Multiclass G/M/1 Queueing System with Self- Similar Input and Non-Preemptive Priority

Transcription

1 Muicass G/M/ Queueing ysem wih ef- imiar Inpu and Non-Preempive Prioriy Mohsin Ifikhar, Tejeshwar ingh and Bjorn Landfed choo of Informaion Technoogies Universiy of ydney ydney, NW, Ausraia Mine Cagar Deparmen of Mahemaics Koc Universiy Isanbu, Turkey Absrac In order o deiver innovaive and cos-effecive IP muimedia appicaions over mobie devices, here is a need o deveop a unified service paform for he fuure mobie Inerne referred as he Nex Generaion NG a-ip nework. I is convincingy demonsraed by numerous recen sudies ha modern muimedia nework raffic exhibis ong-range dependence LRD and sef-simiariy. These characerisics pose many nove and chaenging probems in raffic engineering and nework panning. One of he major concerns is how o aocae nework resources efficieny o diverse raffic casses wih heerogeneous Qo consrains. However, much of he curren undersanding of wireess raffic modeing is based on cassica Poisson disribued raffic, which can yied miseading resus and hence poor nework panning. Unike mos exising sudies ha primariy focus on he anaysis of singe-queue sysems based on he simpes Firs-Come-Firs-erve FCF scheduing poicy, in his paper we inroduce he firs of is kind anayica performance mode for muipe-queue sysems wih sef-simiar raffic schedued by prioriy queueing o suppor differeniaed Qo casses. The proposed mode is based on a G/M/ queueing sysem ha akes ino accoun muipe casses of raffic ha exhibi ong-range dependence and sef-simiariy. We anayze he mode on he basis of non-preempive prioriy and find exac packe deay and packe oss rae of he corresponding casses. We deveop a finie queue Markov chain for non-preempive prioriy scheduing, exending he previous work on infinie capaciy sysems. We exrac a numerica souion for he proposed anayica framework by formuaing and soving he corresponding Markov chain. We furher presen a

2 comparison of he numerica anaysis wih comprehensive simuaion sudies of he same sysem. We aso impemen a Cisco-rouer based es bed, which serves o vaidae he mahemaica, numerica, and simuaion resus as we as o suppor in undersanding he Qo behaviour of reaisic raffic inpu. Keywords: Qo, 3G, UMT, GGN, ef-imiar I. INTRODUCTION Over he pas en years, he subjec of undersanding he naure of Inerne raffic has sparked considerabe research aciviy and i has been shown ha Inerne raffic exhibis sef-simiariy and bursiness over a arge range of ime scaes. The firs sudy, which riggered he aenion of he Inerne research communiy owards sef-simiariy phenomena was based on he measuremens of Eherne raffic a Becore []. As a resu of deaied invesigaions performed on wide-area TCP raffic [, 3, 4] and earier sudies conduced on LAN raffic [5, 6, 7] by using an exensive se of acua races, i was shown ha he disribuion of packe inerarrivas ceary differs from he cassica exponenia disribuion and hese sudies argued convincingy ha boh oca-area and wide-area nework raffic appear o be beer modeed by using saisicay sef-simiar processes as compared o Poisson modes. ubsequen saisica anaysis has provided much experimena evidence ha many oher ypes of Inerne raffic incuding WWW raffic [8], VBR video [9] and ignaing ysem No. 7 [] aso exhibi sefsimiariy. In addiion, a deeper invesigaion of Inerne raffic has ed o he discovery of various properies such as sef-simiariy [], ong-range dependence [] and scaing behavior a sma ime-scaes [3]. Furher informaion on he modeing and anaysis of sef-simiar raffic is found in [4, 5] which cover boh heoreica and appied aspecs of sef-simiariy and ong-range dependence. Concurreny, hird Generaion ysems 3G are being depoyed and growing in popuariy. One of he disincive objecives of 3G sysems is o provide voice, graphics, video and oher broadband services direc o he end-user over mobie devices. The Universa Mobie Teecommunicaion ysem UMT is one of he major proposed sandards for 3G, deveoped by Third Generaion Parnership Projec 3GPP [6]. A simpified UMT nework archiecure is shown in Fig.. Evoved from GM and GPR, he Core Nework CN of UMT consiss of wo service domains, a Circui wiched C service domain and a Packe wiched P service domain, which is of ineres in his paper. In he P service domain, UMT connecs o a Packe Daa Nework

