Discrete Dynamics in Nature and Society

Volume 2016, Article ID 4135056, 11 pages

http://dx.doi.org/10.1155/2016/4135056

## A Network Traffic Prediction Model Based on Quantum-Behaved Particle Swarm Optimization Algorithm and Fuzzy Wavelet Neural Network

School of Mathematics and Statistics, Chuxiong Normal University, Chuxiong, Yunnan 675000, China

Received 2 November 2015; Revised 14 January 2016; Accepted 3 February 2016

Academic Editor: Ahmed Kattan

Copyright © 2016 Kun Zhang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Due to the fact that the fluctuation of network traffic is affected by various factors, accurate prediction of network traffic is regarded as a challenging task of the time series prediction process. For this purpose, a novel prediction method of network traffic based on QPSO algorithm and fuzzy wavelet neural network is proposed in this paper. Firstly, quantum-behaved particle swarm optimization (QPSO) was introduced. Then, the structure and operation algorithms of WFNN are presented. The parameters of fuzzy wavelet neural network were optimized by QPSO algorithm. Finally, the QPSO-FWNN could be used in prediction of network traffic simulation successfully and evaluate the performance of different prediction models such as BP neural network, RBF neural network, fuzzy neural network, and FWNN-GA neural network. Simulation results show that QPSO-FWNN has a better precision and stability in calculation. At the same time, the QPSO-FWNN also has better generalization ability, and it has a broad prospect on application.

#### 1. Introduction

With the rapid development of computer network technology, network applications have infiltrated every corner of human society and play an important role in various industries and situations. Since the network topology structure is gradually complicated, the problem of network’s emergencies and congestion are more and more serious. Through monitoring and accuracy prediction of network traffic, it can prevent network congestion and can effectively improve the utilization rate of the network [1].

In general, the network traffic data is a kind of time series data and the problem of network traffic prediction is to forecast future network traffic rate variations as precisely as possible based on the measured history. The traditional prediction model, such as Markov model [2], ARMA (Autoregressive Moving Average) model [3], ARIMA (Autoregressive Integrated Moving Average) model [4], and FARIMA (Fractional Autoregressive Integrated Moving Average) [5] model, has been proposed. As the network traffic is affected by many factors, the network traffic time series show quite obvious multiscale, long-range dependence, and nonline characteristic. The methods mentioned above have the weakness of low-level efficiency [6].

An artificial neural network (ANN) is an analysis paradigm that is roughly modeled after the massively parallel structure of the brain. Artificial neural networks can be thought of as “black box” devices that accept inputs and produce outputs and are able to give better performance in dealing with the nonlinear relationships between the output and the input theoretically [7]. Although artificial neural networks have been successfully used for modeling complex nonlinear systems and predicting signals for a wide range of engineering applications, artificial neural networks (ANNs) have limited ability to characterize local features, such as discontinuities in curvature, jumps in value or other edges [8]. These local features, which are located in time and/or frequency, typically embody important process-critical information such as aberrant process modes or faults.

The fuzzy neural networks (FNN) are the hybrid systems which combine both advantages of the fuzzy systems and artificial neural networks. The FNN possesses the merits of the low-level learning and computational power of neural networks, and the high-level human knowledge representation and thinking of fuzzy theory [9]. A fuzzy wavelet neural network (FWNN) is a new network structure that combines wavelet theory with fuzzy logic and NNs. The synthesis of a fuzzy wavelet neural inference system includes the determination of the optimal definitions of the premise and the consequent part of fuzzy IF-THEN rules [10]. However, many fuzzy neural network models, including FWNN, have common problems derived from their fundamental algorithm [11]. For example, the design process for FNN and FWNN combined tapped delays with the backpropagation (BP) algorithm to solve the dynamic mapping problems [12]. Unfortunately, the BP training algorithm has some inherent defects [13, 14], such as low learning speed, existence of local minima, and difficulty in choosing the proper size of network to suit a given problem. Thus the systems which employ basic fuzzy inference theory make the degree of each rule extremely small and often make it underflow when the dimension of the task is large. In such a situation, the learning and inference cannot be carried out correctly.

As a variant of PSO, quantum-behaved particle swarm optimization (QPSO) is a novel optimization algorithm inspired by the fundamental theory of particle swarm and features of quantum mechanics such as the use of Schrödinger equation and potential field distribution [15]. As a global optimization algorithm, the QPSO can seek many local minima and thus increase the likelihood of finding the global minimum. This advantage of the QPSO can be applied to neural networks to optimize the topology and/or weight parameters [16].

