Innovative Cross-Layer Optimization Techniques for the Design of Optical Networks

Abstract

Optical networks have become indispensable in the era of 5G-and-beyond communications, supporting applications that require unprecedented capacity, reliability, and high Quality-of-Transmission (QoT) of lightpaths. To meet these requirements, network operators strive to provide innovative solutions while managing network costs effectively. This work summarizes the main findings of my Ph.D. thesis Innovative Cross-Layer Optimization Techniques for the Design of Filterless and Wavelength-Switched Optical Networks, that has been conducted in partnership with an industrial partner, SM-Optics. The main objective of the Ph.D. thesis is to investigate solutions to reduce network costs while enabling network expandability through novel network architectures. To ensure cost savings and scalability, (1) we optimize the deployment of Optical Amplifiers (OA) while accurately modeling physical layer impairments in filterless networks, (2) we propose a modular node architecture relying on pluggable devices and a scalable add/drop section at the node level for traffic grooming and capacity increase, and (3) we investigate the application of Machine Learning (ML)—regression approaches to estimate lightpaths’ QoT as they allow to make informed decisions about how conservative or aggressive a network operator can be when taking network planning choices, i.e., deploying a new lightpath. Numerical evaluations show that our proposed approaches achieve significant cost savings compared to benchmark approaches: (1) ${\sim }50\%$ savings in OA cost, (2) ${\sim }50\%$ savings in node architecture equipment cost, (3) ${\sim }70\%$ in penalty costs for deploying wrong lightpath configurations.

1 Introduction

To cope with high capacity demands and reliable connectivity due to 5G-and-beyond applications while keeping network costs to minimum, novel network architectures must be adopted. Filterless Optical Networks (FONs) are emerging as a cost-effective and scalable solution, and currently being deployed by network operators. In particular, horseshoe filterless networks have been proven as a practical solution [1, 8, 12]. The main benefits brought by the deployment of FON are (i) low CapEx costs as costly Wavelength Selective Switches (WSS) are replaced by passive splitters and combiners; (ii) a compact shelf configuration is possible due to the low power requirements of passive splitters and combiners, and a modular architecture relying on pluggable devices, i.e., optical amplifiers; and (iii) a modular, i.e., scalable, add/drop section at node level and deployment of equipment supporting traffic-grooming ensure a capacity increase to cope with new traffic requirements. To this end, the objective is to investigate and develop novel optimization techniques to minimize the cost of Optical Amplifiers and the cost Optical Transport Network (OTN) traffic-grooming boards in FONs.

Moreover, the continuous growth in network design complexity has brought the need for new enabling methodologies such as Machine Learning (ML) to address the challenges in network control, design and management [4, 13]. A continuous challenge for network designers remains the accurate estimation of physical layer impairments. Compared to traditional approaches, e.g., closed-form formulas, that lead to an under-utilization of spectral resources due to design margins, ML has been shown to be an efficient solution capable of capturing the complex non-linear nature of signal propagation and estimating uncertainties introduced by time-varying penalties.

To this end, the objective is to develop novel ML-regression approaches that estimate the probability distribution of unestablished lightpaths’ Signal-to-Noise Ratio (SNR), i.e., quantifying if and how far the SNR is from the threshold (which is pivotal in presence of uncertainties introduced by fast time-varying penalties) [7].

The following sections are organized as follows: Sect. 2 describes the placement of Optical Amplifiers in horseshoe FONs. Section 3 describes the problem of minimizing OTN traffic-grooming equipment cost in mixed 10G/100G/200G FONs. Section 4 describes the application of ML-regression to estimate the SNR distribution of unestablished lightpaths.

2 Optical Amplifier Placement in Metro Networks

2.1 Introduction and Problem Description

Fiber-To-The-Home technology has shown to be crucial in coping with high capacity demands due to an emerging use of applications related to remote working, teleconferencing, video on demand, gaming and online education. As a result, the management of metro networks has been transformed, leading operators to redefine the design process with a main objective to minimize network costs. An alternative to reduce costs is to utilize the short link distances and high number of nodes to optimize the number, location and type of Optical Amplifiers (OAs) in egress of network nodes, i.e., booster amplifiers (boosters), in ingress of network nodes, i.e., pre-amplifiers (pre-amps), and those located along fiber links, i.e., in-line amplifiers (ILAs), while guaranteeing lightpath feasibility in terms of Signal-to-Noise Ratio (SNR) [9].

