we compare the proposed SFONT model with a real-world case considering past real earthquakes [10], and real Indian RailTel optical network topology [START_REF] Pandya | Simultaneous optimization of power economy and impairment awareness by traffic grooming, mixed regeneration, and all optical wavelength conversion with an experimental demonstration[END_REF].

Related Work:

The failure modeling and resilience analyses done by majority of the studies [START_REF] Tran | Geographical route design of physical networks using earthquake risk information[END_REF][START_REF] Msongaleli | Disaster-aware submarine fiber-optic cable deployment for mesh networks[END_REF][START_REF] Saito | Spatial design of physical network robust against earthquakes[END_REF][START_REF] Bao | Fairness-aware degradation based multipath re-provisioning strategy for post-disaster telecom mesh networks[END_REF][START_REF] Agrawal | SLA-aware differentiated QoS in elastic optical networks[END_REF][START_REF] Habib | Design of disaster-resilient optical datacenter networks[END_REF][START_REF] Agrawal | Survivability enhancement of backbone optical networks leveraging seismic zone information[END_REF] are based on one/two network topology, arbitrary disaster zones, and few instances of disaster occurrence. However, for a comprehensive and rigorous assessment of any resilience-improvement scheme, they should be analyzed for a number of randomly obtained networks, since the performance may vary with changes in network topologies. In the proposed SFONT model, random network topologies of varying size and connectivity are obtained in each iteration. In [START_REF] Saito | Spatial design of physical network robust against earthquakes[END_REF][START_REF] Cao | Survivable topology design of submarine networks[END_REF], spatially uniform occurrence of disasters has been assumed in a given geographical region. However, practically, in a large geographical region, the probability of occurrence of natural disasters varies with different geographical locations. We incorporate this characteristic of spatially non-uniform disaster probability in the proposed SFONT model to generate earthquake epicenters. Moreover, we leverage the spatially non-uniform disaster probability in the proposed SRA-OND to locate the nodes in low seismic-risk regions. In [START_REF] Bao | Fairness-aware degradation based multipath re-provisioning strategy for post-disaster telecom mesh networks[END_REF], four arbitrary sample disaster zones have been assumed for resilience-analysis. Random disaster zones have been assumed in [START_REF] Habib | Design of disaster-resilient optical datacenter networks[END_REF] without considering any disaster risk model. However, for a comprehensive and long-term assessment of any resilience-improvement scheme, it should be tested for a large number of possible disaster occurrences.

For the first time in the literature, in the proposed SFONT model, we perform optical network densification in a given geographical region using Matérn hard core point process (MHCPP) and Gabriel graphs (GGs). To generate large number of spatially non-uniformly distributed earthquakes as per seismic-risk, we employ Poisson point process (PPP). The proposed SFONT model is described in the next section. The notations used in this paper are summarized in Table 1.

Proposed Stochastic Failure and Optical Network Topology (SFONT) Model

To obtain network topology, and to generate failures in a geographical region, we consider a bounded set G ⊂ R 2 representing the geographical region of interest. In this paper, we consider a bounded region G of 3240 km × 3240 km representing India and the neighboring regions. The region lying outside the border of India is included in a set n G representing non-feasible regions for node placement. The objective is to obtain random network topology within the Indian terrestrial region (G ∩ n G), and to generate spatially non-uniformly distributed earthquakes as per the seismic-risk. To obtain seismic-risk information, we consider the seismic hazard map of India [START_REF] Murty | Where are the seismic zones in India?[END_REF] (shown in Fig. 2), which is classified in to four types of zones on the basis of maximum Modified Mercalli (MM) intensity of seismic shaking. In this work, to describe the proposed SFONT model and to validate the hypothesis, we consider the seismic hazard map of India. However, the proposed model is generic in the sense that it can be used to obtain failures and network topologies in any geographical region, given its seismic hazard map. It can be observed from seismic hazard maps of various countries/continents [10,[START_REF] Murty | Where are the seismic zones in India?[END_REF]] that seismicity does not change for small distances, and thus seismic-risk of a small local region can be considered as uniform throughout that region. We convert the geographical region G provided as an input into a grid-structure, as shown in Fig. 2, where the grid contains square shaped bounded regions of small area, which we refer as 'sectors'. The whole geographical region is subdivided into n number of sectors, where each sector of area a |B| represents a bounded region in R 2 , and the seismic risk (i.e., seismic zone type) does not change within a sector. We obtain the locations and density of earthquake epicenters in each B i ∈ B using the poisson point process (PPP) [START_REF] Elsawy | Stochastic geometry for modeling, analysis, and design of multi-tier and cognitive cellular wireless networks: A survey[END_REF]. As per the definition of