3 PDN hrough he erving GPR uppor Node GN and Gaeway GPR uppor Node GGN. From he UMT perspecive, 3GPP defines four differen UMT Qo casses conversaiona, sreaming, ineracive and background cassified and ordered by heir deay sensiiviy [6]. On he oher hand, he increasing demand for wireess Inerne access and he wide depoymen of arge ubiquious insaed IP infrasrucure is imposing a major paradigm shif and pressuring he wireess indusry o adop he echnoogies, services and archiecures aready presen in he Inerne and i is now widey recognized ha IP wi be he foundaion for nex-generaion mobie neworks [7]. Three main Qo frameworks, Inerv [8], Differv [9] and MPL [] have been sandardized in order o provide suppor for a variey of raffic casses wih differen demands in he Inerne. Based on hese hree IP Qo modes, various kinds of a-ip archiecures have been proposed for 3G wireess neworks [-7]. 3GPP is aso eaning owards fuure wireess a-ip nework archiecure o deiver innovaive and cos-effecive services e.g. IP eephony, media sreaming and muipary gaming. To suppor hese services over UMT neworks, 3GPP has defined a new domain, IP Muimedia ubsysem IM in is aes specificaion [8]. A hese facors have conribued o arac he aenion and curiosiy of researchers owards undersanding he naure of wireess IP raffic and recen sudies have proved ha wireess daa raffic exhibis sef-simiar behaviour as we [9-34]. However, mos of he exising work on nework raffic modeing is based on he simpified assumpion of Poisson disribued raffic. The Poisson modes fai o capure he aribues of rea nework raffic which is ong-range dependen and saisicay sef-simiar. This necessiaes new raffic modes wih sef-simiar characerisics for opima resource aocaion and bandwidh assignmen o heerogeneous raffic casses. In order o offer guaraneed Qo o differen end-users, here is a need o deermine parameers such as queueing deay, packe-oss rae, and expeced queue engh using reaisic raffic condiions. We sar by giving an overview of reaed work on wireine and wireess IP raffic modeing. Then, we presen our proposed sef-simiar raffic mode and highigh our conribuions o nework raffic modeing. II. RELATED WORK In his secion, we firs discuss reaed work, which has been done in he area of performance evauaion of wired IP and Wireess IP neworks under sef-simiar inpu and hen we compare our mode wih he previous work. 3

4 A. Reaed Work on Wireine IP Traffic Modeing During he as en years, much research has been dedicaed o Inerne raffic modeing based on queueing heory in he presence of sef-simiar raffic [35-44]. Here we discuss some of he avaiabe resus. In [35], i was shown ha, wih sef-simiar raffic, shared oupu buffering provides higher hroughpu and ower ce oss probabiiy as compared o dedicaed oupu buffering sraegies a he cos of higher ce deay. An empirica demonsraion was provided in [36] o prove ha ong-range dependence is a dominan characerisic for a number of raffic engineering probems and has considerabe impac on queueing performance beyond is saisica significance in raffic measuremens. A Markovian Moduaed Poisson Process MMPP is used in [37] as raffic inpu o compue he numerica resus for a wo cass Differv ink on he basis of a Marix Geomeric anayica mehod. The oss probabiiy of MMPP/D/ was invesigaed in [38], where MMPP is generaed so as o mimic he variance-ime curve of he sef-simiar process over severa ime-scaes. The major weakness of MMPP modes is ha MMPP may require an esimaion of a arge number of parameers. A neura-based echnique was proposed in [39] for esimaing queueing aency of sef-simiar packe raffic. To see he impac of sef-simiariy on he performance of Differv neworks, on OPNET based simuaion anaysis was done in [4] and performance measures in he form of expeced queue engh were found in reaion o he Hurs parameer and server uiizaion. I is hard o offer guaraneed Qo parameers on he basis of such anaysis. The offered queueing based resus in [35-44] ack he capabiiy of offering differenia reamen o muipe casses of inpu raffic because he majoriy of he anaysis is based on FIFO scheduing and furher he resus are asympoic. To provide an overview of work in he area of IP neworks performance evauaion, he readers are referred o [45-49]. The major drawback of he exising work is ha he queueing modes considered are no abe o capure he sef-simiar characerisics of IP raffic. Furhermore, i is imporan o noe ha mos of he previous work is focused on he anaysis of ony one ype of raffic wihou discussing is effec on he performance of oher kinds of nework raffic. B. Reaed Work on Wireess IP Traffic Modeing Here we discuss he mos reevan work in he area of wireess raffic modeing. According o 3GPP, UMT-o-IP Qo mapping is performed by a ransaion funcion in he GGN rouer/server ha cassifies each UMT packe 4