FONs are emerging as a promising technological direction to reduce cost in optical networks as costly WSS are replaced by broadcast-and-select switching architectures composed by passive splitters and combiners. However, due to channels’ broadcast beyond lightpath termination, FONs suffer a reduced spectral efficiency. Moreover, when deploying OAs in FONs, the absence of WSSs causes propagation of Amplified Spontaneous Emission (ASE) noise beyond lightpath termination, and accumulation of ASE generated before a lightpath is initiated. Hence, FON architectures may become more sensitive to lightpath degradation due to higher ASE noise compared to WSS-based node architectures (WSS serves to block unintended ASE accumulation).

We consider a practical case of a metro network composed by several interconnected filterless branches, where each branch is constituted by a horseshoe as shown in Fig. 1. These horseshoe topologies typically contain two types of nodes: Terminal (T) nodes that interconnected to the rest of the metro network and are equipped with filters that block ASE noise propagation, and Filterless (F) nodes that are placed along the optical line and are equipped with passive splitters, combiners and variable optical attenuators.

Fig. 1

Metro network composed of interconnected horseshoe topologies

Fig. 2

Example of ASE noise accumulation in a filterless horseshoe topology

Figure 2 shows an example of signal propagation and ASE noise accumulation in a filterless horseshoe. Lightpath-1 ($L_1$) is generated at T-1 on wavelength $\lambda _1$ (red colored) and destined to node F1 and lightpath-2 ($L_2$) is generated at node F1 on wavelength $\lambda _2$ (blue colored) and terminated at node F2. Due to the broadcast feature of FON, $\lambda _1$ propagates beyond F1 and $\lambda _2$ propagates beyond F2. As a result, ASE noise generated by OA-A, i.e., ASE-A, propagates beyond destination and accumulates with ASE noise generated by OA-B, i.e., ASE-B, and impact the SNR of $L_2$. Given the additive nature of the ASE noise generated by amplifiers [9], the SNR due to ASE contribution for $L_2$, i.e., ${SNR_{LP-2}^{ASE}}$, can be expressed as:

$$\begin{aligned} \frac{1}{SNR_{LP-2}^{ASE}} = \frac{1}{SNR_{A}^{ASE}} + \frac{1}{SNR_{B}^{ASE}} \end{aligned}$$

(1)

where ${SNR_{A}^{ASE}}$ is the accumulated $SNR_{ASE}$ contribution due to OA-A amplifier, and ${SNR_{B}^{ASE}}$ is the $SNR_{ASE}$ contribution due to OA-B amplifier. Note that, the same can be generalized for any lightpath i that traverses M optical amplifiers from source to destination and is impacted by the accumulated ASE noise of N optical amplifiers located before the source node [1].

Problem statement. The problem of OA placement in filterless metro networks can be stated as follows: Given a horseshoe FON, a set of traffic demands, and a set of candidate locations to place OAs, decide the OA placement (location, i.e., booster, pre-amplifier and ILA) and decide the Route and Spectrum Allocation (RSA) for each traffic demand, constrained by a required QoT for each lightpath (SNR and received power thresholds), spectrum continuity and contiguity constraint, network capacity constraint, with the objective of minimizing the overall cost, constituted by the deployed OAs.

2.2 Genetic Algorithm for OA Placement

Due to high combinatorial complexity of the problem (considering x OA candidate locations, there are $2^x$ combinations of OA placement) and, its non-linear nature which requires to consider several cross-layer design parameters, we have developed a Genetic Algorithm (GA) to solve this problem. The evolutionary process of the GA is driven by competition among members (solutions) of the population and genetic operations, such as mutation and crossover. Each solution of the population is encoded as a string of binary values, i.e., the genes, which represent the candidate locations for OA placement, and assume value “1” if an OA is placed and “0” if the OA is not placed.

