An adaptive fuzzy model using orthonormal basis functions based on multifractal characteristics applied to network traffic control

Neurocomputing 74 (2011) 1894–1907

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Neurocomputing journal homepage: www.elsevier.com/locate/neucom

An adaptive fuzzy model using orthonormal basis functions based on multifractal characteristics applied to network traffic control F.H.T. Vieira, F.G.C. Rocha n ´s, 74605-010 Goiˆ School of Electrical and Computer Engineering (EEEC), Federal University of Goia ania, GO, Brazil

a r t i c l e i n f o

a b s t r a c t

Available online 14 April 2011

In this paper, we present an adaptive predictive orthonormal basis functions (OBF)-fuzzy model that considers the multifractal behavior of network traffic flows. To this end, we model the traffic flows using orthonormal basis functions obtained through multifractal analysis. We insert the orthonormal basis functions into a fuzzy model trained with an adaptive clustering algorithm. Further, we propose a predictive flow control scheme for broadband networks. Also, using the fuzzy model parameters, we derive an expression for the optimal traffic source rate. Comparisons to other predictive control schemes prove the efficiency of the proposed adaptive OBF-fuzzy based control and training algorithm. & 2011 Elsevier B.V. All rights reserved.

Keywords: Fuzzy model Orthonormal basis functions Adaptive learning Congestion control Multifractal traffic

1. Introduction Some researches have revealed that multifractal models are adequate in describing traffic characteristics of some network traffic flows [1,9,14,29]. Compared to monofractal models, multifractal processes have in addition to long-range dependence, different scaling laws [25]. The slow decay of the autocorrelation function of some network traffic traces indicates that they present long-range dependence which greatly influences network performance [16]. Predictive traffic control can be very efficient once it tries to reduce congestion before it happens and adapts according to changes in network traffic conditions [2,33]. Some of the proposed resource allocation models for network traffic flows based on the neural network and fuzzy predictions deserve attention. Many studies show that the fuzzy models have advantages over linear models in describing the nonlinear time-varying characteristics of unknown real process such as network traffic flows [5,40]. In fact, the fuzzy modeling is capable of representing a nonlinear complex system through the combination of linear local models [41]. Several congestion control schemes for computer networks have been proposed in the literature [24,13,22,30,45]. Among the proposed congestion control using fuzzy logic, some are not adaptive such as in Hu and Petr [21] and others are based on specific protocols or network technologies [6,30]. In the first case, many of the schemes are not sufficiently accurate in predicting the traffic variation behavior generated by real-time applications

n

Corresponding author. Tel.: þ55 62 3209 6400. E-mail addresses: fl[email protected] (F.H.T. Vieira), fl[email protected].(F.G.C. Rocha) 0925-2312/$ - see front matter & 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.neucom.2010.07.038

due to their failure to adapt the model parameters [37]. For example, in Hu and Petr [21], the authors propose a fuzzy based algorithm for predictive traffic flow rate control. However, in this approach the membership functions that characterize the fuzzy predictor inputs are fixed, i.e., such functions are not updated as new inputs are available. In the second case, we can cite control schemes that are exclusives to the protocols of ATM networks (Asynchronus Transfer Mode) [6,23] or based on the congestion control mechanisms of Transmission Control Protocol/Internet Protocol (TCP/IP), as in [3,45,46]. Among the congestion control proposals that do not depend on specific network mechanisms, we highlight the scheme presented by Adas et al. [1,29], which uses the LMS adaptive algorithm [28] for network traffic prediction and control of traffic flow rates. In [8,42], another work in this direction, the authors use the RLS algorithm that has faster convergence, to predict the necessary traffic rates [11]. Adaptive control algorithms are the most appropriate for realtime multimedia applications due to on-line processing capability [40]. Taking this information into account, in this work we present a novel fuzzy model based on orthonormal basis functions (OBF) that adaptively creates fuzzy clusters as traffic data input are available. The OBF functions are obtained through an analytical expression for the autocorrelation function of multifractal flows. In order to adaptively adjust the fuzzy model parameters, we propose a slightly modification in the traditional fuzzy on-batch training. The proposed adaptive regressive fuzzy clustering (ARFC) algorithm enables that the OBF-fuzzy model efficiently forecast the network traffic rates. Further, we apply the OBF-fuzzy modeling in a traffic flow control scheme that maintains the queueing length in buffer at a desired level. The paper is organized as follows: in Section 2, we describe the Laguerre orthonormal basis functions. In Section 3, we introduce

F.H.T. Vieira, F.G.C. Rocha / Neurocomputing 74 (2011) 1894–1907

some multifractal concepts and we also present an analytic expression for the orthonormal basis functions pole. In Section 4, the corresponding OBF-fuzzy model is introduced as well as the adaptive training algorithm. In Section 5, the proposed OBF-fuzzy model is evaluated and compared to some adaptive predictors existing in the literature. In Section 6, we propose an OBF-fuzzy model based congestion control scheme is proposed and we derive an expression for the optimal flow control rate in function of the fuzzy model parameters. In Section 7, the proposed congestion control method is compared to others congestion control schemes and some performance measures are evaluated. Finally, we conclude in Section 8.

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the mapping between lj(k) and the system output y(k). Hence, in this way, a more accurate model than the model NonLinear Moving Average (NLMA) with the same number of functions is obtained [13,44]. Given a number of Laguerre functions n (order of the model), an appropriate value for the pole p results in a better representation of the system being modeled. We aim to introduce information about the multifractal behavior of network traffic processes in the calculation of the pole b(k). Thus, in the next section, we briefly introduce the multifractal behavior of stochastic processes.

3. Multifractal analysis 2. Orthonormal basis functions

3.1. Multifractal network traffic

In this section, we discuss some concepts involving orthonormal basis functions in order to further establish a relationship to fuzzy modeling. The Laguerre basis is used in many contexts of identification and control of nonlinear systems [12,22,32]. In this work, we adopt the Laguerre basis especially because it is completely parameterized by a single pole, the Laguerre pole. We also chose the Laguerre basis due to its relation to the state equation formulation and, as a consequence to fuzzy modeling. Notice that the Laguerre pole is related to the autocorrelation function of the multifractal traffic process. That is, the Laguerre pole can be given in function of the multifractal traffic parameters, allowing us to consider the multifractal properties of the traffic trace into the fuzzy modeling. The set of transfer functions associated with this basis is given by the following equation: qffiffiffiffiffiffiffiffiffiffiffiffi 1 1 q ðq pÞj1 Fmag,j ðq1 Þ ¼ 1p2 , j ¼ 1, . . ., n ð1Þ ð1pq1 Þj

