Heuristic Chaotic Hurricane-aided Efficient Power Assignment for Elastic Optical Networks

In this paper we propose a dynamical transmission power allocation for elastic optical networks (EONs) based on the evolutionary hurricane search optimization (HSO) algorithm with a chaotic logistic map diversification strategy with the purpose of improving the capability to escape from local optima, namely CHSO. The aiming is the dynamical control of the transmitted optical powers according to the each link state variations due to traffic fluctuations, channel impairments, as well as other channel-power coupling effects. Such realistic EON scenarios are affected mainly by the channel estimation inaccuracy, channel ageing and power fluctuations. The link state is based on the channel estimation and quality of transmission (QoT) parameters obtained from the optical performance monitors (OPMs). Numerical results have demonstrated the effectiveness of the CHSO to dynamically mitigate the power penalty under real measurements conditions with uncertainties and noise, as well as when perturbations in the optical transmit powers are considered.


I. INTRODUCTION
The growth of the traffic demand with heterogeneous characteristics associated to the increment of the SNR rate requirements has pressing the development of dynamical optical networks. Currently, the technological maturity of devices, equipment and protocols provides the use of dynamical flexible grid-rate elastic optical network (EON). In the EONs, the lightpaths with adjustable bandwidth, modulation level and spectrum assignment can be established according to actual traffic demands and quality of service (QoS) requirements [1], [2]. In addition, the quality of transmission (QoT) of each lightpath is evaluated previously to resources allocation purpose, as well as to obtain reliable optical connectivity [2], [3]. The best knowledge of the QoT is needed in the design and operation phases, owing to the margin has to be added in the network when the QoT is not well established [4]. The QoT prediction can utilizes different methodologies based on sophisticated analytical models, The associate editor coordinating the review of this manuscript and approving it for publication was Rentao Gu . approximated formulas and optical performance monitors (OPMs) [4], [5]. The QoT estimation with OPMs distributed in the route or in the coherent receiver can be appropriated in term of precision and computational complexity (CC) when integrated in to the active control plane to provide the link conditions in real time [1], [3]. However, it is important to consider the limited accuracy of the OPMs that increase the measurements uncertainty considering the channel impairments (including linear and nonlinear effects), receiver architecture and noise, which decrease the performance of the channel state estimation [5], [6]. In addition, the power dynamics related to the channel-power coupling effects, which are influenced by the network topology, traffic variation, physics of optical amplifiers and the dynamic addition and removal of lightpaths can cause optical channel power instability and result in QoT degradation [1]. Moreover, the interactions between lighpaths in some routes of the network can generate fluctuations to form closed loops and create disruptions.
The power, routing, modulation level and spectrum assignment (PRMSA) problem is usually determinate in the VOLUME 8, 2020 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ planning stage of the network and margins are included considering the QoT inaccuracies, equipment ageing, inter-channel interference, as well as the uncertainties of the optical power dynamics [7], [8]. However, there are some investigations to the development of resource allocation algorithms based on OPMs with reduced margins, which have considered ageing and inter-channel interference to configurable transponders with launch powers [5], regenerator placement [4] and the optimization of the physical topology for power minimization [9]. These algorithms can be based on derivative-free optimization (DFO), constrained direct-search algorithms [5], and mixed integer linear programming (MILP) [9]. Furthermore, in [6] an adaptive proportional-integral-derivative (PID) with gains auto-tuning based on particle swarm optimization (PSO) to dynamically control the transmitted power according to the OPMs measurements for mixed line rate (MLR) was proposed. Previous investigations for the legacy single rate network for power control adjustment to the optical-signal-to-noiseratio (OSNR) optimization considering the physical impairments were conducted based on a game-theory-based [10] and (PID) back propagation (BP) neural networks [11]. Moreover, the power allocation (PA) optimization aiming at obtaining energy-efficient optical CDMA systems using different programming methods is carried out in [12]. Such optimization methods, including augmented Lagrangian method (ALM), sequential quadratic programming method (SQP), majoration-minimization (MaMi) approach, as well as Dinkelbach's method (DK) were compared under the perspective of performance-complexity tradeoff. The findings reported in the previous papers assume that there are no impact of queuing issues on the optical network convergence and performance. To highlight this important aspect, in [13] the authors carried out a review on the role of the queuing theory-based statistical models in wireless and optical networks.
The PRMSA is an NP-hard problem, which can be decomposed into several sub-problems with lower complexity, e.g.: PA, routing; spectrum assignment; and others [21], [25], [36]. Between them, the PA sub-problem is an important procedure: to combat the main impairments of the EONs, including the amplified spontaneous emission (ASE) and non-linearities interference (NLIs); to guarantee QoT in dynamic scenarios, minimizing the waste of resources during the network's lifetime; to increase the network capacity; to limit the power budged in the sense of maximizing the energy-efficiency (EE); to mitigate the effects caused by fluctuations in the amplifiers. Analytical and heuristic methods have seen proposed in the sense of the better performance and complexity trade-off [1]- [6], [10]- [15], [33]. In [18], the hurricane searches optimization (HSO) was proposed as a candidate heuristic for the solution of NP-hard problems with large-scale, but it has not seen investigated in optical scenarios. Recently, the HSO was considered as a promising alternative algorithm for solving problems in practical large-scale power systems [19]. In this context, the contributions of this work include: a) proposing an effective, efficient PA strategy based on the HSO and its variation based on the insertion of a chaotic map, named PA-HSO and -CHSO, respectively; b) investigating systematically the input parameter optimization (IPO) for both PA algorithms, aiming at improving the performance-complexity tradeoff of the proposed algorithm; c) validating both PA algorithms for different realistic EON channel conditions, i.e., non-perfect monitoring of the OPMs, channel ageing effects, dynamical scenarios, including power instability, particularly in EONs. Moreover, comparisons have been performed assuming a convex optimization through the gradient descent (GD) [14], [15].
Channel estimation is evaluated in terms of Qot parameters, which are obtained from OPMs. OPMs measure the connection status and update the PA algorithms as illustrated in Fig. reffig:1. Thus, PA algorithms perform dynamic power control. To corroborate the effectiveness and efficiency of the proposed resource allocation strategy in EONs, it is evaluated: normalized mean square error (NMSE); convergence; power penalty (PP); probability of success; CC; and performance-complexity tradeoff.

