The Power Allocation in PON-OCDMA with Improved Chaos Particle Swarm Optimization

In this work, it is investigated an improved chaos particle swarm optimization (IC-PSO) scheme to refine the quality of the algorithm solutions regarding to solve the optimal power allocation in next generation passive optical networks (NG-PON)s. The proposed IC-PSO scheme utilizes the Beta distribution instead of uniform distribution of the traditional PSO. A factor of damping based in the chaotic logistic map related to the updating of the best global val ue is successful introduced. The numerical results corroborate the best relation between the performance-complexity tradeoff and the quality of the algorithm solutions for the proposed IC-PSO when compared with the classical PSO power allocation scheme.


Introduction
The resource allocation in passive optical network (PON) is utilized for dynamic bandwidth allocation (DBA), power allocation (power control), multiple bit rate control and the adjustment of the number of actives optical network units (ONUs), such as sleep mode, to improve the network capacity, flexibility and energy efficiency [1][2] [3].In the power allocation problem the aim is to obtain the optimization of the transmitted power to minimize the interference between the users and maximize the energy efficiency, considering the quality of service (QoS) restrictions in terms of signal-to-noise-plus interference ratio (SNIR) of each optical user class [4] [5].The power allocation problem are related to not convex cost and constraint functions, therefore this problem in not straight to be solved [6] [7].In this context, there are several approaches to solve the power allocation problem, such as analytical-iterative algorithms, matrix inversion, numerical procedures and meta-heuristic schemes [4] [7][8] [9].The metaheuristics methods are very promissory approaches to perform the power allocation considering its performance-complexity tradeoff and fairness features regarding the previous cited approaches [9] [10].In addition, the bio-inspired meta-heuristics have been presented relevant results to solve the power allocation problem [6][9] [10].In this work, the meta-heuristic of particle swarm optimization (PSO) and its variations are considered in the investigation of the power allocation problem in context of the next generation of PONs (NG-PONs) [11].The Power Allocation in PON-OCDMA with Improved Chaos Particle Swarm Optimization the advanced modulation format [1][2] [3].PONs based on OCDMA (PON-OCDMA) technology presents characteristics such as asynchronous operation, high network flexibility, protocol transparency, simplified network control, quality of service (QoS) in the physical layer and improvement in security aspects [4] [19].In this work, the PON-OCDMA with advanced modulation format is selected to our investigation about resource allocation considering the competitive cost and flexibility of the PON-OCDMA scheme [11][19].In this scheme the multiport encoder/decoder at the OLT are based on multi-port arrayed waveguide gratings (AWG) to generate and recognize multiple time spreading optical codes in a single device simultaneously [20].Besides, the encoder/decoder at the ONUs is based on super-structured fiber Bragg grating (SSFBG) that is independent of the code length and polarization [11][20].The code generated in the OLT and ONUs is a coherent code phase-shift-keying (PSK), in which the code information is transmitted in the phase.
In the encoder/decoder at OLT, a set of optical codes are generated considering the AWG with N inputs/outputs in the time domain and each PSK code is obtained through a combination of light pulses with different phase [20].The chip period ( ), that represents the amount of time interval between two consecutive pulses in each optical code is defined as , where is the effective refractive index, is the differential path length and is the light speed [11].
The code cardinality is obtained by the binomial , where is the code length, thus the maximum cross-correlation is given by and the autocorrelation peak is represented by [20].In the SSFBG at the ONU, an optical code sequence is obtained by reflecting back optical chip pulses from each fiber Bragg grating (FBG) chip.To generate time-spreading PSK code in the SSFBG, it is required to adjust the number of FBG chips and the phase-shift level for desired code sequences.The SSFBG also acts as the decoder, resulting in either the autocorrelation or cross-correlation waveform according to its FBG chip arrangement.

