Performance Evaluation of ATM Networks in Computer Simulation

Transcription

1 A. Ghaa et al / (IJCSIT) Intenational Jonal of Compte Science an Infomation Technologie, Vol. 2 (6), 2, Pefomance Evalation of ATM etwo in Compte Simlation A. Ghaa, Y. Mohanna, M. Dbo, O. Bazzi Phyic an Electonic Laboatoy (LPE) Lebanee Univeity, Faclty of Science (I), Haath, Beit, Lebanon Abtact One of the pincipal phae in the eign poce of the infomation ytem i the pefomance evalation of thee ytem. Qeing netwo an Maov chain ae commonly e fo pefomance an eliability evalation of compte, commnication, an ytem manfacting. In the context of thi wo, thi phae of analyi i bae on a mathematical moelling an eoltion in tem of Maov chain. Seveal eptable fomalim wee evelope to ae the poblem aiing fom the ize (potentially vey lage) of thee chain. In thi pape, we ae inteete in the geneation of compte an telecommnication netwo ing the Aynchono Tanfe Moe (ATM). ATM netwo e mall fixe ize cell to tanmit infomation. Thi allow them to hae the ame netwo fo voice, vieo, an ata at a wie ange of itance. Mot compte an telecommnication companie ae woing on ATM poct an evice. The eqie pefomance elate in thi ty to the tationay itibtion of the Maov chain. Iteative algoithm wee aleay evelope, howeve, fo vey lage moel, obtaining the tationay itibtion emain ifficlt when applying thee algoithm. Keywo ATM, Maov Chain, Moeling, Simlation, Pefomance evalation, meical metho. I. ITRODUCTIO De to inceaing nmbe of netwo in exitence an thei geate complexity, eigning new ytem an impoving the pefomance of exiting one become moe an moe ifficlt an time conming. Coneqently, it i o impotant to e moelling an imlation tool to eal with thi complexity. ATM (Aynchono Tanfe Moe) i a netwoing technology in high pee local an wie aea netwo []. ATM banwith on-eman feate mean that a ingle netwo can cay all type of taffic-voice, vieo, image an ata. In mmay, ATM i an efficient, flexible technology with the ability to ppot mltimeia taffic at extemely high ate. Compte netwo to a cla of phyical ytem that can be tie effectively by mean of icete event imlation moel. The main objective of thi wo i to gie the ATM netwo to povie itable contol inpt leaing to poce a eie epone. If the compte etwo Simlation plant moel i capable of appoximating well an with fficient accacy, then it may be e within a moel bae contol tategy. etwo entitie ch a the acce behavio to the netwo may be ecibe in the fom of icete chain event. By imlating an ATM moel, the pefomance of the imlation ha been compae in tem of convegence by Maov chain analyi. In thi pape we peent a nmeical oltion fo ATM etwo, which we valiate ing an analytical metho. Sch analyi, allow inee, to evalate the thoghpt an elay pefomance of the ytem. Two metho fo analyi ae popoe: the taitional Maov chain analyi an the nmeical oltion. The Maov bae metho povie the exact nmeical elt. A a elt, the nmeical oltion can be e fo implifying the nolve complex cae, moe nmeical example ae given. The pape i oganize a follow: Section I give an intoction to ATM an the commnication netwo achitecte. An oveview of elate wo i ice in Section II. Section III eal with moelling, imlation an analyi bae on Maov chain moel fo ATM netwo. Section IV peent the nmeical oltion. Section V ice the conclion an in VI we peent ecommenation fo fte wo. II. RELATED WORKS Seveal pape have been pblihe concening pefomance evalation of etwo. Mot of thee wo foc on the taitional Maov Chain analyi (Bith Death poce) to evalate the thoghpt, elay pefomance an tability of the ytem. In [] an [2] two analytical oltion ae ecibe. Howeve, many elate wo ae epote in jonal an confeence poceeing, following ae ome example:. In 994 Gang W, Kaiji Mmoto, an Aia Fa, Analyi the pefomance evalation of eeve ile ignal 2632

