Pade and Laguerre Approximations Applied. to the Active Queue Management Model. of Internet Protocol

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1 Applied Mahemaical Sciences, Vol. 7, 013, no. 16, HIKARI Ld, hp://dx.doi.org/ /ams Pade and Laguerre Approximaions Applied o he Acive Queue Managemen Model of Inerne Proocol Tolaimae Ichrak and Elalami Nourredine Laboraoire d Auomaique e Informaique Indusrielle Ecole Mohammadia d Ingénieurs EMI, Raba, Morocco olaimae.ichrak@gmail.com, elalami@emi.ac.ma Copyrigh 013 Tolaimae Ichrak and Elalami Nourredine. This is an open access aricle disribued under he Creaive Commons Aribuion License, which permis unresriced use, disribuion, and reproducion in any medium, provided he original work is properly cied. Absrac This paper considers he problem of pure ime delay in he Acive Queue Managemen model, he delay elemen is approximaed by using Padé and Laguerre presenaions. Hankel norm is used o deermine he order of runcaion. Quaniaive error bounds on delay approximaions are given and calculaed. The approximaed models are compared wih he exac model wih delay o deermine he efficiency of he presened mehods. The example presened in illusraion is he model of Congesion conrol in Inerne Proocol. Keywords: Delay, Hankel, Laguerre, Padé. 1 Inroducion Traffic congesion of he Inerne is one of he major communicaion problems lived by millions of users. Many works have been devoed o improve he inerne congesion conrol and several mahemaical models of AQM Acive Queue Managemen schemes supporing TCP flows in communicaion neworks have been proposed, [1].

2 664 Tolaimae Ichrak and Elalami Nourredine In fac, a fluid model of TCP dynamical behavior was developed; i uses he heory of sochasic differenial equaions. The model describes he evoluion of he variables on he nework such as TCP Window size and Queue lengh. Based on some reasonable assumpions, we ge he following relaions [1]-[]-[3]: + + +, 0 :, 0, max 0, :,, 1 T p C q q when W N C q when W N C q p W W W f & & Where: TABLE I Descripion of Nework parameers Parameer Descripion W Window lengh W Time Derivae of Window lengh q Queue lengh C Transmission Capaciy R RTT made up of wo iems Tp Propagaion delay N Number of TCP connecions P Probabiliy of packe dropping 3 1

3 Pade and Laguerre approximaions 665 The equilibrium poin is defined by 0 1 and a his poin gives: w& and q& 0 Nw Rc ² δ ẇ p R R² C N ² 4, and he linearizaion of Nw q δ q. 5 R R We apply Laplace Transformaion o 4 and 5, and, hen we ge: sw s NW s RC ² e R ² C N ² -Rs p s NW s Q s sq s R R -Rs This leads o: Q s ke G s 6 p s T s + 1T S + 1 Where: 3 RC R ² C k ; T1 ; T R 4N ² N 1 So, he linearizaion of equaions can be modeled as a wo-order wih delay plan ransfer funcion. We will apply he Padé and Laguerre approximaions o he delay in he ransfer funcion, and we will ry o ge an approximaion near he rue sysem hrough he choice of adequae degree. Noions The approximaion mehods used mos widely in pracice are based on he presenaion of he delay elemen as a raio of wo polynomials where q n x is a sable polynomial of degree n. An r h -order Laguerre approximaion of e s is : / / 7

4 666 Tolaimae Ichrak and Elalami Nourredine General r h -order randfer funcion : 8 All pass r h -order ransfer funcion of he form 9 An r h -order of Pade approximaion is : Where!!!!! 10.1 Padé approximaion [1] The Padé approximaion is ofen used o approximae a pure ime delay by a raional ransfer funcion. As we know, mos conrol design algorihms canno handle ime delays direcly. A common echnique is o replace delays by heir Padé approximaions all-pass filers,bu because his approximaion is valid only a low frequencies, i is imporan o compare he rue and approximae responses o choose he righ approximaion order and check he approximaion validiy [1]. The Padé approximaion for he erm e sl is given by: Nr sl e sl Dr sl Where: r k! Nr sl sl k! r k! r k 0 r r k! k Dr sl sl k 0 k! r k! r represens he order of he approximaion. k. Laguerre approximaion The Laguerre polynomials, named afer Edmond Laguerre , are soluions of Laguerre's equaion: 1 0 which is a second-order linear differenial equaion. This equaion has nonsingular soluions only if n is a non-negaive ineger. The associaed Laguerre polynomials are soluions of 1 0

5 Pade and Laguerre approximaions 667 The Laguerre polynomials are also used for Gaussian quadraure o numerically compue inegrals of he form These polynomials, usually denoed L 0, L 1,..., are a polynomial sequence which may be defined by he Rodrigues formula:! They are orhogonal polynomials wih respec o inner producf,g fxgxe dx The delay elemen can be presened by Laguerre approximaion by : An r h -order Laguerre approximaion of e s : / / Hankel minimum degree approximaion A Hankel marix, is a square marix consan skew diagonals posiive sloping diagonals: If he i,j elemen of A is denoed A i,j, hen we have,, The error bound of a sysem is compued based on Hankel singular values. For a sable sysem, Hankel singular values indicae he respecive sae energy of he sysem. Hence, reduced order can be direcly deermined by examining he sysem Hankel Singular values. This mehod guaranees an error bound on he infiniy norm of he addiive error G-G RED for well-condiioned model reduced problems.4 Error bounds [5]-[6] Le e -sd be approximaed by an r h -order ransfer funcion Grs: For he r h-order Padé approximaion, he weighed infiniy norm error is given by: For he r h-order Laguerre approximaion, he weighed infiniy norm