II. PROPOSED SCHEME
The proposed scheme utilizes the information collected from the OPMs to control dynamically the level of the launch power of the lightpaths. The power adjustment considers the QoT inaccuracies, equipment ageing, inter-channel interference, as well as the variation of the optical power dynamics. Differently of the others approaches based on intelligent systems [14], it is not necessary the training phase and the proposed scheme can be performed in near-real time. In addition, the power budget is determinate in the planning stage of the network and margins are included [2], [4] and the proposed scheme will act during the regular operation of the EON. For the proposed scheme it is considered that the lightpaths were previously established from the resource allocation algorithms associated with route, modulation, bandwidth and spectrum.
The proposed scheme continuously update the transmitter launch power for each lightpath in response to dynamic OPMs information, it is considered a communication delay between the OPMs in the receiver node, control plane and transmitter adjustment. This process can encompass the delay related to the duration of the OSNR estimation in the OPM, the control message transmission duration, the processing time, the actuation phase in the transmitter and the round-trip delay. In this sense, considering the current technology, each algorithm updating can be estimated in 100 ms or less [4]. Therefore, the time needed to close the loop related to the signal latency and the other operations needed for controlling the transmitted power is assured.
The proposed PA algorithm utilizes the chaotic hurricane, which is a new hybrid algorithm based on the hurricane search optimization (HSO) associated with probability distribution from the chaotic maps instead of uniform distribution of the traditional HSO. The objective is obtain an algorithm with balance between the exploration (diversification) and the exploitation (intensification) to improve the algorithm capability to escaping of the local solutions and the amelioration of the velocity of the convergence without affecting the quality of the algorithm solutions [16], [17].
The HSO is a metaheuristic algorithm for global optimization considering single-objective [18] and multiobjective [19] optimization problems, inspired by natural phenomena on the hurricanes behavior, where wind parcels move in a spiral course moving away from a low-pressure zone called the eye of the hurricane. These wind parcels search for possible new eye position, which represents a lower pressure zone to find out the optimal solution. The performance is very competitive compared to others metaheuristics optimization algorithms, such as gravitational search algorithm (GSA) and PSO. Although there is a variety of optimization algorithms, the development of new optimization algorithms have been motivated by the no free lunch (NFL) theorems for optimization, which have proved that an universally efficient optimization algorithm does not exist. Moreover, the particularities and characteristics of the optimization problem strongly affects the capacity of the optimization algorithm to finding the optimal solution in global optimization problems [18]. Herein, it is important investigate several distinct optimization algorithms for different optimization problems considering the related aspects. In addition, the application of chaos theory alone or jointly with other algorithms such as ant colony algorithm (ACO) [16], firefly algorithm (FA) [20] and PSO [17] have improved the optimization algorithms. Chaos presents a non-repetitive nature that increase the random search characteristics of the optimization methods, as well as increases the ability to get away from local solutions. In general, chaotic maps based on the complex behavior of a nonlinear deterministic system are utilized to optimization goal.