n G ⊂ R 2 , n G ⊂ G, index g (here, n is prescript) B a bounded set representing the geographical region of sec- tors, B ⊂ G, B ⊂ R 2 , index i (i.e.,
⊂ R 2 , X ⊂ G V set of points representing node locations in original topo- logy, V ⊂ R 2 , V ⊂ G V set of points representing locations of nodes in the SRA- OND topology V set of nodes, index v, V ⊂ N W set of links, index w, V ⊂ N D set of link lengths, index w, V ⊂ N D set of link lengths in the SRA-OND topology, index w, V ⊂ N K set of sectors, index k, K ⊂ N K s sector of type s seismic zone K(v)
set of sectors to be searched for shifting of v th node (Vv), PPP Φ of density λ, the number of points (q) in the bounded set B ⊂ R 2 has a poisson distribution with mean a |B|λ, i.e.,

K(v) ⊂ K, index p K(n) set of sectors non-feasible for node placement, K(n) ⊂ K K(r) k set of sectors used to calculate SR k , K(r) k ⊂ K SR k seismic-
P(Φ(B) = q) = exp (-a |B|λ) ( a |B|λ) q q! , (1)
where the density λ is given by

λ = E[Φ(B)] a |B| . (2)
In PPP, the number of points (epicenters) in a given bounded area is given by ( 1), and the location of points (X) is uniformly distributed in each B i ∈ B. Since earthquakes are more frequent in high seismic zones [10], we generate different number of earthquake epicenters using different values of density (λ s ), as given in Table 2. Large number of earthquake epicenters obtained in the given geographical region are shown in Fig. 3. It can be observed from Fig. 3 that the number of earthquakes per unit area decreases with seismic-risk in the order: Zone V→ Zone IV→ Zone III→ Zone II [START_REF] Murty | Where are the seismic zones in India?[END_REF].

We consider significant earthquakes [10] (M ≥ 5) for optical network resilience evaluation. To assign magnitude to the earthquake epicenters obtained in Fig. 3, we consider real statistics from the United States Geological Survey (USGS) [START_REF]Earthquake Statistics[END_REF], where the approximate percentage distribution of earthquakes of different range of magnitudes is: (M ≥ 8) = 0.0624%; (7 ≤ M < 8) = 0.8737%; (6 ≤ M < 7) = 8.3717%; and (5 ≤ M < 6) = 90.6922%. We denote the above ranges of magnitudes by M 8+ , M 7-8 , M 6-7 , and M 5-6 , respectively.

To design the physical network topology in the given geographical region, we generate random Gabriel graphs (GGs) [START_REF] Szcześniak | Itinerant routing in elastic optical networks[END_REF][START_REF] Cetinkaya | On the fitness of geographic graph generators for modelling physical level topologies[END_REF], modified by Matérn hard core point process (MHCPP) [START_REF] Elsawy | Stochastic geometry for modeling, analysis, and design of multi-tier and cognitive cellular wireless networks: A survey[END_REF]. In the literature [START_REF] Szcześniak | Itinerant routing in elastic optical networks[END_REF][START_REF] Cetinkaya | On the fitness of geographic graph generators for modelling physical level topologies[END_REF], random GGs have been generated to design the network topologies, where a fixed number of nodes are located with uniform spatial distribution in a given geographical area. However, using uniform spatial distribution, it is possible that two or more nodes may get placed in the same location, which is impractical. Hence, we modify the GGs by MHCPP, that eliminates the impractical probability of two or more nodes being located at the same location, or very close to each other.

MHCPP is obtained from PPP, where points are first generated using PPP [START_REF] Elsawy | Stochastic geometry for modeling, analysis, and design of multi-tier and cognitive cellular wireless networks: A survey[END_REF]. Then, some of the points are deleted such that no two points (nodes) with a separation less than r h coexist in the constructed MHCPP. An instance of the proposed SFONT model is shown in Fig. 3, where r h = 200 km, and a |G|λ m = 25. It can be seen from Fig. 3 that there is at least a distance of r h km amongst different nodes. The fiber-links amongst various node-pairs are considered to be of shortest physical route. To rigorously assess various network-resilience schemes for physical network design, logical topology design, protection, restoration, early warning data backup, survivable routing, etc., networks of different size and topology can be obtained by tuning different parameters, and evaluated for a large number of possible failures using the proposed SFONT model.