5 fow and maps i o a suiabe IP Qo cass. The principe of fows aggregaion beween end users and GGN eads o an increase of he oad on he nework eemens whie moving owards he GGN. Thus, as can be seen from Fig., he GGN is he node mos exposed o sef-simiariy infuence in UMT [5]. In his paper, a FBM/D/ queueing sysem has been used o anayze he performance of GGN whie aking ino accoun sefsimiar inpu. The submied approach enabed he deerminaion of differen probabiisic and ime characerisics: upper and ower bounds of he GGN service rae, he average queue engh in he server buffer and average service ime of informaion unis. A Qo framework for heavy-aied raffic over he wireess Inerne is proposed in [5]. A simuaion sudy ha has been conduced o anayze he performance of he Foreground-Background scheduer and Round-Robin RR scheduer and he resuing insigh shows ha a FB scheduer requires much ess nework resources o aain a given Qo. There are no anayica proofs of he simuaion resus. The aggregaed connecioness raffic is modeed wih Fraciona Brownian Moion FBM in [5]. This sudy indicaes hree major conribuions characerizaion of connecioness raffic, bandwidh aocaion formua and 3 shorerm raffic predicion. An aggregaed raffic mode for UMT is presened in [53]. The key idea is based on cusomizing he bach Markovian Arriva Process BMAP such ha differen packe sizes of IP packes are represened by rewards. Modeing and simuaion of he Ceuar Digia Packe Daa CDPD nework of Teus Mobiiy a commercia service provider are performed by using he OPNET oo in [54]. The race-driven simuaions wih genuine raffic race exhibiing ong-range dependen behaviour are used o evauae he performance of he CDPD prooco. The resus indicae ha genuine raffic races, compared o radiiona raffic modes such as Poisson, produce onger queues. The references [55, 56] provide a deaied discussion on pracicay usabe raffic modes for emerging daa appicaions in GPR neworks. The readers are furher referred o [57-6] o ge an overview of he anaysis ha has been done in wireess IP raffic modeing. These sudies are merey based on characerizaion of wireess raffic and he issue of providing Qo guaranees o differen users wih diverse Qo demands has no been addressed propery. C. Our Proposed ef-imiar Traffic Mode and is Comparison wih Prior Work Compared o prior work done for wireess environmens, he presen sudy brings a cerain eve of novey and overcomes he major imiaions in he fied of raffic modeing wireine and wireess IP boh by offering guaraneed Qo parameers o heerogeneous raffic casses. We are presening a reaisic and nove anayica 5

6 mode by considering wo differen casses of raffic ha exhibi ong-range dependence and sef-simiariy. Our mode impemens wo queues based on a G/M/ queueing sysem and we anayze i on he basis of prioriy wih no preempion. The raffic mode considered is parsimonious wih few parameers o mach measuremens and has been sudied in [6]. The mode is anayica sovabe when fed ino queueing modes, fexibe one mode bu many varians for differen appicaions, impemen-abe ess ime consuming for simuaion and exhibis absoue accuracy criica for business case sudies. The mode is furhermore simiar o an on/off process, in paricuar o is variaion N-Burs mode sudied in [6] where packes are incorporaed. However, ony a singe ype of raffic is considered in [6]. The work in his paper exends on an earier conference paper from 6 [63]. In his paper, we make he foowing major conribuions o IP raffic modeing wireine and wireess: Inerarriva Time Cacuaions: For he paricuar sef-simiar raffic mode [6], we cacuae he packe inerarriva ime disribuions. The disribuion of cross inerarriva ime beween differen ypes of packes is derived on he basis of singe packe resus. Qo Parameers for Muipe ef-imiar Traffic Casses: We consider a G/M/ queueing sysem which akes ino accoun wo differen casses of sef-simiar inpu raffic denoed by /M/ and anayze i on he basis of non preempive prioriy and find exac packe deays and packe oss rae for corresponding sef-simiar raffic casses. For he firs ime, we presen cosed form expressions for G/M/ wih prioriy. Embedded Markov Chain Formuaion: We deveop he finie Markov chain for he non-preempive prioriy scheduing discipine, exending he previous work on infinie capaciy sysems and derive he corresponding ransiion probabiiies. Numerica ouion of Markov Chain: We exrac a numerica souion for he above menioned queueing sysem by numericay formuaing and soving he corresponding Markov chain. Impemenaion of imuaor: We impemen a discree even simuaor for modeing a G/M/ queueing sysem under sef-simiar raffic ha is readiy exendibe o any scheduing discipine. We presen a comparison of simuaor and numerica resus o verify our anayica modeing. Tes bed Impemenaion: We impemen a rea raffic generaor, which reaizes he sef-simiar raffic mode [6] described above. We run and impemen his raffic generaor on a rea es bed consising of Linux worksaions 6