III. SYSTEM MODEL AND PROBLEM FORMULATION
The EON physical layer is composed by transmitters with adjustable modulation format, SNR rate and level of launch power, an erbium-doped fiber amplifier (EDFA) per span, ROADMs and receivers with digital signal processing capability to compensate the dispersion effects. The ROADMs present equalization to compensate undesired spectrum tilting due to EDFAs. In addition, the EDFAs operate in an automatic gain controlled (AGC) mode according to each ROADM to achieve spectral tilt correction. A connection between ith and jth node, i.e., N (i, j), is called communication link. The ith link from ith channel is formed by N span i spans, being each span formed by: a fiber length L in Km; and an EDFA. The lightpaths are represented as Nyquist wavelength division multiplexing (WDM) superchannels with bandwidth where ξ i is the traffic demand data rate (Gbps) of the ith channel, M is the number of channels and c i is the spectral efficiency defined by modulation format of the ith channel, Table 1. The γ * B2B,i is the back-to-back signal-to-noise ratio target for the ith channel required to achieve error-free considering forward-error-correction (FEC) codes with bit-errorrate requirement of b * = 4 · 10 −3 . Hence, to obtain an appropriate QoT, the effective back-toback signal-to-noise ratio for the channel i (γ B2B,i ) must be γ B2B,i ≥ γ * B2B,i . This formulation can be defined as a problem of residual margin (RM). The RM in the ith channel can be defined as [14]: while the RM in vector form is represented as = [ 1 , 2 , · · · , M ] 1×M . The concept of residual margin in WDM systems can be formulated as an optimization problem, which the objective is to minimize the RM of all the M WDM channels in the sense of * = [1, · · · , 1] 1×M , while guaranteeing the QoT.
Such RM optimization problem can assume an optical network topology as static in a short-time due to the multi-stage traffic demand. Therefore, the optimization problem reduces to the efficient power assignment problem [5]: where p = [p 1 , p 2 , · · · , p M ] T is the optical power vector and p i the transmitted optical power for the ith lightpath, subject to the constraints related to power budget and SNR required to achieve QoT [3]; moreover, ξ min i is the requirement minimum bit-rate, p min and p max is the minimum and maximum value considered as allowable transmitted power, respectively.
The quality of the RM optimization attained by different methods can be evaluated via the Euclidean distance between and * . Mathematically, this is expressed as: From eq. (3), the problem in the original form is not convex. Therefore, it do not guarantee the minimum global of applying nonlinear programming (NLP), because can to result in a solution that is far from global solution * . In this sense, heuristic evolutionary optimization methods, such as the HSO, have the advantage of achieving in a polinomial processing time an high quality solution, which is not necessary the optimum solution. For comparison purpose, in the numerical results it will be evaluated descent gradient (GD) proposed [15] with convex formulation [14]. The first constraint (C.1) from eq. (3) is based on a Gaussian noise (GN) model to establishment the QoT in the lightpath [21], [22].
For the proposed evaluation scenario in Fig. 1, it is considered that the ith lightpath characteristics such as modulation formats, routes, and spectral orderings of all the connections were previously determined, and it includes the design margin M d (τ ) due to the QoT model inaccuracies and ageing margin of the transponder M t (τ ) on its sensitivity, modeled as a function of time τ . Basically, the design margin M d (τ ) is included to overcome the GN model inaccuracies and avoid the EON operating at the limit. M d (τ ) decreases along with the time τ (in years), since a better EON knowledge is obtained by increasing of τ [4], [7], [9], [25].
Hence, the γ B2B,i for the ith channel is modeled as: where each parameter from eq. (5) can be modeled as a linear or nonlinear function of time τ [4]. Herein, we have adopted the following linear function of τ : where M t (τ end ) and M t (τ 0 ) are the transponder margin for End-of-Life (EoL) and Begin-of-Life (BoL) time, respectively, while τ = τ end − τ 0 is the network's lifetime. Besides, the first term γ i (τ ) assumes GN model, while includes the linear and nonlinear noise effects for the ith channel [21]: where p ASE i (τ ) is the power of the amplified spontaneous noise (ASE) noise and p NLI i (τ ) is the power of the non-linear interference (NLI) noise.
The power of the ASE noise is given by [4], [21]: A span e,i (τ ) = L e,i · α(τ ) + c e,i · c loss (τ ) + s e,i · s loss (τ ) (9) where for the eth span of the ith user, the L e,i is span length; c e,i is connection number, and s e,i is the splice number; while α(τ ), c loss (τ ) and s loss (τ ) can be modeled as functions of time, representing the fiber attenuation, the connector's loss and the splice loss, respectively [4].
The power of the NLI noise results of the self-and cross-channel interference, SCI and XCI, respectively. It is given by: where η i (τ ) + η i (τ ) defines the non-linear factor of the transmitted signal spectrum of the ith channel, denoted as η i . ζ is the non-linear parameter, β 2 is the group velocity dispersion and p j is the power of jth interfering channels. Therefore, it can be obtained a bit error rate (BER) expressed as a function of the SNR, which takes into account the baudrate, FECs limit BER and the modulation format of the ith channel [23], [24], as follows: where the function ϑ(·) is defined by modulation format [4]. The QoT prediction consists of developing a systematic procedure for the evolution of the vector p in order to reach the optimum value p * , based on the γ B2B, values. Theses values are monitored by OPMs at add, through and drop node by channel estimation and reported to the control plane to guarantee the QoT. The channel estimation quality is affected by three main assumptions: 1) non-perfect monitoring of the OPMs considering their limited accuracy due to channel impairments (linear and nonlinear effects) and the receiver architecture, as well the noise measurement and peaks occurrence caused by polarization mode dispersion (PMD) effects [2], [4], [6], [25], [26]. Theses uncertainties can be modeled as a random variable δ i added to the γ B2B,i ,which follows a Log-Normal distribution LN (µ, σ ). Therefore, the estimated γ B2B,i can be modeled by [6]: 2) ageing resulting from increases fiber losses due to splices to repair fiber cut, detuning of the lasers leading to misalignment with optical filters in the intermediate and add/drop nodes. These values can be modeled by eqs. Theses values can be modeled as a power perturbation in the input power of the ith user: [dBm] (13) where the power perturbation function is modeled as with A n being the peak of the perturbation in [dB], n is a discrete-time index, and p i the nominal transmitted power for the ith lightpath. This model assumes power fluctuations propagation across the network nodes [6]. The full knowledge of the QoT parameters during the estimation of the ith channel increases reliability and enables design solutions considering the p * for different bit rates requirement in the lightpath. In this sense, mixed line rates (MLR) networks have focused on optimum launch power, obtaining suitable cost minimization [27], combined to the maximization of the number of established connections [10], while reducing the transponder cost [28] and improving the launch power versus regenerator placement tradeoff [29]. However, when the QoT parameters for the ith channel is not known perfectly, PP occurs, being modeled as: where the p * i value is defined considering the perfect knowledge of the QoT parameters [5]. Negative values ofp i , i.e, p (−) i , mean that the measured BER did not reach the p * ; while positive values ofp i , i.e,p (+) i , mean that the b * is reached with energy waste. Therefore, for availability of the ith lightpath, the margins (m i ) should satisfying the condition of m i ≥p In context of margins, in [8] a system margin (SM) is adjusted by a ML based on the maximum-likelihood principles to improve the QoT prediction of new lightpaths. The predict parameters can provided more accurate QoT of not-already-established lightpaths compared to the limited amount of information available at the time of offline system design. In [11] is proposed a ML-based classifier to predict if the candidate lightpath presents suitable bit error rate (BER) considering the traffic volume, modulation format, lightpath total length, length of its longest link, and number of lightpath links. To train of the ML classifier is based on the OPMs or in the BER simulation, which is utilized in the absence of real field data. In [14] is performed the optimization of transmitted power to maximize minimum margin and to maximize a continuously variable data rate. The Gaussian noise nonlinearity model is utilized to expresses the SNR in each channel as a convex function of the channel powers. Convex optimization is performed with objectives of maximizing the minimum channel margin.
Therefore, the progress in the network planning, design and active operation control has become margins an important resource to be optimized [2], [4]. In this sense, the margin in each lightpath should be as little as possible to ensure guarantee reliable optical connectivity. The reducing of the excess margin can be utilized to increase the maximum transmission distance, reduce the number of regenerators, as well as postpone the installation of more robust transponders than are closely necessary in the beginning of the network operation [25]. Several efforts have been made to become the margins variable and adjustable to increase the network capacity and decrease the costs of the network implantation and operation [1]- [6]. In this sense, the determination of the level of transmitted power is performed in the planning stage of the network and a SM is included considering the uncertainties of the OPMs measurements and optical power dynamics [7], [21].