2 A. Ghaa et al / (IJCSIT) Intenational Jonal of Compte Science an Infomation Technologie, Vol. 2 (6), 2, mltiple-acce cheme fo wiele commnication netwo [3]. 2. In 997 R. W. Dmitoca an San G. Gibon, ice a new poct fo HP Boaban Seie tet ytem. The HP E429 ATM netwo impaiment emlato allow telecommnication netwo an eqipment manfacte to emlate Aynchono Tanfe Moe netwo in the laboatoy []. 3. In 28 Jia Li, Shnxiang Li, an Shheng Jia (in oe to olve the poblem of anom an flctation of expeiment eo an peication eo of neal netwo, neal netwo moel moifie by a fzzy Maov chain wa intoce, When neal netwo wa e to peict, the peiction eo between actal vale an otpt vale of the netwo wee itibte anomly. That can be imlate by a Maov chain [4]. 4. In 2. Sinno, H. Yoef, an A. Ghaa ty the evalation of the Maov Moel in compte etwo Simlation [5]. III. MARKOV CHAI-BASED MODEL FOR ATM ETWORK We now evelop a Maov chain moel fo ATM etwo. Eentially, we ame a Maov chain bae on the ize of cell (tate) [2]. The cell elay vaiation mole emlate cell bffeing, netwo congetion, an mltiplexing elay in ATM netwo by vaying the amont of elay applie to cell, epening on the line ate. Each cell in the team i given a iffeent amont of elay. A citical feate i that in an ATM netwo, cell eqence integity i gaantee. Thi poe a technically inteeting poblem, ince imply elaying each cell to ceate the eie tatitical itibtion wol pt cell ot of eqence. The netwo impaiment emlato cell elay vaiation impaiment mole i implemente a a bitheath Maov poce. Each matching cell i elaye by a pecifie amont. Thi amont i etemine by the cent tate of Maov chain. The teay-tate itibtion of the Maov chain can be e in the cell elay vaiation itibtion. Only the eivation of the binomial itibtion with paamete an P will be hown in etail, whee i the total nmbe of inepenent tial (o, eqivalently, the maximm nmbe of cell-time elay) an p the pobability of elay on any ingle tial. Othe itibtion can be eive ing the ame techniqe by changing the bith ate an eath ate appopiately. A tanition to next tate i calle a bith, while a tanition on the pevio tate i calle a eath. Thi poblem wa olve by the e of a Maov chain moel, which i hown: Fig.: Maov chain moel fo Bith-Death poce: i the nmbe of tate; i the maximm nmbe of cell time elay; the, ae bith, eath ate an S the teay-tate. A. Bith-Death Poce Bith-eath pocee ae Maov chain whee tanition ae allowe only between neighboing tate. We teat the contino time cae hee, bt analogo elt fo icete-time cae ae eaily obtaine. A one-imenional bith-eath poce i hown Fig. an it tanition matix i hown a: Q Fig 2: Tanition pobability matix The tanition ate, ae tate epenent bith ate an,, ae efee to a tate epenent eath ate. In the limiting cae when = we have p,, an. Similaly, fo = we have, p, an p. The tanition ate, ae tate epenent bith ate an,, ae efee to a tate epenent eath ate. B. Cloe-Fom Soltion We now eive a cloe-fom oltion fo the Maov moel eive in the pevio ection. Let be the teay-tate pobability of being in tate, we baically compte fo each poible tate the pobability. A the vey fit tep, we wite the following fixe point eqation [6] fo the Maov chain in Fig.. :, () Which expee the teay-tate pobability (of being in tate ) a the nion of thee event: (i) eaching (bith) tate fom tate -, with pobability, (ii) emaining in tate with pobability ; (iii) eaching tate fom +, with pobability, by mean of a eletion (eath) event. We can ewite the ame eqation a: 2633