A multifractal network traffic flow presents a structure of strong dependence among the samples with incidence of bursts at various scales [31,45]. These features can degrade network performance more than those of Gaussian traffic flows and short-range dependence [23,36]. Multifractal processes are defined by a scaling law for the statistical moments of the processes’ increments over finite time intervals. This means the network traffic has complex and strong dependence structures inherently, appearing very bursty and the burstiness looks similar over many scales [39]. Traffic flows with such properties make the network performance much worse than those of Gaussian and short-range dependent traffic types [36]. Now, let us formally define the concept of a multifractal process.

where pA{P: 1opo1} is the pole of Laguerre functions (Laguerre basis) and where q  j is the shift operator. It may be noticed that making p¼0 in (1), results in the basis Fmag,j ðq1 Þ ¼ qj . Therefore, the basis Fmag,j ðq1 Þ ¼ qj is a special case of the Laguerre basis. The output of an input–output model can be written as yðkÞ ¼ Hðl1 ðkÞ, . . ., ln ðkÞÞ

ð2Þ

Definition 1. A stochastic process X(t) is called multifractal if it satisfies q

Eð9XðtÞ9 Þ ¼ cðqÞt tðqÞ þ 1

ð7Þ

for tAT and qAQ, where T and Q are intervals on the real line and t(q) and c(q) are functions with domain Q, t(q) is the scaling factor and c(q) is the moment factor of the multifractal process. Furthermore, we assume that T and Q have positive lengths, and that 0AT, [0,1] DQ. If t(q) is linear in q, the process X(t) is called monofractal; otherwise, it is multifractal [39].

lðkþ 1Þ ¼ AlðkÞ þ buðkÞ

ð3Þ

From the multifractal model properties, an equation for the autocorrelation function of a multifractal process can be obtained. Let X(n) be a multifractal process with parameters a, r and g as described in [10]. The scaling function t0(q) and the moment factor c(q) can be, respectively, written as the following [10]:   GðaÞGð2a þqÞ ð8Þ t0 ðqÞ ¼ log2 Gð2aÞGða þqÞ

yðkÞ ¼ HðlðkÞÞ

ð4Þ

and

1

where lj(k)¼ Fmag,j(q )u(k) is the jth Laguerre function at time k, n is the number of basis functions, u(k) is the input signal and H is a nonlinear operator. Note that the nonlinear operation corresponding to H can be accomplished by fuzzy modeling. Laguerre functions lj(k) are recursive and can be obtained through a state equation formulation as follows [22,32,48]:

where l(k) ¼[l1(k)yln(k)]T. The matrix A and vector b depend on the order n of the model and the value of the pole p as follows [13]: 0 1 p 0 0 ... 0 B 2 1p p 0 ... 0C B C B C 2 2 C Þ 1p p . . . 0 ðpÞð1p A¼B B C B ^ ^ ^ & ^C @ A ðpÞn2 ð1p2 Þ ðpÞn3 ð1p2 Þ ðpÞn4 ð1p2 Þ . . . p

cðqÞ ¼ erq þ g

2 q2 =2

ð9Þ

where G(  ) is the Gamma function. The autocorrelation function of the process X(n) for the time instants n and k is given by the following equation [10,20,27]: ! N1   2r þ g2 aða þ1Þ log2 ðða þ 1Þ=ða þ ð1=2ÞÞÞ E XðnÞ,Xðn þkÞ ¼ e ð10Þ k ða þ 1=2ÞN where N ¼ log2 ðNa Þand Na is the number of process samples.

ð5Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffih b ¼ ð1p2 Þ 1

p

ðpÞ2

...

ðpÞn1

iT

ð6Þ

The nonlinear model represented by Eqs. (3) and (4) is a linear mapping between the input u(k) and Laguerre functions lj(k), plus

3.2. Pole model obtained from the autocorrelation function for multifractal processes In this section, we introduce an analytical expression for the pole p used in the calculation of the orthonormal basis functions.

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Proposition 1. The pole p used for obtaining the orthonormal basis functions for the OBF-Fuzzy model can be given by: 1 p ¼  log ðða þ 1Þ=ða þ ð1=2ÞÞÞ : ð11Þ 2 2

Proof. See Appendix A.

&

Note that we can adaptively estimate the pole p through the adaptive estimation of a parameter. If the traffic process {Xk} has scaling properties, then the absolute moments E(9X9q) versus m on a log–log plot should be a straight line as follows [39]: q

log Eð9X9 Þ ¼ t0 ðqÞlog m þ log cðqÞ

ð12Þ

The slope of this straight line provides an estimate of t0(q) and its interception point on the vertical axis is the numerical value of log c(q). Therefore, we propose the following algorithm to adaptively estimate the a parameter: Algorithm 1. Adaptive estimation of the a parameters 1) 2)

q Let p0 ¼1, a^ 0 ¼[0 0], q Compute a^ k ¼ ½ ^ 0 ðqÞ

k¼1, y, 2 , j¼ 1, 2, y, N and q ¼1, y, q2. t logc^ ðqÞ for each q value using the following recursive equations [18]: ð13Þ

q q q a^ k ¼ a^ k1 pk ½xk xTk a^ k1 x k yqk 

where yqk ¼ log Eð9X k 9Þq , h i 1 log 2 . . . log k .

ð14Þ 

X k ¼ X1

X2

...

Xk



and

xk ¼

3) Estimate the a parameter of t0 using the Levenberg–Marquardt algorithm according to the following updating rule [28]:

ai þ 1 ¼ ai ðHes þ ZdiagðHÞÞ1 rfðai Þ

Fuzzy models have been effectively used in systems modeling since they are universal approximators [41]. These models can be divided into three classes: language models (Mamdani’s Models) [47], Fuzzy relational models [47] and Takagi–Sugeno models [34,41]. In fuzzy modeling, the time series information is divided into ‘clusters’ (groups), where each cluster is described by a local model. The fuzzy model Takagi–Sugeno–Kang (TSK) is a particular process of combining local models AutoRegressive (AR) via fuzzy rules [41]. We can extend the notion of fuzzy interpolation of local models, which is the central idea of the TSK model [18,41], to the context of orthonormal basis functions. Our proposal is based on a state space version of the TSK model, i.e., each fuzzy model rule Ri represents a different state space model as the following: Ri :

If (

j

pq,k ¼ pq,k1 pq,k1 x k ½1 þ x Tk pk1 x k x Tk pq,k1

4. OBF-fuzzy model

Then

l1 ðkÞ

Fig. 1 shows the value of the pole d as a function of time for a real Internet traffic trace.