IV. PA SCHEMES
In this section, the PA-HSO and -CHSO schemes are proposed for PA in EONs. Firstly, the recently proposed heuristic HSO [18] is revisited in the sequel, and modifications are implemented to solve suitably the PA-EONs optimization problem. Hence, the original HSO is modified further, with the insertion of a chaotic map [16], [17]. Moreover, the PA-GD in [14], [15] are applied aiming at comparing the complexity-performance tradeoff with the proposed PA methodology. Indeed, the PA-GD is assumed as an analytical method, which result in a global optimum solution in a cost of high complexity.

A. PA-HSO
In the HSO, the eye (lower pressure zone) is related to the best solution of the hurricane structure and can be repre- The kth wind parcel on the nth iteration moves around the eye according to: where r k [n] and θ k [n] are respectively the radial and angular coordinate of the power increasing of the kth wind parcel at the nth iteration. r 0 is the initial value of r k . And, z k [n] is the rate of the increase of the spiral at nth iteration, which is defined by a random variable with uniform distribution, i.e., z k [n] ∼ U ∈ U[0, 1]. Indeed, the behavior of the kth wind parcel in the nth iteration follows a logarithmic spiral pattern. The system evolves looking for a lower pressure zone (new eye position) in the search space. Once a new lower pressure is discovered, its position (P k ) becomes the eye (p) and the process start over and the process start over again [18]. From eq. (16) and (20), the power updating of two consecutive channels associated to the kth wind parcel at the nth iteration is given by: where i = (k mod h) + 1 corresponds to the ith channel from the kth parcel updating, that represents K wind parcels. Each group is denoted by G i , representing the power updating of two specific channels from p k , as in eq. (17), resulting p k ⊂ G i . The θ k [n] updating from eq. (17) is defined by concept of velocity variation of the kth wind parcel in the nth iteration, which is given by: if r k [n] > p max (18) where ω WP k [n] is a tangential velocity of the kth wind parcel at nth iteration, ω WP max is the maximum tangential velocity adopted for all the wind parcels, z k [n] performs ω WP k [n] fit at nth iteration [18]. Thus, the θ k [n] updating at the nth iteration is given by: where ω WP k is assumed as a fixed value ∀k, because p max , r k [n] and z k [n] are sufficient to update the kth wind parcel.
In addition, the initial power vector of the PA-HSO is defined as p 0 while the component p k,i ∈ [ p min ; p max ]. Therefore, when p k,i / ∈ [ p min ; p max ], the function ϕ(p k,i ) is true; thus, the initial and current angular coordinates of the kth wind parcel, θ k [1] and θ k , respectively, must be updated as: [1] and θ k = 0.
The stopping criterion is defined by the number of iterations N f . A pseudo-code for the PA-HSO is described in Algorithm 1. for k = 1 to K 5: The PA-CHSO and -HSO algorithm are similars. The updating of z k [n] is the unique difference between them. As described in subsection IV-A, the PA-HSO assumes z k [n] as a random variable with uniform distribution. However, aiming to escape of local minimum and to diversify the candidate solutions, the PA-CHSO assumes z k [n] as a random variable assigned by a chaotic mechanism [16], [17]. Such chaotic logistic map is related to the dynamics of the biological population with the chaotic distribution features and it is obtained by the recursive equation: where z k [n] ∈ [0, 1] is the chaotic variable and µ is the control parameter in the range 0 < µ ≤ 4 [16], [17].
The assumed z k [n] values brings randomness to the search step when compared with uniform distribution. Therefore, the same PA-HSO algorithm can be adopted, i.e., Algorithm 1, but z k [n] is updated by the eq. (20).

C. PA-GD
The PA-GD algorithm is based on the outline of a general GD method [14], [15], which defines a descent direction p and a suitable step size selection using backtracking line search method (from Algorithm 9.1 and 9.2 in Ref. [15]). Here, p is normalized by ||J 1 (p)||. Summarizing, the main parameters are the number of iterations (N GD f ) from Algorithm 9.1 and the number of iterations from the backtracking search (N GD bt ) [15]).