3 A. Ghaa et al / (IJCSIT) Intenational Jonal of Compte Science an Infomation Technologie, Vol. 2 (6), 2, ( ), (2) Which eqate the pobability of leaving tate, compte a ( ) to the pobability of enteing tate, compte a. By eplacing,,, with the actal vale we obtain (3): p- an p p ( p) Whee p i the pobability of elay on ingle tial, i the total nmbe of inepenent tial (o, eqivalently, the maximm nmbe of cell-time elay). Eqation 3 i a econ oe iffeence eqation whoe paamete ae epenent on, i.e., on the cent tate. We e Eqation 3 an the conition fo a pobability itibtion: (4) To eive the following cloe-fom eqation fo pobability : p ( p ) (5) The eqation how that the contant pobability of bith p with a linea inceaing pobability of eath /, elt in a binomial itibtion with mean. p an vaiance. p( p). C- Application: We tat fom the following eqation, intoce in ection III:, By eplacing the pobabilitie,, an, with thei actal vale: p, ( p ) p an ( p ) IV. p p.umerical SOLUTIO A icete time Maov moel i efine in tem of a et of tanition pobabilitie between the icete tate of the moel. Tanition ae aociate with tanition pobabilitie between tate. The x tanition pobabilitie between tate ae calle the tanition pobability matix Q. If Q i initial pobability vecto then the tate pobability afte tep (6) i given by the vecto: Q * Q, the matix Q i the -th tanition matix afte mltiplying Q by itelf, time. The teay tate pobability vecto in all tate of the Maov moel i ch that: * Q. Thi eqation will be e along with the conition:. In oe to etemine all the teay tate pobabilitie fo the netwo moel [5, 9, ]. To valiate the popoe Maov chain fo ATM etwo, we pefom two et of expeiment. In the fit et expeiment, we apply a nmeical oltion [7, 8, 9] ing Gaient Conjgate Metho. Fo each expeiment, we collect the tatitic egaing the ize nmbe of teay tate (), an compae the Binomial itibtion of teay tate with the moel itibtion povie by eqation 5. We applie thee iffeent ize, = 5, =, =5 with the pobability of elay on ingle tial an 32 p, (Fig.3, 4 an 5) fo o moel we ame that the ATM etwo. Fige 6 how cve of mean pobability ve the nmbe of tate (22). It can be obeve that thi mean pobability eceae a the nmbe of tate inceaing. Fige 7 how fo thee nmbe of tate (5, an 5) the mean pobability. Alo, thi mean pobability eceaing a the total nmbe of tate inceaing. The elt we epote o fa how that o moel fo ATM etwo in Maov Chain can peict the Binomial itibtion an howe that the teay-tate pobability. To tet whethe the teay-tate pobability itibtion of the nmeical oltion ata i actally fit the theoetical itibtion (Fig. 3, 4 an 5) povie by the moel we employ the goone of fit fo icete itibtion. In all the expeiment epote the meical oltion ata ae compatible with a Binomial itibtion, the eo between nmeical an analytical oltion i le than.e-6 fo = 5, an 5. To mmaize, it i elatively eay fo the netwo to extact the global popetie fom the total nmbe of tate, the pobability of elay on ingle tial p, the bith ate an eath ate. V. COCLUSIOS In thi pape we have invetigate the pefomance evalation in Compte imlation. We eive a Maov chain moel fo ATM etwo an howe that the teaytate pobability can be appoximate by a Binomial itibtion. Each tet, the itibtion of the nmeical oltion (icete itibtion) ata i compatible with the teay-tate Binomial itibtion (the theoetical itibtion povie by the moel). 2634

4 A. Ghaa et al / (IJCSIT) Intenational Jonal of Compte Science an Infomation Technologie, Vol. 2 (6), 2,

5 A. Ghaa et al / (IJCSIT) Intenational Jonal of Compte Science an Infomation Technologie, Vol. 2 (6), 2, A a conclion, we ame that the elt epote in thi pape can be e in the eign of compte netwo ytem. The analyi incle, inee, the qalitative evalation of the pefomance, an qantitative analyi coniting of the teay-tate pobability itibtion of the Maov chain. We obeve that if we apply moe analyi, the elt can lea to a new moel. Thi moel col imlate tempoal vaiation on the inheent elay. Sch wo eqie moe mathematical an tatitical moeling effot. The pefomance of ATM etwo i evalate analytically an via nmeical imlation. The imlation wee pefome on the Unix C++ pogamming langage. VI. FUTURE WORKS Ot of the elt of the peent wo, ome ecommenation fo fte wo ae popoe:. Apply iteative metho to evalate netwo by Maov Chain a ontationay iteative metho ch a (BiConjgate Gaient (BiCG), BiConjgate Gaient Stabilize Metho, Conjgate Gaient Sqae Metho, an Conjgate Gaient fo paallelim [7]. 