Li1 ,

...

and

! li ðk þ1Þ ¼ Ai li ðkÞ þ bi x ðkÞ yi ðkÞ ¼ Hi ðli ðkÞÞ

ln ðkÞ

is

Lin

ð16Þ

where the matrix Ai and vector bi depend on the pole pi(k) and Hi(li(k)) is the mapping that relates the output yi(k) of the local model i to its corresponding state of Laguerre functions (ortho ! normal basis) li ðkÞ ¼ l1 ðkÞ l2 ðkÞ . . . ln ðkÞ, and x ðkÞ ¼ x1 ðkÞ x2 i ðkÞ . . . xn ðkÞ the input vector and Lj the fuzzy membership function for the i rule associated with the jth premise variable. The premise variables are the Laguerre states of the resulting OBFfuzzy model. The OBF-fuzzy model output is given by PC yðkÞ ¼

i¼1

PC

yi ðkÞwi ðli ðkÞÞ

i¼1

ð15Þ

where Hes is the Hessian matrix (Hes ¼ r2f(ai)) of the quadratic error function f and Z is a control parameter at the ith iteration of the Levenberg–Marquardt algorithm. 4) Apply again the Levenberg–Marquardt algorithm to estimate the r and g parameters of the c(q) function.

is

wi ðli ðkÞÞ

ð17Þ

where C is the number of rules (local models) and the weights wi(li(k))of the i rule are given by wi ðli ðkÞÞ ¼

n Y

Lij ðlj ðkÞÞ

ð18Þ

j¼1

The OBF-fuzzy model can be represented by the Eqs. (3) and (4) and H given in accordance with Eqs. (17) and (18).

Fig. 1. Adaptive pole estimation for the internet traffic trace dec-pkt-3.

F.H.T. Vieira, F.G.C. Rocha / Neurocomputing 74 (2011) 1894–1907

4.1. Adaptive training algorithm for the OBF-fuzzy model

and

In this section, we propose an adaptive regressive fuzzy clustering (ARFC) algorithm to the OBF-fuzzy model. The proposed training algorithm ARFC consists of two parts: an adaptive training procedure based on a clustering method that divides the traffic data into several linear clusters by fuzzy interpolation, each one described by a Laguerre OBF local model. In the second part, a learning algorithm based on the descent gradient is used to refine the obtained model and to improve the modeling accuracy. In the ARFC algorithm, we take into account the spatial distribution of the data considering the regression error and the distance between the input data and the clusters. Let the cost function of the algorithm ARFC be defined as:

Pðk þ1Þ ¼ PðkÞ

C X N X

J¼

u2ik ðrik dik Þ2

ð19Þ

i¼1k¼1

Subject to C X

uik ¼ 1,

for

1 r kr N

ð20Þ

i¼1

where uik is the ‘firing strength’ of the ith rule for the kth training pattern, C is the number of fuzzy rules and N is the total number of training data. In Eq. (19), rik is the error between the kth desired output y(k) of the system modeled and the output of the ith rule with the kth entry, i.e., !i

! rik ¼ yðkÞfi ð x ðkÞ; a ðkÞÞ

ð21Þ

where i¼1, 2, y, C and k¼1, 2, y, N. Also in Eq. (19), dik is the ! distance between the input vector x ðkÞ in discrete time k and the center of ith cluster bi, i.e., ! dik ¼ x ðkÞbi

ð22Þ

In order to minimize the cost function J in (19), the Lagrange multiplier method is applied [4] resulting in a Langrange function given by ! C X N N C X X X 2 2 L¼ uik ðrik dik Þ  lk uik 1 ð23Þ i¼1k¼1

k¼1

i¼1

where lk are Lagrange multipliers and k¼1, 2, y, N. The minimization of the cost function J implies in the imposition of the following conditions: N 2 X @rik @L ðuik Þ2 ðdik Þ2 ¼ ¼0 !i !i k¼1 @ a ðkÞ @ a ðkÞ

ð24Þ

@L ¼ 2uik ðrik dik Þ2 lk ¼ 0 @uik

ð25Þ

N X @d2 @L ¼ ðuik Þ2 ðrik Þ2 ik ¼ 0 @bi @bi k¼1

ð26Þ

Let XAPN  (n þ 1) be a matrix where its elements are the values ! x ðkÞ in its jth column j ¼1, y, nþ 1 (the first column X is all composed by 1), YAPN a vector where the kth element is the value of y(k) and QiARN  N a diagonal matrix where the kth diagonal is given by the term qðkÞ ¼ u2ik d2ik . We propose a recursive algorithm for the model parameter updating represented by the following equations (Proof in Appendix B): !i !i a ðk þ 1Þ ¼ a ðkÞ þ ðPi ðk þ1Þxðk þ 1Þqðk þ 1ÞÞ !i ðyðkþ 1Þxðk þ1ÞT a ðkÞÞ

ð27Þ

1897

qðk þ1ÞPi ðkÞxðk þ1Þxðk þ 1ÞT Pi ðkÞ 1 þqðk þ1Þxðk þ 1ÞT Pi ðkÞxðk þ1Þ

ð28Þ

where x(kþ1) is the (kþ1)th row of the matrix X and q(kþ1) is the (kþ1)th element of the matrix diagonal Qi(kþ1). In order to minimize the cost function (19), the second condition (Eq. (25)) necessary to minimize the cost function J must be satisfied. Thus, the firing strength ui,k of the ith rule must be given by: 2 2 1=2ðrik dik Þ uik ¼ PC 2 d2 Þ 1=2ðr i¼1 ik ik

ð29Þ

The third condition, which is represented by Eq. (26), can be simplified since the partial derivate @dik =@bi in this equation can be evaluate through the distance between the kth desired output and the center of the ith rule. By using Eq. (22), we have the following: @d2ik @d ! ! ¼ 2dik ik ¼ 2ð x ðkÞbi Þð1Þ ¼ 2bi 2 x ðkÞ @bi @bi

ð30Þ

Substituting (30) into (26), we obtain the following equation for the center bi of the ith cluster: PN 2 2! ¼ 1 riz uiz x ðzÞ bi ¼ zP ð31Þ N 2 2 z ¼ 1 riz uiz In this first part of the ARFC algorithm, the consequent parameters are also obtained. We adopted Gaussian functions as membership functions of the antecedent parts, that is 8 !2 9 i < = x ðkÞ y j j1 i i Aij ðyj1 ; yj2 Þ ¼ exp  ð32Þ i : ; y j2

For the calculation of the membership functions, we must i i obtain the parameters yj1 and yj2 , which correspond, respectively, to the mean and the standard deviation of the jth membership function of the ith fuzzy rule, where 1 rjrn and 1 rirC. Thus, we have the following equations for the calculation of the i i parameters yj1 and yj2 : PN 2 z ¼ 1 ðuiz Þ xj ðzÞ yij1 ðkÞ ¼ P ð33Þ N 2 z ¼ 1 ðuiz Þ vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi uPN i 2 2 u z ¼ 1 ðuiz Þ ðxj ðzÞyj1 Þ PN 2 z ¼ 1 ðuiz Þ

yij2 ðkÞ ¼ t

ð34Þ

As an example of the algorithm operation, Fig. 2 shows the clusters created by the OBF-fuzzy model with 2 rules and 1 input element for a real network traffic data. In this case, we considered as the desired target output for the model the next step ahead sample of the traffic trace. 4.2. Fine tuning algorithm In the second part of the ARFC clustering algorithm, the con!i i i sequent (yj1 and yj2 ) and premise ( a ðkÞ) parts are adjusted by an adaptive fine tuning algorithm based on a supervised learning procedure to improve the modeling accuracy. The supervised learning algorithm used in this work is the descent gradient [19], whose cost function ER is given by ER ¼

N 1X e2 ðkÞ Nj¼1

ð35Þ

where eðkÞ ¼ yðkÞy^ ðkÞ, y(k) is the desired output and y^ ðkÞ is the output of the OBF-fuzzy model.