7 and Cisco 84 moduar rouer. We impemen non-preempive prioriy scheduing on he Cisco rouer and find he queueing deays for corresponding sef-simiar raffic casses. The Cisco es bed serves o vaidae he mahemaica, numerica and simuaion resus as we as o suppor in undersanding he Qo behaviour of reaisic raffic inpu. The res of he paper is organized as foows. ecion III is devoed o he expanaion of sef-simiar raffic mode wih muipe casses and he derivaion of inerarriva imes. ecion IV describes he procedure of formuaing he embedded Markov Chain aong wih a derivaion of he imiing disribuion and Qo parameers. In ecion V we exrac he numerica souion of he queueing sysem by numericay formuaing and soving he corresponding Markov Chain. In secion VI and VII we cover he simuaion anaysis and es bed impemenaion respecivey aong wih a comparison of anayica, simuaion and es bed resus. The appicaions of he mode are discussed in secion VIII. Finay, we concude he paper wih fuure work in ecion IX. III. ELF-IMILAR TRAFFIC WITH EVERAL CLAE In his secion, he sef-simiar raffic mode is reviewed as necessary for he derivaion of he inerarriva ime disribuions. The inerarriva ime of packes for a singe cass is considered in deai. Then, he disribuion of cross inerarriva ime beween packes of differen casses is derived on he basis of singe packe resus. A. Traffic Mode We use a raffic mode ha capures he dynamics of packe generaion whie accouning for he scaing properies observed in eecommunicaion neworks [6]. I beongs o a paricuar cass of sef-simiar raffic modes caed infinie source Poisson modes. A common feaure in such modes is a heavy-aied disribuion for he sessions ha occur a he fow eve and arrive according o a Poisson process. On he oher hand, he oca raffic injecion process over each session is a disinguishing feaure. The Hurs parameer is impici in he disribuion of he sessions and is esimaion has been sudied receny in [64]. Our raffic mode is ong-range dependen and amos second-order sef-simiar as he auo-covariance funcion of is incremens is equa o ha of fraciona Gaussian noise for sufficieny arge ime ags. The raffic can be approximaed by FBM when he rae of packe arrivas ends o infiniy [6]. In fac, wo oher heavy raffic imis are aso possibe depending on he increase of he arriva rae as shown receny in [65, 66]. One of hese is a Levy process, which does 7

8 no accoun for packe dynamics ike FBM. Anoher imi is a variaion of he Teecom process which appears in he anaysis of anoher infinie source Poisson mode [66]. The Teecom process represens a fuid ype raffic injecion raher han individua packes. Bordered by such various imiing sef-simiar and/or ong-range dependen sochasic processes for daa raffic, our packe generaion mode covers a wide range of saisica disribuions hrough he choice of is parameers. The raffic is found by aggregaing he number of packes generaed by severa sources. In he framework of a Poisson poin process, he mode represens an infinie number of poenia sources. Each source iniiaes a session wih a heavy-aied disribuion, in paricuar a Pareo disribuion whose densiy is given by g r = δ b r δ δ, r > b, where δ is reaed o he Hurs parameer by H = 3 δ /. The sessions are assumed o arrive according o a Poisson process wih rae λ. Locay, he packes generaed by each source arrive according o a Poisson process wih rae α hroughou each session. The oca packe generaion process coud be aken as a compound Poisson process which woud hen represen packe sizes as we [6, 66]. For a singe cass of raffic, he raffic Y measured as he oa number of packes injeced in [, ] can be wrien as Y = U i Ri i i where U i denoes he oca Poisson process over session i, R i and i denoe he duraion and he arriva ime of session i, respecivey, and he vaues of i denoe an enumeraion of he arriving sessions. Here, rea vaued and R i is posiive, i is U i which couns he number of packes of session i is ineger vaued. As a resu, Y corresponds o he sum of packes generaed by a sessions iniiaed in [,] uni he session expires if ha happens before, and uni if i does no. We consider he saionary version of his mode based on an infinie pas. Fig. iusraes he componens of he raffic. The sessions have been arriving for a ong ime and hence he incremena raffic is saionary. The sessions are represened wih horizona ine segmens wih heir enghs equa o he ordinae of heir saring poins s,r. The saring poins of he sessions are indicaed wih a diamond. The verica segmens represen he packes which are paced over each session a he ime of heir arrivas. The componen u, which is no represened 8