Each solution is characterized by its fitness value and feasibility status. The fitness of a member of the population is the cost accounting for the type and location of the placed OAs. The objective is to minimize total OAs cost, therefore a lower fitness value is preferred compared to a higher fitness value. Feasibility represents the extent to which a solution satisfies the constraints, and it is defined as the ratio between the total number of feasible lightpaths and the total number of lightpaths routed. A lightpath i is feasible if its SNR ($SNR_i$) and received power ($P_{rec, i}$) are greater than a pre-defined threshold [9].

We propose two versions of the GA, minCostGA and ConstrainedGA. The objective of minCostGA is to minimize cost, however, this may lead to a drawback on SNR performance. ConstrainedGA overcomes the drawbacks of minCostGA and minimizes OA cost while guaranteeing the SNR performance provided by the benchmark approaches.

Fig. 3

OAs total cost for FON, WSON and Baseline for varying total horseshoe length

2.3 Illustrative Numerical Results

We evaluate the GA performance on realistic 6-node horseshoe topologies. A candidate OA location is assumed every 10 km of fiber and booster and pre-amplifier candidate locations at each node. The length of the horseshoe is varied, simulating small metro (100 km) and regional metro (900 km) networks. Results are averaged for each horseshoe length by considering 20 topologies with random link lengths with a variability of ${\pm } 50$% from average link length. For example, a horseshoe of length 300 km with 5 links, the average link length is 60 km so each link’s length is equal to a random value between 30 km and 90 km.

Five approaches are compared: minCostGA-FON, minCostGA-WSON,

ConstrainedGA-FON, ConstrainedGA-WSON, and Baseline. As a benchmark strategy, i.e. Baseline, we consider an OA placement working as follows: (i) all nodes are equipped with pre-amplifiers and boosters (booster OA gain is set to compensate for the node loss and pre-amp gain set to compensate for the span it terminates), and (ii) inline amplifiers are placed approximately every 60 km (corresponding to an OA gain of 15 dB).

Figure 3 shows the numerical results in terms of total cost of deployed amplifiers in cost units (cu) in case of FON and WSON network architectures. Compared to Baseline approach, minCostGA-FON and minCostGA-WSON achieve OA cost savings up to 60% and 52%, respectively. However, despite the significant cost savings, minCostGA has lower minimal SNR (minSNR) and average SNR (avgSNR) of around 5 dB compared to Baseline.

To overcome the lower SNR performance, we developed ConstrainedGA that minimizes overall OA cost while meeting Baseline SNR. Figure 3 shows that ConstrainedGA achieves cost savings up to 56% (in FON) and up to 41% (in WSON) compared to Baseline. Comparing OA placement in FON vs WSON, we observe that OA placement in FON allows to save up to 25% total OA cost savings compared to WSON.

3 Minimizing Equipment Cost in Mixed 10G/100G/200G Filterless Horseshoes with Hierarchical OTN Boards

3.1 Introduction and Problem Description

An alternative to further reduce costs in metro FON networks is to optimize the deployment of traffic-grooming boards and interfaces deployed in OTN equipment. The practical need to solve this problem comes from the fact that real-world metro networks still employ legacy 10G technology, hence a gradual upgrade that mixes coherent (100G/200G) and non-coherent (10G) transmission technologies is required for cost-efficient short/mid-term network planning. Additionally, we consider real filterless horseshoe networks, that currently represent a prominent candidate for cost-effective optical-network deployment.

However, accounting for a hierarchy of different OTN grooming boards while employing mixed coherent and non-coherent transmission technologies makes the problem extremely complex, as it accounts for the inter-dependency between the deployment of various types of OTN boards and the establishment of lightpaths at different rates. In fact, to the best of our knowledge, no previous works have tackled the grooming problem considering: (i) the hierarchical grooming-node structure consisting of various stacked OTN boards, (ii) the co-existence of coherent and non-coherent transmission technologies (100G/200G and 10G lightpaths), and (iii) filterless node architecture that adds significant complexity to the problem as it impacts wavelength allocation and lightpath establishment.