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Fig. 2. Clusters (2 rules and 1 input): Bc-Octext traffic trace.

From the minimization of the cost function ER (Eq. (35)), it is possible to show that the parameters of the premise parts of the i fuzzy model yjl with l¼1 and 2, can be precisely adjusted through the following equation:

Dyijl ðkÞ ¼ ZðyðkÞy^ ðkÞÞðyi ðkÞy^ ðkÞÞ PC

1

@wi ðkÞ

ð36Þ

i i i ¼ 1 w ðkÞ @yjl ðkÞ

where Z is the learning constant, y(k) is the desired output, y^ ðkÞ is the output of the OBF-fuzzy model and yi(k) is the ith rule output of the OBF-fuzzy model. Let wi(k) be the parameter vector given by min Aij ðxj ðkÞÞ

j ¼ 1,2, ..., n

and j* be the index j when the minimization in wi(k) occurs, i.e.: j ¼ arg min Aij ðxj ðkÞÞ j j ¼ 1,2, ..., n

5. Performance evaluation of the OBF-fuzzy prediction model In the simulations we used real Transmission Control Protocol/ Internet Protocol (TCP/IP) traffic traces, obtained from Digital Equipment Corporation (DEC), Ethernet traffic traces obtained from Bellcore and traces captured between the years 2000 and 2002 on the Petrobras’s network through a data analyzer of ActernaTM model DA350, with resolution of 32 microseconds [5,26]. In this section, we present a comparative performance evaluation of the proposed predictor and three other different predictors, when applied to traces of TCP/IP and Ethernet traffic. The other predictors taken into account were: the Least Mean Square (LMS) [11], the Recursive Least Square (RLS) [11,20] and the TSK Fuzzy model trained with Fuzzy Clustering Regression Model (FCRM) [35,47]. We evaluate the proposed predictor using two error measures, known as normalized mean square errors (NMSE) [6,34].

i

Then, when j¼j* and l ¼1, the term @wi ðkÞ=@yjl ðkÞ of Eq. (36) can be evaluated as the following: @wi ðkÞ i @yj1 ðkÞ

@wi ðkÞ i

@yj1 ðkÞ

¼

¼

@Aij

mean square error of type 1 is the defined as: h i E ðx^ xÞ2 MSE i NMSE1 ¼ 2 ¼ h sx E ðmxÞ2

i

@yj1 ðkÞ i

2

xj ðkÞyj1 ðkÞ

yij2 ðkÞ

yij2 ðkÞ

8 !2 9 i < xj ðkÞyj1 ðkÞ = exp  : ; yij2 ðkÞ

@wi ðkÞ i

@yj2 ðkÞ

¼

¼

i xj ðkÞyj1 ðkÞ @wi ðkÞ yij2 ðkÞ @yij1 ðkÞ

For j aj*, @wi ðkÞ i @yj2 ðkÞ

¼

@wi ðkÞ i

@yj1 ðkÞ

ð38Þ

Definition 3. Let x^ pa be the predicted value of the sample process X, whose value is the same of the sample immediately before the procedure. The normalized mean square error of type 2 is defined as: h i E ðx^ xÞ2 i NMSE2 ¼ h ð39Þ E ðx^ pa xÞ2

And for l ¼2, we have @Aij i @yj2 ðkÞ

Definition 2. Let s2x be the variance of the process X, given by s2x ¼ E½ðmxÞ2  where m is the mean of process X, the normalized

¼0

The parameters of the consequent parts of the OBF-fuzzy model can be adjusted applying the gradient descent algorithm, resulting in the following update equation: wi ðkÞxj ðkÞ

Daij ðkÞ ¼ zðyðkÞy^ ðkÞÞ PC

i¼1

wi ðkÞ

ð37Þ

where z is another learning constant, y(k) is the desired output and y^ ðkÞ is the output of the OBF-fuzzy model.

According to the above definitions, a predictor which has an NMSE1 value equal to or less than 1 will have performance equal to or better than a predictor which estimate the future value as being equal to the process mean. For a NMSE2 close to 1, we can say that the predictor will present a performance close to that of a predictor which estimates the future value as being equal to the sample immediately preceding it. Before starting the comparisons to other predictors, we present some results of the proposed OBF-fuzzy predictor for the Bellcore traffic trace Bc-octint. Fig. 3 compares the predicted values and

F.H.T. Vieira, F.G.C. Rocha / Neurocomputing 74 (2011) 1894–1907

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Fig. 3. QQ-plot for the Bc-Octint traffic trace using the OBF-fuzzy predictor.

Fig. 4. Prediction for Bc-Octint traffic trace using the OBF-fuzzy predictor.

actual values by a quantile-quantile plot known as QQ-plot [34]. It can be noticed an almost linear relationship between the predicted values and real values, which denotes adequate performance prediction. In fact, one can observe that the one-step ahead predicted values are close to the real ones as shown by Fig. 4. One goal of adaptive training is to adjust the algorithm to the dynamic environment of network traffic. Tables 1 and 2 present the EQMN evaluation for different traffic traces and number of rules. In order to compare the adaptive training to on-batch training type, we also inserted in Table 1 the NMSE 1 values obtained with the TSK fuzzy predictor trained with the nonadaptive Fuzzy Clustering Regression Model (FCRM) algorithm [43,47]. The FRCM fuzzy predictor uses all the samples from the traffic series in its training. Once these parameters are determined, the fixed parameter model is applied to predict the next step (one-step ahead prediction) of the considered time series. It can be easily observed from Tables 1 and 2 that we obtained NMSE2 generally lower for the considered traffic traces using the OBF-fuzzy predictor. Therefore, the results show that one can obtain with the knowledge of few past samples (in this test, we used 2 samples), an error as small as that obtained by processing with all samples of the traffic trace. When comparing the OBFfuzzy predictor to a version without the use of orthonormal basis

Table 1 Comparisons using NMSE1. Traffic trace

Interval

Adapt. OBF-fuzzy

Adapt. fuzzy

RLS

LMS

Fuzzy FCRM

Dec-pkt-1 Dec-pkt-2 Bc-Octint BC-Octext

1–2048 1–2048 801–1701 1000–2000

0.6564 0.5758 0.4102 0.4144

0.7366 0.6704 0.4654 0.4355

0.8513 0.7022 0.4114 0.4298

0,9304 0,7614 0,4817 0,5010

0,6987 0,5836 0,3107 0,4654

Table 2 NMSE2: Number of rules ¼number of coefficients. No of rules

Adapt OBF-fuzzy

Adapt fuzzy

RLS

LMS

2 3 4 5

0.6121 0.5762 0.5633 0.5846

0.7208 0.7114 0.7043 0.7282

0.9553 0.8820 0.8739 0.8731

1,0779 0,9779 0,9531 0,9378

functions (denoted as Fuzzy Adapt. in Table 1), it can be observed a lower prediction error for the proposed predictor. In theory, as we increase the number of orthonormal basis functions for the OBF-fuzzy predictor, we could note a lower

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F.H.T. Vieira, F.G.C. Rocha / Neurocomputing 74 (2011) 1894–1907

Fig. 5. p-value. dec-pkt-2 traffic trace.