9 in Fig., is he number of packes over a session. The numbers s, r and u denoe he reaizaion of i, Ri and i U for session i, respecivey. In he presen sudy, we expoi he raffic mode o represen differen casses of raffic sreams. Each sream has is own parameers and is independen from he ohers. The packe sizes are assumed o be fixed because each queue or raffic cass corresponds o a cerain ype of appicaion where he packes have fixed size or a eas fixed service ime disribuion. The imes of arrivas can be visuaized in Fig. as he projecions of he packe arriva imes on a sessions o he ime axis. Ahough he oca packe generaion is assumed o be Poisson over each session, he aggregaed packe arriva process is ceary no Poisson. This aspec is consisen wih he ong-range dependence of he packe arrivas. We sudy he disribuion of he ime beween consecuive packes nex. In conras o oher infinie source Poisson modes or on/off processes, our mode ends isef o such a compuaion under cerain simpificaions. B. Inerarriva Times for a inge Cass In his subsecion, we obain he inerarriva disribuion for a singe cass of raffic by aking advanage of he specific srucure of our raffic mode. Given ha here is a packe arriva a an insan in ime, we find he disribuion of he ime uni he nex arriva T hrough deermining P { T > } for >. This is a condiiona probabiiy concerning wo consecuive packes. Therefore, i can be safey used in he cacuaion of he ransiion probabiiies of he embedded Markov chain for G/M/. Ceary, he imes beween differen pairs of consecuive packes of he same ype are no necessariy independen. ince he raffic inpu is saionary, he curren ime can be aken as. To find he condiiona probabiiy ha here is no packe arriva in he nex ime unis, P { T > }, his even can be spi as A = Any acive sessions ha expire afer do no incur any new arrivas B = Any acive sessions ha expire before do no incur any new arrivas. C = No new session arrivas in or a eas one session arriva wih no packe arriva in. 9

10 We find he probabiiy ha a hree evens occur a he same ime by using he independence of a Poisson poin process over disjoin ses. The evens A and B are independen from C, as he arriva imes of he sessions invoved in each fa ino disjoin regions on he s, r pane as shown in Fig.3. The evens A and B are associaed wih he regions A = { s, r : s, r > s} and = { s, r : s, s r s}, respecivey, and he sessions of he even C B are in he region C = { s, r : s }. The sessions wih saring ime and duraion in se A are acive a ime and he expiraion ime s+r is afer, hence reaed o he even A. imiar argumens hod for he evens B and C. Reca ha a probabiiies mus be cacuaed condiionay on he even ha a packe arriva occurred a ime which has an effec on he disribuion of he number of acive sessions a ime. Mos imporany, he number of acive sessions mus be sricy posiive in ha case. Tha is why we inerpre he given condiion as here is a eas one acive session a which makes possibe he cacuaion of he firs wo probabiiies. Namey, P A B a packe arriva a ime P A B a eas one acive session a ime I is we known ha he number of acive sessions ha do no and do expire before are independen Poisson random variabes [67, pg. 77] wih respecive means ν A = λ g r dr ds = λ r g r dr = λ r g r dr λg s s g r dr ds = λ r g r dr + ν B = λ λ G s where g andg are he densiy and he compemenary disribuion funcions, respecivey, corresponding o Pareo disribuion. The noaion ν A is chosen o indicae ha i is he measure of Poisson poin process over he se A. imiary, ν B is for B. The condiion ha here is a eas one packe aive vioaes he independence of he wo pars of he acive sessions in a very specific way; heir oa mus be sricy posiive. Oherwise, we do he probabiiy cacuaions as in he uncondiiona case. The as sep is o assign he probabiiy ha no packes arrive in each session, which can easiy be found hrough he oca