Problem statement. The problem of minimizing equipment cost in filterless horseshoe networks with hierarchical OTN boards (minOTN) can be summarized as follows: Given a filterless horseshoe topology, a set of traffic requests between node pairs, a set of candidate OTN boards and interfaces to be placed at each node, decide jointly: (i) the deployment of OTN boards and interfaces (including location and type), DCM modules and filters for non-coherent traffic, and (ii) the route and wavelength assignment (RWA) and traffic-grooming of traffic requests, constrained by (i) traffic-processing capacity of each OTN board and interface type, (ii) maximum number of client interfaces given for each board, (iii) wavelength capacity, (iv) filterless networks constraints on wavelength assignment and (v) ensuring dedicated path protection for traffic requests, with the objective of minimizing equipment cost of deployed equipment (OTN boards and interfaces, transponders, DCM modules and filters).

Fig. 4

Structure of the node with OTN traffic-grooming boards

Problem modeling. Figure 4 shows the hierarchical structure of the node, and it is composed of up to three stacked OTN boards: OTU2-ADM, OTU4-ADM, OTU-TPD. Each node may be equipped with a pair of OTU2-ADM, OTU4-ADM, and OTU-TPD boards. Note that, at the optical layer, ROADM is not equipped with WSS, but rather with passive splitters and combiners [6]. Each OTU2-ADM supports add/drop and performs multiplexing, and has ten access interfaces, each with a maximum capacity of 10 Gbit/s, allowing clients to connect directly to the board. The access interfaces support various types of client signals such as SDH (i.e., STM1, STM4, STM16, STM64), Ethernet (i.e., 1GbE, 10GbE), and OTN (i.e., OTU1). Each OTU2-ADM board has four Small Form-factor Pluggable (SFP) interfaces that can be used as a 10 Gbit/s OTN point-to-point (p2p) line interface connecting to OTU4-ADM or a 10 Gbit/s OTN optical interface connecting to ROADM, capable of establishing a non-coherent 10G lightpath. OTU4-ADM performs add/drop and multiplexing, and clients can connect to OTU4-ADM boards through ten 10 Gbit/s access interfaces. Moreover, each OTU4-ADM board can receive/send traffic from/to OTU2-ADM via its four 10 Gbit/s line interfaces. OTU4-ADM can groom traffic of connected clients and OTU2-ADM into an OTU4 signal at the p2p line interface connected to an OTU-TPD board. An OTU-TPD board provides an interface for at most two OTU4-ADM boards and has a colored output interface connected to a ROADM and establishes a coherent 100G/200G lightpath. Optimizing the architecture of such hierarchical OTN boards enables a cost-efficient traffic-grooming, leading to overall network cost reduction.

3.2 Strategies to Solve MinOTN

To solve minOTN we have developed an ILP model and, to scale with larger problem instances, we developed a Genetic Algorithm (GA). We benchmark ILP and GA to a state-of-the-art approach, referred to as Omnibus (OB) [6].

ILP model. The objective function ($\min \sum _{j\in N} \delta _j * \mu _j + \sum _{i\in N_3, w \in W} \tau _{i,w} * \mu _i$) is to minimize the cost ($\mu _j$) of logical nodes ($\delta$), i.e., OTN boards and interfaces, and cost ($\mu _i$) of transponders ($\tau _{i,w}$) used to establish lightpaths on wavelength w.

Main constraints of the problem refer to the capacity of OTN boards and their interfaces, transponder interfaces, wavelength capacity, filterless constraints, and other constraints referring to the physical constraints of OTN boards.

Genetic Algorithm (GA). The encoding of minOTN is done such that OTN board/interface placement and routing of demands are encoded in gene clusters and genes: a gene cluster is defined for each demand and, in each gene cluster, the genes represent candidate paths to route the demand.

Fitness value is defined as the total cost of deployed equipment (we minimize cost, so a lower fitness value is desirable). Feasibility refers to constraints violation: a solution has feasibility equal to one, if it satisfies all constraints of the problem.