NMSE for a certain traffic trace. However, we observed for all tested predictors that the NMSE1 and NMSE2 decrease to a certain number of coefficients. After that point, it is not always possible to obtain lower NMSEs. Table 2 corroborates this statement, in which aiming to simplify the analysis, we use a number of rules equal to the number of coefficients (equal to the number of past samples) of the models to predict the traffic trace dec-pkt-1. The OBF-fuzzy predictor started to show a deterioration of NMSE1 and NMSE2 with 5 rules. For RLS and LMS algorithms, the same happens with a number of coefficients equal to 7. However, note that even these two algorithms being in its best configuration, they did not provide lower NMSEs than the OBF-fuzzy model with 2 rules and 2 coefficients. It is also important to evaluate the performance of the OBFfuzzy predictor for larger prediction horizons. To this end, we analyzed the prediction errors of the OBF-fuzzy model with the T-test [38]. The T-test is a hypothesis test that can be used to determine whether a statement about a time series characteristic is true. This test provides the probability of the confidence degree of this statement through the variable known as p-value [38]. The p-value corresponds to the probability of observing a certain result since the null hypothesis is true. The null hypothesis H0 is that the average error is zero (m ¼0) and the alternative hypothesis H1 is that the error be different of zero (m a0). The significance level of a statistical test is the probability of rejecting a true hypothesis. We set the significance level of the performed test at 0.05, which corresponds to a confidence interval of 0.95. Fig. 5 shows the p-values of the T-test applied to errors sequences for different prediction steps for the TCP-IP trace dec-pkt-2. In this case, we fixed the number of coefficients of the local AR models as 2 to the proposed fuzzy model (with 2 fuzzy rules), as well as to the coefficients of the RLS predictor and the adaptive fuzzy model without OBF. In all considered prediction steps h¼0 were obtained for the OBF-fuzzy model. That is, one should not reject the null hypothesis considered with a significance level of 0.05. Table 3 summarizes the p-values obtained in on step-ahead prediction for different traffic traces and adaptive predictors. Note that the higher p-values for the OBF-fuzzy predictor indicate higher security degrees to grant that the mean prediction errors are zero by using this predictor Fig. 6 compares the prediction performance measures (NMSE1 and NMSE2) among the proposed predictor and the LMS and RLS predictors. We verify that the proposed predictor obtain lower values for NMSE1 and NMSE2, i.e., the best prediction performance between

Table 3 p-values. Traffic trace

Adapt OBF-fuzzy

Adapt fuzzy

RLS

LMS

Dec-pkt-1 Dec-pkt-2 Bc-Octint Bc-Octext lbl-pkt-5 10-7-S-1

0.5204 0.9109 0.8103 0.4771 0.9071 0.0241

0.2681 0.2113 0.7926 0.0256 0.8713 0.0193

0.0081 0.0274 0.0362 0.0066 0.1089 0.0211

5,0752.10  5 1,0800.10  4 3,0357.10  5 3,0800.10  6 1,8130.10  5 2,5657.10  9

the considered predictors. With the use of few initial samples, the mean square error of the proposed predictor decays and keeps next to zero. In the same figure, one can note that the others predictors require a larger time interval to reach similar results. Thus, we can conclude that the application of adaptive OBF-fuzzy model for traffic prediction obtain expressive performance gains when compared to other predictors as well as faster convergence. In Figs. 7 and 8, we show the effect of the fine tuning algorithm in the prediction error. In fact, the NMSE1 and NMSE2 are clearly decreased by applying the fine tuning algorithm. Therefore, the fine tuning algorithm can be seen as an important part of the adaptive fuzzy model training.

6. Adaptive OBF-fuzzy based congestion control In this section, we present the considered congestion control system where the proposed adaptive fuzzy algorithm is applied. In this control approach, we aim to adaptively predict the queueing behavior in the buffer and, by using the OBF-fuzzy model parameters to control the source rate so that the buffer queue length is maintained at a desired reference level. The basic structure of the considered congestion control system is shown in Fig. 9, in which is identified all the elements of the control system:

      

Controllable traffic flow m(k). Uncontrollable traffic flows n(k). Buffer. Feedback delay (round-trip time) d. Buffer queue length b(k) at instant k. Output link capacity Z. Desired level for the buffer queue length bt.

F.H.T. Vieira, F.G.C. Rocha / Neurocomputing 74 (2011) 1894–1907

Fig. 6. Comparison of errors between the proposed predictor and the other predictors. Trace dec-pkt-2.

Fig. 7. NMSE1—comparison of the error measures before and after of the fine tuning algorithm for traffic trace dec-pkt-3.

Fig. 8. NMSE2—comparison of the error measures before and after of the fine tuning algorithm for traffic trace dec-pkt-3.

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F.H.T. Vieira, F.G.C. Rocha / Neurocomputing 74 (2011) 1894–1907

6.1. Optimal flow control rate In this section, as part of the proposed adaptive congestion control scheme, we present an expression for the calculation of the optimal control rate to regulate the controllable source rate m(k) from the minimization of cost function J below   1 l Jðk þdÞ ¼ E ðbðk þdÞbt Þ2 þ m2 ðkÞ ð41Þ 2 2

Fig. 9. Congestion control system model.