Proposed Seismic Risk Aware Optical Network Densification (SRA-OND) Scheme

The proposed SRA-OND scheme can be used to perform OND so as to make the future TCI robust against earthquake induced GCMFs. We assume that the desired region for each new node to be deployed is a square area of size 5×5 sectors. The randomly obtained node locations in the proposed SFONT model are assumed as the center of the desired regions for node deployment. Node shifting is then performed to calculate the seismicrisk (SR) for all the nodes to be placed within their desired regions. In the proposed SRA-OND, a bounded geographical region G ⊂ R 2 (square or rectangular) subdivided into sectors B, and a network topology with locations of all the nodes, are provided as input. The current locations V of all the nodes are also provided as input. The proposed SRA-OND is described in the following steps as follows:

Step 1 : The set of non-feasible sectors K(n) is found, where all the sectors having a common region to that specified in n G, are included in the set K(n).

Step 2 : For each V v ∈ V , node shifting is performed from the current (i.e., central) sector to all other sectors lying within the desired region (of size 5 × 5 sectors) for node deployment.

Step 3 : The SR k corresponding to each sector K k is calculated. The SR k for k th sector is calculated using the seismic zone type of nearby n r k sectors, and the Euclidean distance of these sectors from the k th sector. We calculate SR k using (3) as

SR k = Ω k • dist(c k , c l ), ∀ K l ∈ K(r) k , (3)
where, c l represents the center of sectors in K(r) k .

Step 4 : Amongst all the sectors in K(v) for the node V v ∈ V , the sector with minimum value of SR k is chosen for node placement (provided a sector of lower seismic zone type is found in the desired region).

Step 5 : The modified node locations (V ), and link distances (D ) are returned by the proposed SRA-OND scheme.

Performance Evaluation

In this section, resilience-evaluation of network topologies has been performed considering network densification in various stages. We consider four stages of network densification, namely, Stage 1 (S 1 ), Stage 2 (S 2 ), Stage 3 (S 3 ), and Stage 4 (S 4 ). At each stage, the number of nodes is gradually increased by varying the value of λ m . We generate 100 instances of the proposed SFONT model for the parameter values given in Table 2, except r h , which is considered as 180 km, 150 km, 120 km, and 100 km for S 1 , S 2 , S 3 , and S 4 , respectively. The decreasing values of r h indicate the decrease in minimum link-length with increased network densification. The effect of generated earthquakes on various random network topologies is then calculated.

To compare the proposed SRA-OND, we study another scheme, namely, SRU-OND, where network densification is seismic-risk unaware (SRU). To realize SRU-OND, the nodes have been located in the central sector of the desired regions for each node to be deployed. To evaluate the proposed SRA-OND scheme under real-world network scenario, all the significant past earthquakes that occurred since the year 1900 in India [10] have been simulated. We use the Richter logarithmic scale to generate GCMFs based on the distance between epicenter and node [START_REF] Ammon | Earthquake Size[END_REF]. Accordingly, we use the distance ratio n f k : 3n f k : 6n f k : 10n f k to generate GCMFs due to various earthquakes of magnitude M 5-6 , M 6-7 , M 7-8 , and M 8+ , respectively. The average number of nodes corresponding to S 1 , S 2 , S 3 , and S 4 are a |G|λ m = 40, a |G|λ m = 60, a |G|λ m = 80, and a |G|λ m = 100, which shows gradual network densification.

Fig. 3 shows the initial network topology (which represents the existing network to be densified) obtained using the proposed SFONT model, and Fig. 4 shows the dense network obtained at S 4 by adding nodes in the initial topology shown in Fig. 3. The 19-node RailTel network topology is shown in Fig. 5, and the dense network at S 4 for the topology shown in Fig. 5 is shown in Fig. 6. The SRA-OND topology corresponding to Fig. 6 is shown in Fig. 7, where the nodes interconnected by blue solid lines represent the SRA-OND topology obtained by seismic-risk aware node shifting, whereas the initial randomly obtained nodes interconnected by black dashed lines represent the part of SRU-OND topology which has been modified in the SRA-OND topology. Here, the nodes which are either already located in the lowest seismic zone, or could not be relocated in a lower seismic zone within the desired region (of size 5 × 5 sectors) for that node are kept at their original locations. Similar to Fig. 7, SRA-OND is performed for each of the 100 generated instances of the proposed SFONT model.