D. INPUT PARAMETERS OPTIMIZATION (IPO)
In this subsection, two input parameter optimization (IPO) strategies are proposed: IPO based on the golden-section (GS), which is a numerical method for uni-modal optimization, hereafter named IPO-GS [35]; and the IPO based on the conditional probability of success (CPoS), which is a empirical method based on the combination of the input parameters inside theirs range, hereafter named IPO-CPoS. Firstly, the IPO-GS and IPO-CPoS optimize the input parameters that affect dramatically the performance. After, the IPO-CPoS optimizes the input parameters that affect dramatically the CC. The IPO-GS is not adopted to optimize the CC, because it increases a larger CC to minimize the objective, i.e., eq. (3). More details about them are described below.

1) IPO-GS
The framework of the IPO-GS is similar to the systematic proceeding proposed in [32], in which only the main input parameters that affect dramatically the performance of the PA algorithms are optimized, i.e., for the PA-HSO and -CHSO are r 0 and ω WP . After that, the input parameters directly related to the algorithm's complexity are optimized regarding the performance-complexity tradeoff, i.e.; for the PA-HSO and -CHSO are K and N f . The IPO-GS is presented for the PA-HSO, but it also is applied on the PA-CHSO. The IPO-GS procedure consists of two steps: a) keep ω WP fixed and optimizes r 0 ; b) r 0 (from first step) is hold fixed while ω WP value is optimized. The optimized input parameter values are found by golden-section search method, which finds the minimum of an objective function by successively narrowing the range of values inside feasible range; in other words, it estimates the maximum and minimum values of the input parameter until the best value of r 0 and ω WP have been found. Both optimization input parameter procedure adopt the same steps; for this reason Algorithm 2 details only the r 0 optimization. From Algorithm 2, analogous the golden section search algorithm [35], the golden-section value is g s = 1+ √ 5 2 , while r min and r max are minimum and maximum value of r 0 , respectively; N lps is the number of loops for reduction of the Algorithm 2 IPO Procedure: PA-CHSO and -HSO Input: N lps , tol r 0 , tol WP ω , p min , p max , ω WP min , ω WP max , g s , N f , K , ω WP , r max , θ k [1], p min , p max , θ k [n] = 0, p 0 ; Output: ω WP , r 0 ; 1: for n lps = 1 to N lps 2: if n lps = 1 3: (a) r l = log(p min ); 4: (b) r u = log(p max ); 5: else 6: (a) I n lps = min(|r 0 −r l |,|r 0 −r u |) (0.5g (n lps −2) s )

2) IPO-CPoS
The IPO-CPoS re-optimizes the performance input parameter to corroborate with the IPO-GS. For each PA algorithm, i.e., PA-HSO or -CHSO, it assumes final values from the IPO-GS, i.e, ω WP and r 0 , so it fixes the parameter of lower impact, i.e., ω WP , and re-optimizes the other, i.e., r 0 . The r 0 parameter has higher impact than the ω WP , because the power increasing granularity is given by it. In this context, the IPO-CPoS measures the probability of M channels to achieve the b * in the n iteration with the lower energy waste given n and r 0 , denoted by Ps 1 .
The Ps 1 formulation follows the RM concept discussed in eq. (2). Then, given r 0 and n, it can be defined as the probability of M -users to satisfy two conditions: Λ 2 assures the b * . In this context, Ps 1 is given by: Ps 1 is evaluated over an average behavior of N r realizations and its target value is defined as Ps * . After optimizing the performance input parameters, those that dramatically affect the complexity of the algorithm are also evaluated, i.e., for the PA-HSO and -CHSO, N f and K . Therefore, analogous to the Ps 1 , the optimization of theses parameters is modeled as: where all (K * ι ; N f * ι ) result of the increasing of i = [1, · · · , N N f ] and j = [1, · · · , N K ], that represent the decreasing of K and N f , respectively, with N N f = |N f | and N K = |K|.
In this context, the performance-complexity tradeoff of the input parameters also is evaluated, e.g., ω WP , r 0 , K and N f for the PA-CHSO or -HSO. Mathematically, it can be modelled as: where C(·) is the CC for the PA-CHSO or -HSO. The feasible solutions result in a Pareto front, which is given by the pairs (K , N f ).

E. Performance AND COMPLEXITY Analysis
The metrics for performance and CC analysis and tradeoff between them are presented. These metrics are used to measure the quality of the PA algorithm solutions during their operations.

1) Performance Analysis
The quality of the PA algorithms is measured based on the normalized mean square error (NMSE), and maximum and minimum PP for M channels. Both are related to the optimal solution vector p * . The NMSE in the nth iteration is given by: where E is the expectation operator and · is the Euclidean distance to the origin. And, the maximum and minimum

2) Complexity Analysis
The computational complexity (CC) is evaluated to measure the performance of the algorithms. Here, the CC is computed during the PA of each algorithm based on the floating-points operations (flop)s count, denoted as C. C is affected by the number of active channels (M ), by the size and number of routes, i.e., N ROADM i and N span i , which are related to measured SNR, from eq. (8), as well as by the number of iterations from algorithms N f . The most relevant metric of CC is C, because it does not depend on the machine, operating system, and programming structure and style.
C is evaluated for each PA algorithm as a function of the number of mathematical operations necessary to run until convergence. These operations include addition, subtraction, multiplication, division (or mod operator), natural logarithm, power or exponential and trigonometric functions, where each is assumed as one floating-points operation (flop). Logical (i.e., and, or) and comparison (i.e., if, else, else if, ≤, etc. . . ) operations, and variable assignment were considered irrelevant time-consuming operations.
In this context, the C of each PA algorithm in the nth iteration is modeled. In case of the PA-HSO, from Algorithm 1 it is given by (26) in case of the PA-CHSO, from Algorithm 1 and chaotic map in eq. (20), it is given by and, in case of PA-GD, from section IV-C, C n is given by: Asymptotically, the C of the PA-HSO, -CHSO and algorithms is of order of O(M 2 ), while C of the PA-GD is of order of O(M 3 ).