2. Compaion between tationay iteative metho an ontationay iteative metho. 3. Apply paallel Compting fo vey lage moel. REFERECES [] Robet W. Dmitoca, San G. Gibon, Tevo R. Hill, Lia Mola Moale, an Chong Tean Ong. Emlating ATM etwo Impaiment in the Laboatoy. Apil 997 Hewllett-Paca Jonal, Aticle 9. [2] Qeeing etwo an Maov Chain Moeling an pefomance evalation with compte cience application, G. Bolch, S. Geine, H. e Mee, an K. Tivei. A wiley-intecience pblication, John Wiley & Son. IC. [3] Sahne R., Tivei K., an Pliafito A., Pefomance an Reliability Analyi of Compte Sytem An Example-Bae Appoach Uing the SHARPE Softwae Pacage. Klwe Acaemic Pblihe, Boten, M.A., 996. [4] JIa Li,Shnxiang Li, an Shheng Jia, A Peiction Moel Bae on eal etwo an Fzzy Maov Chain, IEEE ,28. [5] Evalation of the Maov Moel in Compte etwo Simlation,. Sinno, H. Yoef, an A. Ghaa, IAEG Tanaction on Engineeing Technologie Volme 4, eite by S.-I. Ao, 2 Ameican Intitte of Phyic //$3. [6] D. G. Lenbege. Intoction to ynamic ytem: Theoy, Moel, an application. John wiley & Son, Y, USA, May 979. [7] Iteative Metho fo Spae Linea Sytem, Yoef Saa, copyight 2 Yoef Saa. Secon Eition with coection. Janay 3RD,2. [8] M. Panini an J. H, A nmeical appoximation cheme fo each ability analyi of tochatic hybi ytem with tate-epenent witching, in In Poc, IEEE Int. Conf. Deciion an contol, ew Olean, LA, Dec. 27,pp [9] H. J. Khne, Appoximation an Wea Convegence Metho fo Ranom Pocee with Application to Stochatic Sytem Theoy. Cambige, Maachet: MIT Pe, 984. [] M. Kwiatowa, G. oman, an D. Pae, Pim: Pobabilitic moel checing fo pefomance an eliability analyi, ACM SIG- METRICS pefomance Evalation Review, 29.. Ahma Ghaa eceive a Dipl. Ing. In 986 fom the ESICA (École ationale Spéiee Ingénie e Contction Aéonatiqe e Toloe Fance), an in 992 he eceive a Ph.D. egee in mechanic fom the niveity of Lille Fance. He joine the epatment of applie mathematic at the Lebanee Univeity. Hi eeach inteet incle moeling poblem in mechanic, imlation compte netwo, an Maov Chain. Yae Mohanna eceive hi BE egee in electical engineeing fom the Ameican Univeity of Beit (AUB) in 986 an hi Mate an Ph.D. egee in Optoelectonic fom the ational Polytechnic Intitte of Genoble (I..P.G.) Fance in 987 an 989 epectively. Hi Ph.D. an mate eeach wo wa completely one at ALCATEL FIBER OPTICS Fance. He i a pofeo of electonic engineeing at the Lebanee Univeity (L.U.), Beit, Lebanon. He joine L.U. in 994 an contibte to the evelopment of it Electonic B.S. an Mate pogam. Fom 992 to 994 he wa with the Liban Cable, Lebanon whee he chaie the electonic ection. Fom 989 to 99 he wa with AOIP intmentation Fance whee he contibte to the eign of infae pyomete. Pof. Mohanna i a membe of the Lebanee oe of enginee. Hi actal fiel of inteet ae Digital ignal an image poceing an poceo, contol ytem. Mohame Dbo i a fll time Aociate Pofeo at the Lebanee Univeity (Beit) whee he cooinate a mate-2 eeach egee M2R-SI: Infomation Sytem. He eceive hi PhD fom Pai-S, Fance, 997. Hi eeach inteet incle oftwae engineeing an infomation ytem, geogaphic infomation ytem, ata-waehoing an ata-mining. Oama Bazzi eceive hi BE egee in electical engineeing in 987 fom the Ameican Univeity of Beit an hi MS an PhD egee in electonic in 988 an 992, epectively, fom the Univeity of Valencienne, Fance. He joine the Lebanee Univeity in 996, whee he i cently pofeo. Hi eeach inteet ae in ignal poceing an commnication. 2636