We perform routing, modulation, and spectrum allocation (RMSA) for each request in a set of 1000 randomly generated connection requests (in each topology) having heterogeneous bandwidth requirements uniformly selected from the set {40, 200, 500, 1000} Gbps. We perform distance adaptive modulation, first-fit spectrum allocation, and k-shortest paths routing. After establishing all the requests, the spectrum state is freezed, and the effect of earthquakes is obtained (using n f k = 1) in terms of cumulative bandwidth loss (BL) in Tbps, and connection drop ratio (CDR). CDR is defined as the ratio of the number of connections dropped due to earthquake to the number of connections established before the earthquake arrival.

Fig. 8-11 show the simulation results in terms of BL and CDR. In Fig. 8, BL due to a large number of earthquakes (18000) generated using the proposed SFONT model is shown. Simulations at the first (S 1 ) and the fourth (S 4 ) stage of network densification are shown to compare the proposed SRA-OND with the SRU-OND. It can be observed from Fig. 8 that at both the S 1 and S 4 stages, significant reduction in BL due to earthquakes has been achieved. An average reduction of 14.3%, and 29.9% in BL at S 1 , and S 4 , respectively, has been achieved using the proposed SRA-OND, which demonstrates the advantage of seismic-risk aware physical network design. Besides, it is observed that with increased network densification, the imrovement in BL using the proposed SRA-OND increases, which shows the long-term advantage of the proposed SRA-OND scheme. Fig. 9 shows the BL obtained due to the 87 past earthquakes occurred since the year 1900 in India at different locations. The initial topology considered in this case is the 19-node Indian RailTel network topology [START_REF] Pandya | Simultaneous optimization of power economy and impairment awareness by traffic grooming, mixed regeneration, and all optical wavelength conversion with an experimental demonstration[END_REF]. The earthquakes have been simulated randomly from the set of 87 earthquakes at each of the 100 iterations for every densification stage. Simulation results obtained in Fig. 9 show a similar trend with that obtained under the proposed SFONT model in Fig. 8. As the size of earthquake data set is smaller in Fig. 9, the amount of BL is also smaller as compared to that in Fig. 8. However, the average improvement in BL using the proposed SRA-OND is 17.1% and 38.2% at S 1 and S 4 , respectively, which highlights the advantage of considering seismic-risk while densifying optical networks. In Fig. 10-11, CDR has been evaluated at S 1 , S 2 , S 3 , and S 4 after simulating all the earthquakes. In Fig. 10, CDR under the proposed SFONT model has been evaluated. It is observed from Fig. 10 that CDR decreases with increased network densification, since the percentage amount of traffic per unit node/link decreases with network densification (i.e., the number of connections dropped per unit node/link-failure decreases with network densification). It is also observed that at all the four stages, the proposed SRA-OND performs better than the SRU-OND. In Fig. 11, CDR for past real earthquakes in India for densification of 19-node RailTel network has been evaluated, where a similar trend of results has been observed as that obtained under the proposed SFONT model in Fig. 10. The similarity observed in the trend of results obtained under the proposed SFONT model and that obtained for the RailTel network densification and real past earthquakes show that the proposed SFONT model can be used to emulate earthquake failures and network topologies, given the seismic hazard map any region. Thus, the proposed SFONT model can be used to estimate the impact of various network-resilience schemes on optical networks for a large number of possible earthquakes.

Conclusion

We proposed an SFONT model using PPP, MHCPP, and GGs that can be used to comprehensively evaluate various network-resilience schemes for optical networks. Further, we proposed an SRA-OND scheme that can significantly reduce the BL due to earthquake induced GCMFs. The proposed SRA-OND scheme has been evaluated and compared with the SRU-OND scheme under the proposed SFONT model as well as for a real-world case from India. The similarity observed in the trend of results obtained under both the cases validates the capability of the proposed SFONT model to emulate real-world networks and failure scenarios. In future, we plan to design resilient DC interconnection network with a content-centric approach.