V. NUMERICAL RESULTS
In this section, the performance of the PA-CHSO and -HSO are analyzed and systematically compared. Section V-A presents network's scenario and parameters, while section V-B describes the input parameters optimization (IPO). Sections V-C and V-D analyse the PA performance  for both PA-CHSO and conventional -HSO methods considering perfect and non-perfect channel estimation, respectively. CC assuming different system loading is discussed in section V-E. The numerical simulations were performed with MATLAB (version 7.1) in a computer with 32 GB of RAM and processor Intel Xeon E5-1650@ 3.5 GHz.
A. NETWORK PARAMETERS Fig. 2 illustrates a virtual network topology for the transmission routes from source (S) to destination (D). The span length is 100 Km, channel spacing ( f ) of 50 GHz and guard band of 6 GHz. This topology was chosen to concentrate the routes R in some links, thus the effects of interference, as well as the effects of nonlinearities are more prominent. The EON transmission capability is in range of 100 to 300 Gbps. The routes and spectrum assignment procedure is out of the scope of this work, as these are considered to be stablished by a routing and spectrum assignment (RSA) algorithm. Bit rate requirement, routes, distance and modulation format are listed in Table 2. The physical layer parameters values of the elastic optical network are illustrated in Table 3, [4], [21], [22], [26], [36], [37]. Herein, we evaluate only twelve channels, including the channels with higher and lower power transmitted power, to avoid burden information.

B. IPO PROCEDURE UNDER PERFECT CHANNEL CONDITIONS
This step is very important for EON operation under all the operation conditions, such as uncertainty of SNR monitoring, effects of ageing and power instability. For this reason, the IPO-performance and IPO-Complexity are treated in the next subsections (V-B.1 to V-B.3), assuming the EON operating under perfect channel conditions, which is given by: perfect estimation of SNR, operation at the BoL and static scenario, following the Table 2 and 3, for operation at any conditions. The round-trip delay are compensated from traditional Smith predictor [31].
Basically, there are four main input parameters, which can be divided into two groups: input parameters that affect directly the performance, given by initial value of the power increasing r 0 and the tangential velocity for the power increasing ω WP ; and input parameters that affect directly the CC of the PA algorithms, given by wind parcels K and iterations number N f . The optimization of both groups is discussed in the subsections V-B.1 and V-B.2, respectively. Others input parameters are described in Table 4. Finally, the IPO under the perspective of complexity-performance tradeoff is elaborated in subsection V-B.3.

1) IPO-PERFORMANCE UNDER PERFECT CHANNEL CONDITIONS
In this context, r 0 and ω WP affect drastically the algorithm's performance. The optimized values are obtained by the IPO-GS framework previously described in section IV-D.1. It assumed ω WP = 1.5708 as an initial value for the tangential velocity, number of wind parcels K = 180 and number of iterations equal to N f = 250, all defined empirically. Fig. 3 illustrates the r 0 and ω WP optimization across the loops, in such a way that all the optimized parameters reach full convergence. Different ω WP and r 0 values were VOLUME 8, 2020  obtained for both algorithms in Fig. 3.a) and 3.b), demonstrating that the higher parameters values from PA-CHSO perform more accelerated and exploitive (via map chaotic) searches. Consequently, the PA-CHSO found a better solutions, measured by the cost function J 1 (p) during N r realizations and their respective standard deviation, as depicted in Figs. 3.c) and 3.d). More details are listed in Table 5, considering three loops that describe the optimization trend, i.e., n lps ∈ [1; 15; 30]; the finals optimized parameters is highlighted by bold face, while the parameters kept fixed at each loop is underlined.
In addition to the proposed optimization by the IPO-GS framework, we perform a numerical analysis IPO-CPoS for the performance input parameters, previously described in section IV-D.2, specifically in eq (21). In this numerical analysis, n ∈ [1; N f ] and r 0 ∈ [10 −8 ; 10 −4 ], both from Table 5. The ω WP parameter is not evaluated, so it assumes the optimized value from the Table 5.
As can be observed in Fig. 4, both strategies have attained success, defined as Ps 1 ≥ Ps * . In the case of PA-CHSO, a wider range of success regarding PA-HSO has been achieved, defined by r 0 ∈ [5 · 10 −6 ; 5 · 10 −5 ], and showing that the algorithm presents robustness and lower sensibility during the IPO-CPoS procedure. The best value for the PA-CHSO input parameter is obtained as r * 0 = 5·10 −6 , achieving fast convergence (n = 50) and superior performance, i.e., Ps 1 = 1. On the other hand, under PA-HSO, the CPoS is found for a narrow range of power increment, r * 0 ∈ [6 ± 0.5] · 10 −7 , because adopting similar values, such as r 0 = 6 · 10 −7 or r 0 = 8 · 10 −7 did not allow PA-HSO achieve Ps 1 ≥ Ps * . Hence, PA-HSO presented lower robustness and greater sensibility in adjusting its input parameter in the IPO step. Besides, the PA-HSO found slower convergence and worse performance: Ps 1 (50) = 0; and Ps 1 (250) = 0.98, both at r * 0 . Summarizing, the best r 0 and ω WP parameters found are registered in the last row of Table 5. Varying r 0 with fixed ω WP , Ps 1 found a range of r 0 that achieved success for both algorithms. This range defines the ability of updating power, which is directly related to robustness from both algorithms. Hereafter, we adopt for any condition of network's operation: r 0 ∈ [5 · 10 −6 ; 5 · 10 −5 ] and ω WP = 1.6975 for the PA-CHSO; and r 0 ∈ [6±0.5]·10 −7 and ω WP = 2.8386·10 −1 for the PA-HSO.