 Functional block that represents the proposed adaptive fuzzy congestion prediction and control, which regulates the transmission rate of the controllable traffic. The sampling time interval of the control system is T seconds, i.e., the traffic rates and the buffer queue length b(k) are updated every T seconds. The buffer occupation of the system can be described through the Lindley’s equation [5] bðkþ 1Þ ¼ minfmax½bðkÞ þ ðmðkdÞ þ nðkÞZÞ  T,0,Bmax g

ð40Þ

where Bmax is the buffer maximum capacity. The input traffic arrived at the multiplexer buffer belongs to two distinct traffic groups. One consists of the controllable traffic m(k) which can adjust its rate according to the network condition status, called Available Bit Rate (ABR) traffic. The transmition rate of the ABR traffic is regulated by the proposed adaptive OBF-fuzzy controller implemented in one switch or network node. The other group consists of uncontrollable traffic v(k), which correpond to services of constant bit rate (CBR) and variable bit rate (VBR) types. This group is delay-sensitive and has a higher priority than controllable type. Therefore, the controllable sources can only share the remaining link bandwidth left by the higher priority sources. The variable bit rate (VBR) and constant (CBR) services are apropriated for a great number of aplications [7]. The VBR service can be used to transmit real-time or nonreal-time video and audio data with variable bandwidth. The ABR service is intended for a class of applications that can adapt to time-varying bandwidth and tolerate unpredictable end-to-end delays. Since the ABR source must adjust its tansmition rate from time to time according to the available bandwidth, the network performance will be affected strongly by the efficiency of the rate mechanism adjustment. In order to attain an adequate efficiency level, the source should be informed about the latest network condition via a feedback channel with delay. This feedback delay together with the forward transmission delay is called the roundtrip delay (it is represented by delay g in Fig. 9). The proposed congestion control procedure takes into account the round-trip delay d by predicting the behavior of the buffer occupation in order to avoid the congestion occurrence. Thus, we ^ þdÞ of the buffer length b(kþd) apply a d-step ahead predictor bðk based on the current and past information of the controllable source rate m(s)s r k and the buffer queue length b(s)s r k. In order to keep the d-step ahead predictive buffer queue length around the desired level bt and minimize the variance of the buffer occupation level, the proposed optimal control rate mo(k) is used to regulate the controllable source rate, that will be presented in the next section. In this way, it is possible to confine QoS parameters such as byte loss rate and byte delay within a desired range that can be established in the network traffic contract.

where l is a weighting factor and E[  ] denotes the mathematics expectation. This cost function J was also used in [5] for ARMAX models. In Eq. (41), the cost function J takes into account the adjustment error of buffer queue length, i.e., considers the difference between the d-step ahead of the buffer queue length value and the desired level of buffer queue length bt. Even though the modeling errors are small, they can cause oscillations in buffer queue length b(k) of such form to exceed the reference level bt. Due to this fact, we introduce the project parameter l in Eq. (41) to limit the value of m2(k), avoiding situation as this. The optimal control rate is given in function of d-step ahead prediction of the buffer queue length value. An estimate of this value is obtained as the output of the proposed fuzzy predictor when applied to the prediction of samples of this process by the following equation: ^ þ dÞ ¼ bðk

C X

bi ðk þ dÞhi ðkÞ

ð42Þ

i¼1

where hi ðkÞ ¼ PC

wiðkÞ

i¼1

wi ðkÞ

!i wi ðkÞ ¼ min Aij ð y j ; xj ðkÞÞ

where

j ¼ 1 ... n

ð43Þ

and bi ðk þdÞ ¼ ai0 þ ai1 bðkpÞ þ ai2 bðk þp þ1Þ þ    þ aip þ 1 bðkÞ i

i

i

þ bp þ 2 mðkqÞ þ bp þ 3 mðkp þ 1Þ þ    þ bp þ q þ 2 mðkÞ ð44Þ Thus, we can enunciate the following proposal for the optimal control rate. Proposition 2. The optimal control rate mo that minimizes the cost function J, which is given by Eq. (41) used to regulate the controllable source rate, is: ! C

X o t i i m ðkÞ ¼ b0 ðkÞ  b  a0 þ bp ðkÞ þ bq ðkÞ h ðkÞ ð45Þ i¼1

where PC

b0 ðkÞ ¼ P C

i¼1

i¼1

bp ðkÞ ¼

pX þ1

bip þ q þ 2 hi ðkÞ

bip þ q þ 2 hi ðkÞ

2

þl

ait bðkp1 þ tÞ

t¼1

bq ðkÞ ¼

q X

bit þ p þ 1 mðkq1 þ tÞ

t¼1

Proof. The proof is in Appendix C.

&

According to the proposed Eq. (45), the optimal flow control ! rate mo(k) is obtained in terms of the input vector x ðkÞ, the !i consequent parameters vector a ðkÞ and the desired level of the buffer queue length bt. That is, in the calculation of the optimal rate, it is considered the parameters of the predictive OBF-fuzzy model and the desired level of the buffer queue length.

F.H.T. Vieira, F.G.C. Rocha / Neurocomputing 74 (2011) 1894–1907

Therefore, the proposed scheme can be applied in a great variety of network types.

7. Evaluation of the adaptive OBF-fuzzy congestion control scheme The OBF-fuzzy based congestion control scheme is constituted of a module of adaptive traffic prediction represented by the proposed fuzzy prediction algorithm and a module that calculates the optimal flows rate. The parameters obtained in the prediction of the buffer queue length are used to calculate the source rate so that buffer queue length is maintained below to the desired level and the minimization of buffer length variance occurs. In this section, we evaluate and compare the OBF-fuzzy based congestion control scheme to other congestion control methods. Initially, we present the other considered control methods and the considered perfomance measures. In order to validate the proposed control scheme, we consider a network node where its traffic input is composed by 2 different traffic sources using the traces dec-pkt-2 and dec-pkt-3 in aggregation time scale of 512 ms. 7.1. Congestion control methods We considered the following methods in order to compare the performance of the proposed congestion control scheme: Binary Feedback method and the Proportional Congestion Control method [7,17]. Such methods have similar chacteristics to the proposed method, i.e., these methods analyze the behavior of the buffer queueing length and estimate a new rate for the controllable source from particular mechanisms established in each method and independent of networks specific protocols.

 Binary Feedback



In the Binary Feedback, the traffic source rate is controlled by monitoring the buffer queueing length [17]. Two thresholds are set: Tl and Th. When the buffer queue length exceeds Th, congestion is detected and the source rate is reduced by a factor of 0.98. When the buffer queue length is below Tl, there is no more congestion and the source rate is increased by a factor corresponding to 1% of the link capacity Z. Otherwise, the source rate is not changed. In the simulation, the two thresholds Tl and Th are set to be 0.95bt and 1.05bt, respectively, where bt is the desired level for the buffer queue length. Inicial source rate is set to be equal to the link capacity Z minus the average of the uncontrollable traffic rate. Proportional Congestion Control The Proportional Congestion Control method monitors the buffer queue length and uses a control variable c(k) to regulate the controllable source rate [7]. Let bt be the reference level of buffer queue length control in one link. The control signal of the traffic rate c(k) is generated through of the following equations: 8 if bðkÞ o 0:998b > < 1:002 t t t if 0:998b o bðkÞ o1:002b cðkÞ ¼ 1:002 bðkÞ0:998b t b > t : if bðkÞ o 1:002b 0:998 where 0.998bt can be seen as the lower threshold and 1.002bt as the upper threshold. Let us denote m(k) as the controllable source rate at instante k, then the next source rate is given by m(kþ1) ¼c(k)m(k). Other congestion control methods that are specific for some network tecnologies can be founded in literature such as the Random Early Detection (RED) [15]. RED is based on the congestion control mechanisms of the TCP/IP protocol. On the other hand, in order to implement the proposed flow control scheme, it is only necessary that the network has feedback mechanisms to control the controllable sources rate.