Omnibus (OB). OB is considered as a benchmark scenario as it represents the real-world OTN boards deployment approach. OTN boards are placed considering 100G lightpaths between all neighboring nodes, hence traffic-grooming is performed at every node. In case traffic cannot be served by an OB with single 100G lightpaths, an upgrade is adopted at nodes generating/handling more traffic by doubling the deployed equipment.

3.3 Illustrative Numerical Results

Experimental evaluations are performed on a real 5-node filterless horseshoe topology for three traffic matrices with incremental traffic, i.e., TM1, TM2 (TM1+45% additional traffic), and TM3 (TM1+60% additional traffic), and compare ILP, GA and OB. The cost model is provided by the industrial partner (SM-Optics) [6]. Figure 5 shows the total cost (and its breakdown) of deployed equipment in terms of equipment type for three traffic matrices (TM1, TM2 and TM3) considering ILP, GA and OB. ILP and GA reach the same equipment cost in case of TM1 and TM2, while in case of TM3, GA reaches a 4% higher equipment cost compared to ILP. In terms of execution time, GA finds a solution in under 5 min while ILP takes up to 9 h.

Fig. 5

Equipment cost and each equipment contribution in case of ILP, GA and OB

Compared to OB, ILP and GA achieve cost savings of 51% and 30% in case of TM1 and TM3, respectively. ILP and GA allow significant cost savings due to a predominant deployment of 10G transponders instead of coherent 100G transponders as in OB. In case of TM2, ILP and GA reach the same solution as OB. The reason is that in case of TM1, 100G lightpaths are populated around 50%, so there is sufficient residual capacity to serve the added traffic for TM2 without additional equipment. In case of TM1 and TM3, equipment for establishing non-coherent lightpaths (transponders, OTU2-ADM boards, filters and DCM modules) deployed by ILP and GA compose 67% and 12%, respectively. Compared to ILP and GA, equipment deployed by OB are only for coherent lightpath establishment.

4 Machine Learning for Quality-of-Transmission Estimation of Unestablished Lightpaths in Wavelength Switched Optical Networks

4.1 Introduction and Problem Description

To ensure effective and optimized design and planning of optical networks, accurate prediction of lightpath QoT prior to deployment is imperative. Traditionally, QoT estimation in optical networks has been addressed using margined formulas (e.g., the GN-model [14]) that are computationally-fast but lead to under-utilization of spectral resources [15]. Alternatively, ML uses historical data and overcomes the short-comings of margined formulas and estimates lightpaths’ QoT in reasonable time by modelling uncertainties not captured by physical layer models [3, 10]. Several studies has tackled the ML-based QoT estimation as a classification problem [11, 16]. However, a classification-based approach has three main drawbacks: (1) it does not convey how close to the system threshold the predicted SNR is; (2) it does not return the predicted distribution of the SNR value; and (3) during training, no distinction is made between a training sample with a SNR slightly above the threshold and a training sample that is way above the threshold, which leads to loss of information. To overcome these drawbacks of ML-based classification, we investigate ML-based regression approaches to estimate lightpaths’ SNR, under the assumption that the SNR measured at the receiver can be modeled as a random variable [5, 7].

We assume that a lightpath is characterized by a number of features, e.g., amount of traffic, modulation format, total lightpath length, number of links traversed by the lightpath, length of longest link. However, for a given lightpath configuration, the SNR may still exhibit variations, as it depends on several other factors not captured by the considered features as, e.g., fast time-varying penalties due to polarization-dependent losses. It follows that the SNR associated to a lightpath configuration can be modeled as a random variable and thus be characterized by a Probability Distribution Function (PDF). We propose three regression approaches that estimate the parameters characterizing the distribution of the random variable, i.e., SNR:

• 1. Matched Gaussian Distribution Regressor (MD-R), returns the mean and variance of a Gaussian distribution modeling the SNR value.

• 2. Moments Estimation Regressor (ME-R), enhances MD-R by predicting the four moments of the probability distribution, i.e., mean, variance, skew, and kurtosis.

• 3. Quantile Estimation Regressor (QE-R), removes the assumption of an underlying Gaussian distribution and estimates the quantiles of the PDF.

To better explain our approach, let us consider a given lightpath configuration as testing instance. MD-R, ME-R and QE-R are used to estimate the parameters of the SNR distribution and serve to answer the question: what is the probability that the SNR for this testing instance is below the system threshold?