2) IPO-COMPLEXITY UNDER PERFECT CHANNEL CONDITIONS
K and N f are the parameters that affect drastically the algorithm's complexity. Then, Ps 2 optimizes K given N f , ; N * f ) assumed as reliable and viable, i.e., eq. (23).
In terms of PF, the PA-CHSO results are better than PA-HSO, showing a wider region for valid pairs (N f ; K ), while providing higher regularity in the plane that corresponds to the reliable and feasible region, combined to lower pairs values.

3) PERFORMANCE-COMPLEXITY TRADEOFF
Under channel perfect conditions, the group of input parameters ω WP , r 0 , K and N f are evaluated in terms of performance-complexity tradeoff, in subsection IV-D.2, specifically in eq. (24). The feasible solutions are given by  Table 6.

C. PA UNDER PERFECT CHANNEL CONDITIONS
Assuming IPO procedure has been performed previously, the PA per channel across iterations can be obtained, as illustrated in Fig. 6. In the simulations, it has been assumed perfect channel estimation, optical network operating at the BoL and static scenario, with routes, distances and bit rates given VOLUME 8, 2020 in Table 2, as well as physical parameters values following Table 3. The general parameters of the PA algorithms are adopted from the Table 4, while performance and CC parameters are adopted from the Table 6, being r 0 = 5.8318 · 10 −6 (PA-CHSO) and r 0 = 6.1873 · 10 −7 (PA-HSO). Indeed, the PA per channel reaches full convergence for both PA algorithms. The horizontal dashed lines represent the PA per channel obtained via GD procedure, which is an analytical method that has been used to validate convergence of the PA-CHSO and PA-HSO.
Regarding the results in Fig. 6, the following metrics have been calculated to the PA-CHSO and -HSO: a) the mean integral absolute value of the residual margin for the M -channels during time-window resulted equal to 19.1287 dB and 23.1334 dB, respectively; the maximum PP all the channels (max(p)) at the last iteration of 3.3811·10 −4 dB and 1.4014·10 −3 dB, respectively; b) mean settling iteration of all the users (i s ), assuming tolerance around 10 −4 for the M channels (i.e., p * − p ≤1 · 10 −7 ), results in ≈ 79 and ≈ 129 iterations, respectively. In this sense, the superiority from the PA-CHSO is evident. Besides, Fig. 6 presents overshooting and undershooting during the PA, which is much    Besides, in range of n = 1 to 53 is evident the instability by PA-HSO, due to its lower exploitative capacity for the power launch (or initial power of the eye) of 0 dbm. Therefore, the best PA capacity from PA-CHSO is clear. In order to evaluate the PA-CHSO and -HSO effectiveness in terms of optimal PA, three analysis for channel conditions were carried out: a) non-perfect monitoring of the OPMs, in section V-D.1; b) channel ageing effects, in section V-D.2; c) power instability, in section V-D. 3. The general parameters values adopted for both algorithms are described Table 4, while input parameters are depicted in Table 6, with the choice of r 0 = 5.8318 · 10 −6 (PA-CHSO) and r 0 = 6.1873 · 10 −7 (PA-HSO).

1) NON-PERFECT MONITORING OF THE OPMs
there is an inaccuracy in the monitoring of the OPMs. Here, it is considered as a random variable i ∼ LN (µ, σ ), where µ = 0 dB and σ = 0.16 dB. These monitoring uncertainties corresponds to a maximum error i max = 0.6 dB with high probability (> 0.9995), commonly adopted in the optical networks considering inaccuracies from the OPMs [38]- [40]. This error is added into ith SNR during the PA procedure. Moreover, the adopted scenario assumes an operation at the BoL without power instability. Fig. 7.b) depicts the velocity and the tendency of convergence, as well as the quality of the solutions. As can be observed, there is a decrease in the NMSE with the increase in the number of iterations. It is noticed that for early iterations the PA-CHSO achieves better convergence performance when compared to PA-HSO. In terms of convergence velocity, the PA-CHSO (at n = 42) is able to attain a NMSE = 3.2 · 10 −2 approximately three times faster than PA-HSO (n = 123). On the other hand, similar NMSE values are found in the later iterations, i.e, n ≥ 125 iteration, where both algorithms achieve an asymptotic NMSE ≈ 3.21 · 10 −2 . Those results are affected by the OPMs inaccuracies. Indeed, comparing both PA algorithms performance operating under perfect monitoring condition, Fig. 7.a), the same asymptotic NMSE value has not been observed in both schemes. In this ideal scenario, the maximum PP resulted in p pa-chso i max = 0.26042 dB and p pa-hso i max = 0.26037 dB.