1903

7.2. Performance of the proposed congestion control scheme We considered the dec-pkt-2 and dec-pkt-3 TCP/IP traffic traces under N timescale aggregation to represent the input traffic flows once it was verified that they present multifractal characteristics [11,26]. The specifications of the components shown in Fig. 9 are described below:

   

Uncontrollable traffic dec-pkt-2 (samples 501-2548). Controllable traffic dec-pkt-3 (samples 501-2548). Link capacity: rik bytes/seg. Sampling time interval (k).

We evaluate the performance of the proposed congestion control scheme based on the following performance measures: byte loss rate (BLR), output link utilization and throughput of the controllable rate. The link utilization is the proportion of the output link capacity that is used in the transmition of traffic flows. It is given by following equation: Utilizationð%Þ ¼

Transmitted Bytes 100% Link Capacity

ð46Þ

The throughput is defined as the amount of data (bytes) arriving that are successfully transmited in a certain time period. Throughput ðbytes=segÞ ¼

Transmitted Bytes Total Time of Simulation

ð47Þ

For the simulations of the proposed congestion control scheme, we set a configuration for the considered network in order to cause losses without applying traffic control. The buffer length is fixed in Bmax ¼ 1.5  106 bytes and the desired level for the buffer queue length is chosen as bt ¼30%Bmax, i.e., bt ¼ 4:5  105 bytes. The parameter l of the calculation of the optimal flow control rate is fixed in dik and the control algorithm is implemented. Notice that we chose the value of the parameter l such that the best performance of the control scheme is provided. Fig. 10 presents a comparison of controllable source rate before and after the application of the proposed congestion control. By analyzing the Fig. 10, it is possible to observe that the controllable rate was reduced and some times led to a null value. As mentioned early, the controllable source uses the bandwidth that is available, i.e., the rate of such source is equal to the bandwidth that is not being used by priority sources. We compare the performance of the proposed congestion control scheme to that of the Binary Feedback and the Proportional Congestion Control methods. Fig. 11 shows the buffer queue length for the Binary Feedback method, the Proportional Congestion Control method and the proposed method from top to bottom, respectively. It can be observed that the proposed method obtain a buffer queue process smoother than the other considered methods and below the desired level (Table 4). We verified that the mean and variance of the buffer queue length significantly decreased when applying the OBF fuzzy based congestion control. Also, The BLR is reduced to zero, offering better performance in terms of loss rate and of buffer queue behavior. However, the link utilization, after the application of the OBF fuzzy based congestion control, is reduced once the control scheme limits the controllable source rate. The simulation results confirm that the proposed congestion control method obtain better results in terms of data loss and buffer queueing behavior, maintaining the queue length below the desired value Bmax.

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Fig. 10. Controllable source rate before and after of application of the OBF-fuzzy based congestion control.

Fig. 11. Comparison of the buffer queue length obtained with the proposed congestion control and other methods.

Table 4 Comparison of performance results between the congestion control methods in the network node. Parameter

Real trace (without control)

Binary feedback

Proportional Cong. control

OBF-Fuzzy Cong. Control

Buffer mean Buffer variance BLR Link utilization Controllable throughput

1109.106 3573.1011 8,9854% 99,9044% 9975.104

4022.105 4176.1011 10,6477% 85,8284% 5567.104

6916.105 4916.1011 10,7568% 100,000% 8748.104

1452.103 7761.107 0,0000% 87,6575% 4388.103

8. Conclusion The characteristics of traffic flows in today’s networks as longrange dependence and bursts at multiple scales make modeling and traffic prediction difficult and challenging tasks. In this paper, we proposed the OBF-fuzzy model whose adaptive training algorithm allows the prediction of network traffic samples with a small number of fuzzy rules.

In order to obtain orthonormal basis functions for the fuzzy model by calculating the pole of the system, we derived an analytical expression for the autocorrelation function of multifractal processes [10,20]. Next, we introduced a procedure to calculate the dominant pole, i.e., the Laguerre pole for the OBF-fuzzy model. We observed an improvement of prediction performance for the fuzzy modeling, i.e., predictions become more accurate and robust with the addition of orthonormal basis functions to the adaptive fuzzy

F.H.T. Vieira, F.G.C. Rocha / Neurocomputing 74 (2011) 1894–1907

modeling. Comparisons to other adaptive predictors showed that in general better prediction performance was obtained for the adaptive OBF-fuzzy model. The proposed congestion control strategy consists of applying the OBF-fuzzy model to predict the buffer queue length and to control the controllable source rate such that the queue length variance is minimum possible. We verified that the application of the proposed control scheme guarantees that the buffer queue process does not exceed the stipulated value bt. Thus, it is possible to obtain a maximum delay and null loss rate for the traffic flows, i.e., some parameters of quality of service are guaranteed for the traffic flows. Some applications in nowadays networks can be benefited from these results, as video traffic, Voice over IP, etc. Therefore, we can conclude that the adaptive predictive control algorithm based on the OBF-fuzzy model is a promising tool for reliable adaptive control of trafic flows in real communication networks.

Appendix A Let X(t) be a process whose autocorrelation function is represented by rX(k)¼E[X(tþk)X(t)] and Gjþ 1(j¼1, y, n) is the reflection coefficient used to find an autoregressive (AR) model of order n [11,20]. The coefficients an(j) of the AR model can be calculated from rX(k) by the following equation [11,20]: rx ðkÞ þ

p X

ap ðlÞrx ðklÞ ¼ 0;

k ¼ 1,2, . . ., p

Pi ðk þ1Þ ¼ ðX T ðkÞQi ðkÞXðkÞ þ xT ðkþ 1Þoðk þ 1Þxðkþ 1ÞÞ1 T

Substituting X ðkÞQi ðkÞXðkÞ ¼ Pi ðk þ1Þ ¼

ðPi1 ðkÞ þ xT ðk þ 1Þ T

Pi1 ðkÞ

ð57Þ

in Eq. (57) we have

oðk þ1Þxðk þ 1ÞÞ1

ð58Þ

ðkÞQi ðkÞXðkÞ ¼ Pi1 ðkÞ

Finally, substituting X in (58), the vector of consequent parameters can be calculated recursively by the following equation:   !i !i !i a ðk þ 1Þ ¼ a ðkÞ þ Pi ðk þ 1ÞxT ðk þ 1Þoðk þ 1Þ yðk þ 1Þxðk þ 1Þ a ðkÞ ð59Þ !i The vector a ðk þ 1Þ is calculated using Pi(kþ1), which is given by Eq. (58). However, the matrix inversion of Pi(k) adds a very high computacional cost. To solve this problem, we apply the matrix inversion theorem [19]. Thus, the Eq. (58) corresponding to the calculation of the covariance matrix Pi(kþ1) can be rewriten as   P ðkÞxT ðk þ1Þoðkþ 1Þxðk þ1ÞPi ðkÞ Pi ðk þ1Þ ¼ Pi ðkÞ i 1 þ oðk þ1Þxðk þ 1ÞPi ðkÞxT ðk þ1Þ

ð60Þ

where x(kþ1) is the (kþ1)th line of matrix X(k) and w(kþ1) is the (kþ1)th element of diagonal matrix Qi(kþ1).