Fig. 6

Example of SNR estimation approaches: a SNR ground truth values; b Point estimate of SNR; c Gaussian distribution, i.e., mean, variance, estimation of SNR; and d SNR distribution estimation based on mean, variance, skew and kurtosis

Figure 6 shows an illustrative example of three SNR estimation approaches. Given an unestablished lightpath described by a set of features, the associated SNR may exhibit different values due to varying network conditions, e.g., noise figure of optical amplifiers, fast time-varying penalties. Such values could be measured only a-posteriori (i.e., after the lightpath establishment) by means of optical performance monitors (OPM) [2] at the receiver node, and constitute the ground truth empirical PDF (see Fig. 6a). A standard ML regression model may be leveraged to estimate a scalar value for the SNR, given the features (Fig. 6b). However, this does not capture the uncertainties in the SNR value, e.g., uncertainty introduced due to time-varying penalties. Conversely, if a regressor is used to predict the parameters of the SNR distribution (e.g., mean and variance of a Gaussian distribution as in Fig. 6c or the first four moments of a Gamma distribution as in Fig. 6d), it is then possible to assess how well the estimated PDF fits the observed ground truth samples. This way, a network operator is allowed to set how conservative or aggressive its planning choices should be, i.e., when deploying a new lightpath. In other words, with the proposed approaches, an operator seeking low-margin operation of its network can more discerningly decide how aggressive such margin reduction should be.

4.2 Illustrative Numerical Result

Synthetic traces of realistic SNR values are generated via E-Tool [16] that assumes frequency slice units of 12.5 GHz, a total 4 THz link capacity and transceivers operating at 28 GBaud with a 37.5 GHz channel bandwidth. We assume a per-link random penalty parameter that accounts for fast time-varying impairments (e.g., polarization effects), according to an exponential distribution (according to the principle of maximum entropy) with a 1 dB average. A dataset of $N=1000$ lightpaths is generated by randomly choosing a bitrate in [50, 500] Gbps range with 50 Gbps granularity and one of the $r \cdot M$ possible combinations ($r=3$ routes $\cdot$ $M=6$ modulation formats, i.e., (DP)-BPSK, DP-QPSK and DP-n-QAM, with n $=$ 8, 16, 32, 64). For each lightpath, the SNR calculation is repeated $k=100$ times under different random penalty samples. For the train/test split, a 80/20 ratio is considered. Standard metrics used in ML, e.g., Root Mean Square Error (RMSE), are difficult to interpret from a network operation point of view. Therefore, we provide a cost analysis which quantifies the penalties in the context of wrong lightpath deployment decisions.

Let us consider a lightpath j which belongs to the set J of candidate lightpaths, characterized by the set of features V. The question to be answered is: is ${SNR}_{j}$ lower than a system defined threshold ${SNR}_{T}$?

Given the set of features V, let $F_{G_i}$ be the estimated CDF of the random variable $G_i$ that models the SNR, according to estimator i, where $i \in \{{{MD-R}, {QE-R}, {ME-R}}\}$. The probability that the SNR is below the threshold $SNR_T$, according to estimator i, can be computed as $p_i = F_{G_i}(SNR_{T})$. Different estimators will estimate different probabilities.

One should then make a decision on the basis of this probability. We consider two different penalty costs associated to the two ways that a decision can be wrong: an underestimation cost $(C_u)$, that is paid when $SNR_{j}$ is estimated to be lower (Below) than ${SNR}_{T}$, but is in fact higher (Above) than $SNR_{T}$; and an overestimation cost $(C_o)$, that is paid when $SNR_{j}$ is estimated to be higher (Above) than $SNR_{T}$, but is in fact lower (Below) than $SNR_{T}$.