In order to predict the network traffic more accurately, a prediction model of network traffic based on QPSO algorithm and fuzzy wavelet neural network is proposed in this paper. The network traffic data is trained by QPSO and fuzzy wavelet neural network and weights are progressively updated until the convergence criterion is satisfied. The objective function to be minimized by the QPSO algorithm is the predicted error function.

The rest of this paper is arranged as follows. Section 2 gives a brief introduction to classical PSO algorithm and quantum-behaved particle swarm optimization (QPSO) algorithm. In Section 3, the fuzzy wavelet neural network is introduced and the fuzzy wavelet neural network based on QPSO (QPSO-FWNN) algorithm is presented in detail. In Section 4, simulation results are presented. Performance metrics of the several prediction methods are analyzed and compared in Section 5. Finally, some conclusions are given in Section 6.

#### 2. Quantum-Behaved Particle Swarm Optimization

##### 2.1. Classical Particle Swarm Optimization

Particle swarm optimization (PSO) is an evolutionary computation technique that is proposed by Kennedy and Eberhart in 1995 [17]. Similarly to other genetic algorithms (GA), PSO is initialized with a population of random solutions. However, it is unlike GA, PSO does not have operators, such as crossover and mutation. In the PSO algorithm, each potential solution, called “particles,” moves around in a multidimensional search space with a velocity constantly updated by the particle’s own experience and the experience of the particle’s neighbors or the experience of the whole swarm [18].

In the PSO, each particle keeps track of its coordinates in the search space which are associated with the best solution it has achieved so far and this value is called . Another best value that is 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 [19]. This location is called .

The process for implementing the global version of PSO is given by the following steps.

*Step 1. *Initialize a population (array) of particles with random positions and velocities in the* D*-dimensional problem space. For a* D*-dimensional problem with number of particles, the position vector and velocity vector are represented as where

*Step 2. *For each particle, evaluate the desired optimization fitness function in variables.

*Step 3. *Compare each particle’s fitness evaluation with the particle’s* pbest*. If the current value is better than* pbest*, then set the* pbest* value equal to the current value and the* pbest* location equal to the current location in* D*-dimensional space.

*Step 4. *Compare the fitness evaluation with the population’s overall previous best. If the current value is better than , then reset to the current particle’s array index and value.

*Step 5. *Update the velocity and position of the particle according to (2) and (3), respectively. One haswhere and are two positive constants, known as the cognitive and social coefficients, which control the relative proportion of cognition and social interaction, respectively, and the values of and were decreased with each iteration [20]. and are two random values in the range . , , and are the velocity, position, and the personal best of th particle in th dimension for the th iteration, respectively. The is the th dimension of best particle in the swarm for the th iteration.

*Step 6. *Loop to Step until a stop criterion is met, usually a sufficiently good fitness or a maximum number of iteration generations.

##### 2.2. Quantum-Behaved Particle Swarm Optimization

Motivated by concepts in quantum mechanics and particle swarm optimization, Sun et al. proposed a new version of PSO, quantum-behaved particle swarm optimization (QPSO) [21]. In the QPSO, the state of a particle is depicted by a wave function , instead of position and velocity. The probability density function of the particle’s position is in position [22].

Assume that, at iteration , particle moves in -dimensional space with a potential well centered at on the th dimension. The wave function at iteration is given by the following equation:where is the standard deviation of the double exponential distribution, varying with iteration number . Hence the probability density function is defined asand the probability distribution function is given by the following equation:

By using Monte-Carlo method, the th component of position at iteration can obtain by the following equation:where is a uniform random number in the interval . The value of is calculated aswhere parameter is known as the contraction-expansion (CE) coefficient, which can be tuned to control the convergence speed of the algorithms [23]. is the mean best position () and is defined aswhere is the size of the population. Hence the position of the particle is updated according to the following equation:

From (4) and (10), the new position of the particle is calculated aswhere is a random number in the range . is linearly decreasing factor from 1.0 to 0.3 with iteration aswhere is the maximum number iteration used in algorithm.

#### 3. Fuzzy Wavelet Neural Network Based on QPSO

##### 3.1. The Wavelet Base Function

In , a wavelet dictionary is constructed by dilating and translating from a wavelet base function of zero average [24]:which is dilated with a scale parameter and translated by

##### 3.2. Fuzzy Wavelet Neural Network

The basic architecture of fuzzy wavelet neural network could be described as a set of Takagi-Sugeno models. Assume that there are rules in the rule base and the Takagi-Sugeno fuzzy if-then rules are usually in the following form:where are input of T-S rule, is the th linguistic variable value of the th input, which is a fuzzy set characterized by wavelet function. is constant coefficients which are usually referred to as consequent parameters determined during the training process.

Figure 1 shows the architecture of the proposed FWNN modeling. The FWNN is a 4-layer feedforward network and detailed descriptions and equations for each layer are given here.