Thus, the pole of a model with (j ¼1) is given by [11,20]

gj ej

ð49Þ

where ej is the modeling error. Inserting the autocorrelation Eq. (10) in (49), and knowing that p ¼ G1 we have p¼

As Pi ðkÞ ¼ ðX T ðkÞQi ðkÞXðkÞÞ1 , then the covariance matrix Pi(kþ 1) is given by 0 # !T " !11 XðkÞ Qi ðkÞ 0 XðkÞ @ A ð56Þ Pi ðk þ1Þ ¼ oðkþ 1Þ xðk þ 1Þ xðk þ1Þ 0

ð48Þ

l¼1

Gj þ 1 ¼ 

1905

1

ð50Þ

2log2 ða þ 1Þ=ða þ ð1=2ÞÞ

Appendix C. Optimal flow control rate Once the estimate of the buffer queue length given by the proposed fuzzy predictor is equivalent to the expected value for ^ þdÞ, we can the d-step ahead buffer queue length b(kþd), i.e., bðk write the cost function (41) as the following: i 1h ^ Jðk þdÞ ¼ ðbðk þdÞbt Þ2 þ lm2 ðkÞ ð61Þ 2

Appendix B. Calculation of the vector of consequent parameters -a i

The optimal traffic control rate can be obtained through of minimization of the cost function J(kþd). For this, we derive the Eq. (61) with respect to m(k) and obtain the following equation:

!i The parameter vector a of the consequent parts of ith rule for the kth training pattern is given by

h i ^ ^ þdÞbt @bðk þ dÞ þ lmðkÞ ¼ 0 bðk @mðkÞ

!i a ðkÞ ¼ Pi ðkÞX T ðkÞQi ðkÞYðkÞ

^ þdÞ=@mðkÞ in (62) corresponds to the derivative The term @bðk of Eq. (42) in relation to m(k) resulting in

ð51Þ !i

Thus, the vector of the consequent parameters a (kþ1)th training pattern can be given by

for the

!i a ðk þ 1Þ ¼ Pi ðk þ 1ÞX T ðk þ 1ÞQi ðk þ 1ÞYðk þ1Þ ð52Þ !i Notice that the vector a ðk þ1Þ can be rewritten as # !T " ! XðkÞ Qi ðkÞ 0 YðkÞ !i a ðk þ 1Þ ¼ Pi ðk þ 1Þ oðk þ 1Þ yðkþ 1Þ xðk þ1Þ 0 ð53Þ or !i a ðk þ 1Þ ¼ Pi ðk þ 1Þ½X T ðkÞQi ðkÞYðkÞ þ xT ðk þ 1Þoðk þ1Þyðkþ 1Þ T

ð54Þ

C ^ þdÞ X @bðk ¼ bip þ q þ 2 hi ðkÞ @mðkÞ i¼1

ð62Þ

ð63Þ

Substituting the Eqs. (63) and (42) into Eq. (62), we obtain "

C X i¼1

!

#

l mðkÞ ¼ 0 i i b i ¼ 1 p þ q þ 2 h ðkÞ

bi ðkþ dÞhi ðkÞ bt þ PC

ð64Þ

Note that bi(kþd) is given by Eq. (44). Isolating the term mðkÞ, the Eq. (44) can be rewritten as

bip þ q þ 2

ðkÞQi ðkÞYðkÞ ¼ Pi1 ðkÞ

It is possible to verify in Eq. (51) that X !i a ðkÞ. Then, substituting this result in (54), we have that

bi ðk þdÞ ¼ ai0 þ

pX þ1

ait bðkp1 þtÞ

t¼1

!i !i !i a ðk þ 1Þ ¼ a ðkÞ þ ðPi ðk þ1ÞPi1 ðkÞIÞ a ðkÞ þ Pi ðkþ 1ÞxT ðkþ 1Þoðk þ 1Þyðk þ1Þ

þ ð55Þ

q X t¼1

bit þ p þ 1 mðkq1 þtÞ þ bip þ q þ 2 mðkÞ

ð65Þ

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F.H.T. Vieira, F.G.C. Rocha / Neurocomputing 74 (2011) 1894–1907

Substituting (65) in (64), we have "

! #

ai0 þ bp ðkÞ þ bq ðkÞ þ bip þ q þ 2 mðkÞ hi ðkÞ bt

C X i¼1

l

þ PC

i i i ¼ 1 bp þ q þ 2 h ðkÞ

C X

bip þ q þ 2 mðkÞhi ðkÞ þ

i¼1

mðkÞ ¼ 0

C X

ð66Þ

ai0 þ bp ðkÞ þ bq ðkÞ

i¼1

!

l

hi ðkÞ bt þ

C P i¼1

mðkÞ ¼ 0

ð67Þ

bip þ q þ 2 hi ðkÞ

Finally, isolating the term m(k) we obtain the optimal control rate, given by ! C

X mo ðkÞ ¼ b0 ðkÞ  bt  ai0 þ bp ðkÞ þ bq ðkÞ hi ðkÞ ð68Þ i¼1

where PC

b0 ðkÞ ¼ P C

i¼1

i¼1

bp ðkÞ ¼

pX þ1

bip þ q þ 2 hi ðkÞ

2

bip þ q þ 2 hi ðkÞ

: þl

ait bðkp1þ tÞ

t¼1

bq ðkÞ ¼

q X

bit þ p þ 1 mðkq1 þ tÞ

t¼1

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F.H.T. Vieira, F.G.C. Rocha / Neurocomputing 74 (2011) 1894–1907

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Fla´vio Henrique Teles Vieira received his B.Sc. degree in Electrical Engineering from the Federal University of Goia´s (UFG) in 2000, the M.Sc. degree in Electrical and Computer Engineering from UFG in 2002 and the doctorate degree in Electrical and Computer Engineering at State University of Campinas (FEEC-UNICAMP) in 2006. Since 2008, he has been a Professor at the Federal University of Goia´s in Brazil. He acts in the following research areas: Network Traffic Modeling and Control, Communication Networks and Soft Computing.

1907

Fla´vio Geraldo Coelho Rocha received his B.Sc. degree in Electrical Engineering from the Federal University of Goia´s (UFG) in 2008 and he is currently working toward the M.Sc. degree in Electrical and Computer Engineering at Federal University of Goia´s. He acts in the following research areas: Network Traffic Modeling and Control, Network Performance Analysis, Wireless Networks and QoS support and video transmission.