The expected penalty for a deployment decision (Above or Below) is the probability that such decision is wrong, times the cost of taking such a wrong decision. Therefore, if the decision is that $SNR_{j}$ is below $\textit{SNR}_{T}$, the estimated probability of being wrong is equal to $(1 - {p _i})$. It follows that the expected cost of deciding that $SNR_{j} < SNR _{T}$, according to estimator i is: $C_{i, \texttt {Below}} = (1 - {p_i}) \cdot C_u$ while if the decision is that $SNR_{j}$ is above $SNR_{T}$, the probability of being wrong is equal to $p_i$. Therefore, the expected cost of deciding that $SNR_{j}$ > $SNR_{T}$, according to estimator i is: $C_{i, \texttt {Above}}$ = $p_i$ $\cdot$ $C_o$ For each estimator i, we make a decision $D_i$ according to the following rules: $D_i$ is Below if $C_{i, \texttt {Below}}< C_{i, \texttt {Above}}$, and Above otherwise. The decision $D_i$ is compared to the decision that would be made by leveraging the ground truth ($D_{GT}$), i.e., the one computed based on actual SNR measurements. The ground truth decision is defined as follows: $D_{GT}$ is Below if $SNR < SNR_T$, and Above otherwise. SNR is the actual sampled value. If $D_i$ $\ne$ $D_{GT}$, then the respective cost associated to the decision is added to the total penalty cost of estimator i is: $PC_{i} = (\sum _{j \in J} min(C_{i, \texttt {Below}}, C_{i, \texttt {Above}})) / |J|$, where |J| is the total number of lightpaths. Note that we add the $min(C_{i, \texttt {Below}}, C_{i, \texttt {Above}})$ to the $PC_i$ since the estimator makes the decision based on the comparison between $C_{i, \texttt {Below}}$ and $C_{i, \texttt {Above}}$, so the minimum cost reflects the cost of the wrong decision of the estimator.

We compare the performance of the three proposed estimation approaches in terms of decision penalties against the following four baselines: (1) always decide Below, (ADB); (2) always decide Above, (ADA); (3) random decision, (RD); and (4) cost-insensitive decision (CI), i.e., make the decision Below (Above) if the SNR mean value estimated by the Gaussian regressor, i.e., MD-R, is below (above) the threshold. We also consider a lower bound of the obtainable cost which reflects the penalty cost incurred by an “ideal” estimator (noted as IE), which always returns as output an estimated CDF identical to GT.

We assign the following numerical value to each cost type: $C_u$ $=$ 1 cu (cost unit) and $C_o$ $=$ 10 cu. These values can be interpreted as follows. If we underestimate the lightpath’s SNR we may erroneously consider as infeasible a lightpath configuration that was in fact feasible. Hence, a lightpath with a less spectrally-efficient modulation format will be deployed, leading to the unnecessary occupation of some spectral resources, yet no service disruption will be incurred. Conversely, in the case of overestimation, we erroneously decide to deploy a lightpath with a modulation format which will lead to a below-threshold SNR, eventually resulting in service disruption.

From a network operator’s point of view, the penalty in case of service disruption is higher compared to the penalty of deploying a lightpath that does not adopt the most spectrally-efficient modulation format. This disparity is captured by our cost values, though they may not exactly reflect the actual economic losses experienced by a network operator.

We perform a set of 100 sequences of deployment decisions, each one including 500 candidate lightpaths. Therefore, the total penalty cost for each estimator is averaged over 100 simulations. Table 1 reports the average penalty per instance for each estimator, in the order of performance from best to worst.

We observe that our proposed estimators all perform much better compared to baseline approaches (as for IE, it provides the lowest cost penalty, but it only represents an ideal lower bound for this cost analysis). Results confirm the importance of estimating the probability distribution instead of using a point estimate. In fact, the cost penalty of MD-R (which assumes a Gaussian distribution of the estimated PDF) is significantly lower compared to CIB (the standard regressor that only estimates the SNR mean value). Utilizing more sophisticated estimators, as, e.g., ME-R, we achieve an even better performance in terms of cost penalty (0.069 < 0.087). Note that ADB baseline provides a better cost result compared to ADA baseline, because we use a 70/30% ratio between the lightpath configurations in the dataset having a SNR value below/above threshold, and hence the estimator is more likely to correctly predict a SNR value being below threshold.

Table 1 Penalty cost (PC) in cost units (cu) for each estimator