Power Allocation Problem Formulation
In the PON-OCDMA the SNIR at the OLT (upstream) is related to the carrier-tointerference ratio (CIR) as [4], ( where N is the code length,  is the Hamming average variance of the cross-correlation amplitude and i is the CIR at the input of ith node, given by [4], ( where Gii is gains of transmitter-receiver pairs, is the transmitted power at the ith node, is the transmitted power from interfering nodes, is the power of receiving noise, and the elements Gij constitute the network interference matrix between the nodes given by , where αf is the fiber attenuation (km -1 ), ac represents the encoder/decoder attenuation and Lc is the total internal losses in the optical path.There is the establishment of the virtual path between the transmitting ONU and receiving OLT based on the code and the total link length that is represented by , where is the link length from the transmitting ONU to the remote node and is the link length from the remote node to the OLT.The amplifier spontaneous emission (ASE) effect in the optical preamplifier is a predominant received noise power when compared to thermal and shot noise [4].Moreover, received noise power is given by , which considers the two polarization mode of a single mode fiber, and where is the spontaneous emission factor, h is Planck's constant, f is the carrier frequency, Gi is the amplifier gain, and is the optical bandwidth.
The optimization of the SNIR is related to the power allocation for each PON-CDMA node.
Therefore, the SNIR optimization is based on the determination of the minimum power restriction, named sensitivity level, ensuring the suitably optical signal detected by all optical devices together with the QoS requirements.Thus, the power control in PON-OCDMA is an optimization problem.Denoting i at the required decoder input, in order to get a certain maximum tolerable bit error rate at the ith optical node, considering K the number of ONUs and the K-dimensional column vector of the transmitted optical power p = [p1, p2,…, pK] T , the optical power control problem consists in finding the optical power vector p that minimizes the cost function ; this optimization problem can be formulated as [7] :  i = 1,.., K, where 1 T = [1, ..., 1] is a one vector and is the minimum CIR to achieve a desired QoS; pmin and pmax is the minimum and maximum value considered as permitted transmitted power, respectively.Through a matrix notations, (3) can be grouped as , where I is the identity matrix, H is the normalized interference matrix, which elements evaluated by for and zero for another case, thus , where there is a scaled version of the noise power.Substituting inequality by equality, the optimized power vector solution can be analytically obtained through the matrix inversion .However, the matrix inversion is not attractive procedure due to its performance-complexity tradeoff [4][7].The optimization method to obtaining the optical power vector p based on PSO and IC-PSO are an expeditious method in order to solve resource allocation problems due to its performancecomplexity tradeoff and fairness features regarding the optimization procedure based on matrix inversion [4].An alternative formulation to the power allocation optimization of ( 3) is discussed in [6], and adopted herein with some adaptation in order to jointly include information rate and power allocation: where L is the number of different group of information rates allowing in the system, and is the number of user belonging to the lth rate group with minimum rate given by .Finally, the threshold function in ( 5) is defined as: ( where is the SNIR for the th user belongs to the lth rate group.Note that the term gives credit to those solutions with minimum power and punishes others using high power levels. The quality of solution achieved by any iterative resource allocation procedure could be measured by how close to the optimum solution is the found solution, and can be quantified by the normalized mean squared error (NMSE) when equilibrium is reached.For power allocation problem, the NMSE definition is given by, where denotes the squared Euclidean distance to the origin, and the expectation operator.In addition, a convergence test is considered 100% successful if the following relation holds: (7) where, is the global optimum of the objective function under consideration, is the optimum of the objective function obtained by the heuristic PSO algorithm after iterations, and , are accuracy coefficients, usually in the range In this study it was assumed that T = 100 trials and .

PSO Principle
The meta-heuristic PSO is based on the particles that keeps track its coordinates in the space of search, which are associated with the best solution (fitness) it has achieved so far.Another best value tracked by the global version of the particle swarm optimizer is the overall best value, and its location, obtained so far by any particle in the population.At each time iteration step, the PSO concept consists of velocity changes of each particle toward local and global locations.
The acceleration is weighted by a random term, with separate random numbers being generated for acceleration toward local and global locations.Let bp and vp denote a particle coordinates (position) and its corresponding flight speed (velocity) in a search space, respectively.In the PSO strategy, each power-vector candidate , with dimension , is used for the velocity-vector calculation in the next iteration [9]; in vector form, the K dimensional velocity-vector where is the inertia weight of the previous velocity in the current speed calculation, the diagonal matrices and with dimension K have their elements as random variables with uniform distribution in the range U [0, 1], generated for the pth particle at iteration t = 1, 2, . . ., ; and are the best global position-vector found until the iteration, and the best local position-vector found at the iteration, respectively; and are acceleration coefficients regarding the best local particles' position and the best global positions; both coefficients influence in the velocity updating and in the algorithm convergence.In our PON-OCDMA power allocation problem, the particle's position at the tth iteration is defined by the power-vector candidate .The position of each particle is updated using the new velocity-vector for that particle: (9) where is the population size, which depends on the PON-OCDMA network dimension, specifically the number of ONUs.