6 668 Tolaimae Ichrak and Elalami Nourredine error is given by: The deviaions from he rue infiniy norm of he above formulae is less han 0.01 over he range 5 0 The lower bound of Pade approximaion is also given by: The lower bound of Laguerre approximaion is also given by:.. / 15 Where M and k are parameers saisfying, and d is he delay 3 Applicaion We will apply he noions aken in secion o he acive queue managemen model, we will calculae he error bounds, made some approximaions of Pade and Laguerre, we will compare hese reduce models wih he rue one in simulaions and we will calculae he Hankel degree adequae o he energy of sysem, 3.1 Calculaion of he error bounds: Pade Laguerre acual Lower bound acual prediced Firs order Padé approximaion: [1]The firs order Padé approximaion of he ime-delay erm is:

7 Pade and Laguerre approximaions e Rs 1 + Rs Rs Using his approximaion in 6, we obain he following raional ransfer funcion G f s: Rs k 1 Gf s Rs T1s + 1 T s Rs k 1 s T1s + 1 T s + 1 T Gf 16 Wih T 3 R/ ; Le s R0.06 s, C15 Mb/s, and N10; We will compare he sep responses of he Firs order Padé approximaion Gfs given by 16 and he rue sysem Gs given by 6.Using Scilab, we ge his figure: Fig.1 comparison beween he sep responses of he exac sysem and Firs Order Padé approximaion We see clearly ha he approximaion error is oo large. To ge a beer approximaion, le s ry a second-order Padé approximaion of he delay.

8 670 Tolaimae Ichrak and Elalami Nourredine 3.3 Second order Padé approximaion: The second order Padé approximaion for delay is given by: Rs Rs ² 1 + e Rs 1 Rs Rs ² So, he new Transfer funcion is: Rs Rs ² k 1 + Gs s 1 17 Rs Rs ² T1s + 1 T s And he comparison beween he sep responses of he Second order Padé approximaion G s s given by 17 and he rue sysem Gs given by 6, gives Fig. comparison beween he sep responses of he exac sysem and Second Order Padé approximaion We see now ha he responses mach closely, and we conclude ha he second order of Padé approximaion is adequae o approach he rue sysem. 3.4 Laguerre approximaion Consider Padé, Laguerre approximaions wih r,5,30, we simulae he responses of hese sysems and compare hem wih he rue one. We see ha similar resuls are observed for he shif Laguerre Padé 5 and Laguerre Padé 30

9 Pade and Laguerre approximaions 671 Fig.3 comparison beween he sep responses of he exac sysem and Padé laguerre approximaions for orders,5,30 approximaion Consider Padé, Laguerre approximaions wih r,5,30 and he rue sysem. The ime responses for he full order sysem, he reduced second, fifh, hiry order sysems are shown. For all approximaion models,, he difference in he ime response appears only in he seady sae value I is observed ha all hree approximaion models give almos he same resuls wih he same rae of decay. These models closely approach he full order model. This resul shows ha in he low pass frequency range he approximaion mehods have similar approximaion capabiliies.

10 67 Tolaimae Ichrak and Elalami Nourredine 3.5 Hankel norm Fig.4 calculaion of Hankel singular values for he sysem The Hankel singular values of he raional approximaion of G s, are shown in Fig. 4. The energy of he sysem is limied o he second order. Tha explain why he approximaions wih r5 or greaer orders figure 3 give he same resuls. We can say ha are no efficien, jus complicaed in calculus. Conclusion The paper considers he problem of he pure ime delay elemen approximaion by Pade and Laguerre, represenaions. I is shown ha ime delays can be approximaed successfully. The Hankel norm is used o reduce he order of he weighed ime delay approximaions. Mehods for model reducion are esed ino he acive queue managemen model in inerne proocol and he corresponding errors are calculaed. The resuls obained confirm he efficiency of he shif operaors Laguerre and Pade approximaion of ime delay elemens. References [1] Ichrak TOLAIMATE, Nourredine EL ALAMI, Kharionov approach and Padé Approximaion applied o he robus conroller design of Acive Queue Managemen rouers for Inerne Proocol, WSEAS, Inernaional conferences in Corfu Island, Greece, July 14-17, 011.

11 Pade and Laguerre approximaions 673 [] Ichrak TOLAIMATE, Nourredine EL ALAMI, Robus Conrol Problem as H and H conrol problem applied o he robus conroller design of Acive Queue Managemen rouers for Inerne Proocol, INTERNATIONAL JOURNAL OF SYSTEMS APPLICATIONS, ENGINEERING & DEVELOPMENT Issue 6, Volume 5, 011. [3] Ichrak TOLAIMATE, Nourredine EL ALAMI, Basic conrol design applied o he TCP IP model, 1 Inernaional Conference on Mulimedia and Conrol Sysems ICMCS, Tangier, May 10-1, 01. [4] Ichrak TOLAIMATE, Nourredine EL ALAMI, Quaniaive Feedback Theory applied o he sae model of congesion conrol, he nd Inernaional Conference on Sysems and Conrol, Marrakech, Morocco, June 0-, 01. [5]Al-Amerand Al-Sunni, H-infiniy error bounds in approximaing ime-delay sysems, Proc.Insn Mech. Engrs., Vol.17 ParI: J. Sysems and Conrol Engineering. [6] Kamen Perev, Approximaion of Pure Time Delay Elemens by Using Hankel Norm and Balanced Realizaions, Problems of Engineering Cyberneics and Roboics, 64, Bulgarian Academy of Sciences. [7] Ichrak TOLAIMATE, Nourredine EL ALAMI, Fragiliy via robusness of conrollers dedicaed o he congesion conrol in inerne proocol Applied Mahemaical Sciences, Vol. 7, 013, no. 88, Received: Sepember 6, 013