Fig. 1 :

 1 Fig. 1: Possibility of GCMFs in a mesh-connected optical network due to earthquakes at different epicenter locations around the network.

Fig. 2 :

 2 Fig. 2: Seismic Hazard map of India [16] subdivided into sectors.

Fig. 3 :

 3 Fig. 3: Network topology and epicenters obtained using the proposed SFONT model. Link lengths are in km. (inset shows the epicenters obtained using PPP in different sectors).

Fig. 4 :

 4 Fig. 4: Seismic risk unaware optical network densification (SRU-OND) of the network topology shown in Fig. 3. Here, the nodes of existing/initial network are represented by yellow color, and the new nodes (initially placed in the center of the desired region) to be deployed for network densification are shown in green color.

Table 2 :

 2 Parameters values used to obtain earthquake epicenters in Fig. 2-3 Parameter Value Dimensions of bounded region G, lG, bG 3240 km Length of edge of sectors H ⊂ G, a h 32.4 km Set of types of seismic zones, S {II, III, IV, V } PPP density for K V , λ V 0.02 PPP density for K IV , λ IV 0.01 PPP density for K III , λ III 0.004 PPP density for K II , λ II 0.001 Range of M for K V , [5, 10) Range of M for K IV , [5, 8) Range of M for K III , [5, 7) Range of M for K II , [5, 7) Hard core parameter, r h 200 km

Fig. 5 :

 5 Fig.5: 19-node RailTel network topology[START_REF] Pandya | Simultaneous optimization of power economy and impairment awareness by traffic grooming, mixed regeneration, and all optical wavelength conversion with an experimental demonstration[END_REF]. Link lengths are in km.

Fig. 6 :

 6 Fig. 6: Seismic risk unaware optical network densification (SRU-OND) of the 19-node RailTel network topology shown in Fig. 5. Here, the nodes of 19-node RailTel network are represented by yellow color, and the new nodes (initially placed in the center of the desired region) to be deployed for network densification are shown in green color.

Fig. 7 :

 7 Fig. 7: Seismic risk aware optical network densification (SRA-OND) of the 19-node RailTel network topology (here, the location of new nodes are identified as the sector with minimum seismic risk (SR k ) within the desired region of node placement)

Fig. 8 :

 8 Fig. 8: Cumulative bandwidth loss (BL) due to earthquakes for SRU-OND and SRA-OND under the proposed SFONT model.

Fig. 9 :

 9 Fig. 9: Cumulative bandwidth loss (BL) due to past real earthquakes for SRU-OND and SRA-OND of 19-node RailTel network.

Fig. 10 :

 10 Fig. 10: Connection drop ratio (CDR) for SRU-OND and SRA-OND under the proposed SFONT model at different network densification stages.

Fig. 11 :

 11 Fig. 11: Connection drop ratio (CDR) for SRU-OND and SRA-OND of the 19-node RailTel network at different network densification stages.

Table 1 :

 1 Notations

	Symbol Description
	Φ	point process (PP)
	R 1 = R field of real number R n n-dimensional Euclidean space
	N	set of natural numbers
	E	expectation
	P	probability
	dist(a, b) Euclidean distance between points a and b, where a, b ∈
		R 2
	G	a bounded set representing the geographical region of in-
		terest, G ⊂ R 2
	n G	set of bounded regions non-feasible for node placement,

  Bi denotes the bounded region of i th sector)

	l G	length of the area of G
	b G	breadth of the area of G
	ai	length of the edge of sector Bi
	Φ(B)	number of points of PP in a bounded set B
	S	set of types of seismic zones, index s
	λ	density of poisson point process (PPP)
	λ s	density of poisson point process (PPP) for zone type s
	λm	density of MHCPP

a |G|, a |B|area of bounded regions represented by set G, H, respectively (here, a is prescript) a |G|λ mean number of points of PP in a bounded region G r h hard core parameter of MHCPP X set of points denoting epicenter coordinates, X

Acknowledgment

This publication is an outcome of the R&D work under the Visvesvaraya Ph.D. Scheme of the Ministry of Electronics & Information Technology (MeitY), Government of India (GoI), being implemented by Digital India Corporation. The authors would like to thank the Indian Institute of Technology (IIT) Indore for all the support and resources.