Improved Chaos Particle Swarm Optimization (IC-PSO) Scheme
The proposed Improved Chaos PSO (IC-PSO) scheme is based on two specific features aggregated to the conventional PSO algorithm: i) it is utilized the Beta distribution instead of uniform distribution to generation of random variables aiming at increasing the diversity while aid the exploration (diversification) of undercover regions in the search space during the transmitted power optimization procedure.
ii) it is introduced a damping factor based on random numbers generated by chaotic maps related with the updating of the best global value.These aspects could limit the dominance of the best global particle value to avoid premature convergence, increasing the randomness (diversification) without loss in the exploitation capability of the algorithm.
Under these aggregated features, the velocity updating equations of the IC-PSO is given by: (10) where [0, 1] is the damping factor generated by chaotic maps.In this work, without loss of generality, it is used the one dimensional logistic map that is related to the dynamics of the biological population [13][14].The logistic map is given by, (11) where a is the control parameter.The variation of X[t] will increase the randomness of the influence of the best global.In addition, and are the diagonal matrices with dimension K, where their elements are random variables with Beta distribution in the range B(p,q) [0, 1] generated for the pth particle at iteration t = 1, 2, . . ., ., where p and q are the shape parameters from the Beta distribution [21].The control of the shape parameters enables Beta distribution simulation with symmetric densities ( = ) and asymmetric densities with shape parameters  .Besides, the uniform distribution is a special case of Beta distribution with = = 1 [21].The particle position updating is performed in the same way of conventional PSO, Eq. ( 9).

Parameters Summary
In this section the scenario studied is described, the values for the optical network devices and standard fiber are summarized.The conventional PSO performance presents high dependence of the control input parameters for each kind of optimization problem, therefore the definition of the parameters for resource allocation in optical networks was performed in [24].The conventional PSO parameters utilized in all numerical simulations are illustrated in Table II  For all the numerical simulations it is performed 100 trials (realizations) to obtain the better solution with significant coefficient of variation (CV), which is given by the ratio of the standard deviation to the mean.Our simulations have presented CV lower than 8%.Data distribution with CV < 25 % is considered a low-variance data distribution [25].

IC-PSO Input Parameters Tuning
The conventional PSO and IC-PSO algorithms for optical power allocation present several equivalents parameters; therefore, the equivalents parameters utilized for the PSO will be also utilized for IC-PSO algorithm.On the other hand, for the IC-PSO will be adjusted the parameters related to the Beta function distribution.Initially, in the numerical results is presented the NMSE for the IC-PSO when different values of shape parameters (p, q) and number of ONUs are utilized.The damping factor ( ) will be generated by the logistic map with the control parameter a = 4 to obtain a chaotic behavior [13].Herein, a vast quantity of combinations of shape parameters were simulated taking the following scheme: Firstly, a q shape parameter was fixed and various simulations were performed with different value of p shape parameter.After that, another q shape parameter was fixed and various simulations were performed with different value of p shape parameter again.This process was performed in the wide range of values, which were refined in each step; however, for practicality purpose, only the more representative values will be presented and discussed.situations with 16 and 32 ONUs.¨In this situation, the values of the NMSE is limited by the nonconvexity of the power allocation problem.For the case of 48 ONUs the lower NMSE is obtained with p = 2.5 and q = 1.9.This behavior is related to variation of the Beta distribution shape with the alteration of the shape parameters.Herein, the Beta distribution shape parameter values that represent the best trade-off between the exploration (diversification) and the exploitation (intensification) for the power allocation problem using IC-PSO for 16, 32 and 48 ONUs were obtained and summarized in Table III.