2) CHANNEL AGEING EFFECTS
Under equipment ageing effects, Fig. 9 proposes analyze the PP trend against a multi-period incremental assuming τ = [0, 2, 4, · · · , 10] years, representing the effect of ageing from BoL to EoL network. It illustrates the expected value of the PP from M -channels (E[p]) across the time, as well as their respective standard deviation (σ p ),p  against E[p] ≈ 4.6223 · 10 −2 dB at τ = 10. In terms of minimum PP, the PA-CHSO found BER * all the time, a consequence of E[p] ≈ −5.6462 · 10 −4 dB ≤ min at τ = 10, while the PA-HSO does not found BER * , a consequence of E[p] ≈ −1.6013 · 10 −1 dB ≥ min at τ = 10. Therefore, the PA-CHSO and -HSO resulted at a margin increasing of 5.6462 · 10 −4 dB and 1.6013 · 10 −1 dB, respectively, and presented a better saving energy. Besides, the σ p values found demonstrated that PA-CHSO is more stable than PA-HSO in terms of minimum energy expenditure to achieve the BER * . In this context, PA-CHSO is effective to mitigate the channel ageing effects.

3) POWER INSTABILITY
Assuming now a dynamic scenario characterized by power instability or perturbation, which can represent dropping or adding channels to the EON. After node add-drop channels an undesired effect reaches the surviving channels, herein modeled as a sine function in eq. (13), where A pert = 0.8 dB and f = 0.5 Hz represents overshoot and undershoot maximum adopted in the project of EDFA compensation of ±1 dB. Theses values assured the drops of the two routes, simultaneously [36].
In simulations of Fig. 10, a dynamic scenario has been modeled assuming a network optimized to operate with 12 users, such as in Table 2 and Fig. 2. Thus, a fast variation is introduced at the node 8, where R 10 and R 11 are dropped at the iteration 30. This dropping results in four surviving channels (R 4 , R 8 , R 9 and R 12 ) forward. These channels are affected by power fluctuations from node 8 to D. The interval of perturbation occurs at 30 < n ≤ 49.
Elaborating furhter, Fig. 10 illustrates the effect of the power perturbation under three situations: with and without compensation from PA-CHSO and -HSO. In case of no compensation, the launch power is assumed as optimized from the PA-CHSO, and power adjustment is not carried out after the drop of the two channels. In other words, the channels R 4 , R 8 , R 9 and R 12 are penalized and theirs transmission power are not re-optimized, resulting in a NMSE[210] = 1.0406 · 10 −1 . However, performance improvment can be attained deploying compensation in PA-HSO and PA-CHSO, resulting in a NMSE[210] = 1.7455·10 −4 and 1.2874·10 −5 , respectively. It is evident the PA-CHSO ability to escape from local minimum around n = 100, as well as the behavior of both algorithms in the sense of following the power perturbation and in achieving the optimal power in latter iterations.
A comparison between the initial and final NMSE value showed that for the PA-HSO, similar values are found, i.e., NMSE 1 = 1.7946 · 10 −4 and NMSE[1] = 1.7455 · 10 −4 ; and for the PA-CHSO, a better final value is found, i.e., a gap of NMSE= 3.76 · 10 −5 . Therefore, the PA assuming fluctuation from drop channels is validated and a better performance is found by the PA-CHSO.

E. COMPLEXITY
The CC is evaluated in terms of mathematical operations and number of channels. In asymptotic terms, the PA-HSO and PA-CHSO have complexity of order of O(M 2 ). On the other hand, the complexity of GD algorithm is of order of O(M 3 ), as described in section IV-E.2. Aiming at attaining more accuracy in the complexity analyses, we have considered the mathematical operations from eqs. (27), (26) and (28). Three different system loadings have been adopted: A has 12 channels (2,2 Tbps), as described in Table 2 and Fig. 2; B has 120 channels (22 Tbps); and C has 240 channels (44 Tbps). B and C have the same topology of A, however theirs routes result of 10 and 20 times of A (R 1 , · · · , R 12 ), respectively. Those scenarios assume perfect channel conditions: operation at the BoL, static operation, and perfect monitoring of channel. Fig. 11 depicts the averaged CC for the three algorithms operating under A, B and C scenarios. It also assumes optimized parameters from the Table 6. Those parameters result the worst-case for the CC, i.e., K and N f can be reduced due to the increasing of M -channels and r 0 , while ω WP can be reduced due to the increasing of non-linear effect. The PA-CHSO has resulted in lower complexity than two methods. In addition, the CC can be reduced by considering re-optimization of input parameters for any network operating conditions.

VI. CONCLUSION
The PA-CHSO method proved to be a promising technique to resource allocation in elastic optical networks, especially by Nyquist wavelength division multiplexing (WDM) super-channels, combining competitive convergence speed, control capacity, non-linear effects mitigation, higher probability of success in lower iterations and lower penalties. The PA-CHSO has demonstrated a higher ability to escape of local minimum caused by non-linear effects in scenarios where higher bit rates are required. The optimized parameters presented robustness considering conditional probability of success. Moreover, it resulted in a CC in the order of O(M 2 ), much lower than the gradient descent method (of order of O(M 3 )), and marginally lower compared to the conventional PA-HSO.
The conventional PA-HSO has presented inferior performance regarding the PA-CHSO. In terms of the optimization of parameters, a narrow conditional probability of success was found, resulting in a low ability for absorption of ageing effects and vast-variations, a consequence of higher sensibility to the parameters variation. Moreover, it was found worse penalties and lower convergence speed in case of dynamic scenarios.
The PA-CHSO performs PA in EONs with better performance-complexity tradeoff regarding both the PA-HSO and the PA-GD, considering non-perfect monitoring of OPMs, channel ageing effects and dynamic scenario, that are the main realistic conditions from EONs operations. Such advantages result in a better margin reduction, energy efficiency improvement, and cost limitations.
In summary, inserting chaotic map procedure into the PA-HSO (or -CHSO) brought better performance-complexity balancing tradeoff.