IC-PSO versus PSO Resource Allocation
In this section, the IC-PSO and PSO resource allocation have been evaluated considering the network parameters and variables described in the previous sections.In order to obtain a fair comparison between the IC-PSO and PSO power allocation algorithms, the same computational effort, herein represented by the run time, was guaranteed for both algorithms.The simulations were performed with MATLAB (version 7.1) in a domestic computer with 4 GB of RAM and processor Intel Core i5@ 1.6 GHz.Besides, the IC-PSO power allocation will be compared to the conventional PSO power allocation scheme, which was previously validated and compared with other methods [9][10].ONUs.In addition, it is illustrated in the horizontal dash line the sum of the transmitted power obtained with matrix inversion procedure.The matrix inversion is effective to obtain the correct value of the transmitted powers; however, it presents high computational complexity when compared with heuristic or meta-heuristic approaches [4] [7].Therefore, the figures-of-merit results obtained with matrix inversion will be utilized to validate the proposed IC-PSO power allocation.Herein, the goal is to evaluate the initial behavior of convergence trend from both heuristic algorithms; therefore, it is considered a maximum number of 800 iterations.depicts the case with 16 ONUs where the SNIR is high; even so, there is a marginal improvement in the convergence performance of the IC-PSO over the conventional PSO optical power allocation procedure.Indeed, under this scenario, the power transmission convergence was obtained with approximately 250 and 350 iterations for IC-PSO and PSO power allocation scheme, respectively.Notice in the Fig. 3 (b), for 32 ONUs, the IC-PSO algorithm was able to achieve convergence after approximately 345 iterations in contrast to the approximately 800 iterations necessary for the PSO power allocation scheme convergence.Besides, the more remarkable situation occurs for 48 ONUs that is illustrated in the Fig. 3 (c), where the convergence of the transmitted power occurs with approximately 450 iterations for IC-PSO power allocation scheme and the PSO power allocation scheme does not present tendency of convergence in the next iterations, here beyond of 800 iterations.estimates in medium and weak signal environments, i.e. in PONs with higher number of ONUs .In addition, the IC-PSO power allocation scheme is effective and able to improve substantially the quality of the solutions with decreasing of approximately 5 decades of normalized mean square error (for the same number of iterations) when compared with the PSO power allocation scheme.Indeed, this convergence increment and quality of the solutions improvement is remarkable, since in the IC-PSO power allocation scheme it does not come with a computational complexity increasing when compared with the PSO power rate allocation.

Fig. 2 .
Fig.2.presents the attained NMSE with IC-PSO considering different value of p and q shape parameter of Beta function for 16, 32 and 48 ONUs.

Fig. 3
Fig. 3 illustrates the sum of the transmitted power versus the number of iterations for IC-PSO and PSO power allocation schemes considering the scenario with (a)16, (b)32 and (c)48

Fig. 3 .
Fig. 3. Sum of the transmitted power versus the number of iterations for PSO and IC-PSO power allocation for (a) 16, (b) 32 and (c) 48 ONUs.

Fig. 3
Fig.3illustrates the convergence of the transmitted power value obtained with IC-PSO and PSO power allocation schemes to the value obtained with matrix inversion procedure.The convergence behavior is a clear improvement on the velocity of convergence for the IC-PSO power allocation scheme when compared with PSO power allocation scheme, mainly when the number of ONUs increase, i.e. for the number of 32 and 48 ONUs the IC-PSO convergence gain improves substantially.Herein, the number of 32 and 48 ONUs representing the situation of medium and weak SNIR estimates environment, respectively.The faster convergence of the IC-PSO power allocation scheme when the SNIR has deteriorated is related to the utilization of the Beta distribution which provides diversity increasing while aid the exploration (diversification) of undercover regions in the search space.In addition, the damping factor

Fig. 4 Fig. 4 .
Fig.4depicts the normalized mean squared error (NMSE) against the number of iterations for PSO and IC-PSO optical power allocation schemes to precisely evaluate the velocity of convergence and quality of solutions.In this sense, we have considered more iterations in such analysis aiming at achieving the condition of the non-improvement on the performance (NMSE floor condition).

Fig. 5 .
Fig.5.Asymptotic computational complexity versus the number of active ONUs for heuristicevolutionary PSO, IC-PSO, as well as the matrix inversion power allocation schemes.
[22]e I presents the main system parameters deployed in the numerical simulations.These parameters are based on network equipment currently available[19][22].The link length between the OLTs and the remote node is 40 km.Moreover, the link lengths from the remote node to the ONUs are uniformly distributed over a distance with a . The IC-PSO input parameters tuning that are different of the PSO are discussed in the Subsection 4.2.