11 compound Poisson process. For he even A, e α is he probabiiy of no packe arriva for each session and can be used in he cacuaion as he sessions and he oca packe arrivas are independen from he mode. For he even B, we need o know he expiraion imes of he sessions. I is aso we known ha for Poisson arrivas which depar he sysem afer a random amoun of ime as in an M/G/ queue, he deparure process is aso Poisson. ince we have furher spi he even B by condiioning on he number of sessions, he expiraion imes are now joiny disribued as order saisics over [,] [67]. Therefore, he probabiiy ha no new packe arrivas occur over m acive sessions is m α m m! α e α m dm dm d e Ke = m m α I m, = K 3 Le ρ denoe he probabiiy ha here is a eas one acive session a any ime, which can be found hrough he anaogy wih he seady sae sysem size of an M/G/ queue as ρ λ G = e where G denoes he mean of he Pareo disribuion. We can now wrie P A B a eas one acive session a ime = ρ e v A ν A n! e n= m= n α n e ν B ν B m! m I m, e v A e ν B 4 where he erm e e v A ν B is subraced o make sure ha here is a eas one acive session. Afer subsiuing 3 and simpifying, we ge v A ν B e ρ e [ exp[ ]exp[ / ] ] α v A e v B e α α The condiion ha a packe arriva occurred a ime has no impicaion on he even C. Therefore, hey are independen and we need o find he margina probabiiy P C. The number of session arrivas in [,] is Poisson wih mean λ. When a eas one arriva occurs in [,], he ime of expiraion of such sessions coud be wihin [,] or aer. However, we ignore hese exac arriva and deparure imes when considering he probabiiy of no packe arrivas over each session which we wrie as he Poisson probabiiy e α approximaey in

12 P C e λ + n= [ e λ n α n α λ / n!] e = exp[ λ e ] 5 We have acuay compared his expression wih a deaied version where he arriva and deparure imes of he sessions are aken ino accoun simiar o he anaysis of A and B. This yieds negigibe difference in he numerica resus. Tha is why ony he simpe formua 5 is repored above. Now, we can muipy PC wih he probabiiy in 4 as hey are independen and pu he expression for ρ and observe v A + v = λ o ge B G α α [ exp[ v A e ]exp[ v B e / α] ] λ G e α P{ T > } = exp[ λ e ] λ G e This can be differeniaed and negaed o find he probabiiy densiy funcion of T.. C. Inerarriva Times for Muipe Casses In his subsecion, we consider wo casses of raffic sreams arriving a a rouer. Le T i denoe he inerarriva ime of cass i packes, i=,. We wi derive he disribuion of he inerarriva ime beween a ype i and a ype j packe when boh ypes of packes arrive a he rouer, for i, j =,. Consider he even of consecuive arrivas of cass packes. More precisey, we wi need o consider he condiiona even ha a ype arriva is foowed by anoher ype arriva in he Markov chain formuaion of he nex secion. Given ha a ype arriva occurred and he nex arriva is again ype, he densiy of he ime uni he nex arriva is jus f T, which is he probabiiy densiy funcion of T. I can be found hrough he differeniaion of compemenary cumuaive disribuion funcion F T = P{ T > }. imiary, he densiy of he ime uni he nex arriva of ype given ha a ype arriva occurred is denoed by f T. We now find he cross inerarriva ime densiy for he arriva of a ype packe given ha a ype arriva occurred. If a ype packe arrived a he curren ime, his informaion has no impicaion on he number of acive sessions of cass. Then, we compue he compemenary probabiiy

13 P F = { no ype packes arrive in ime unis} 6 where we denoe by superscrip he fac ha he possibiiy of no ype sessions being acive is incuded in he derivaion of 6. In conras, he condiion ha an arriva occurred impies ha here is a eas one acive session, when a singe cass inerarriva ime disribuion is considered. Excep for his fac, he derivaion is very simiar o he singe cass case sudied in subsecion III-B. As a resu, we obain F v A v B α = e e exp[ λ e α ] exp[ α v A e ]exp[ v B e / α ] where ν A and ν B are defined anaogousy as in and. Noe ha ρ does no appear in he denominaor and here is no subracion of in he as erm as opposed o F since now boh m=n= is possibe in Equaion 4. We denoe he densiy funcion of he ime uni he arriva of a cass packe nex by f, which can be found hrough aking he derivaive of he compemenary disribuion funcion F. The use of hese densiy funcions in he Markov chain of he nex secion is as foows. Noe ha for a ransiion o occur from a cass arriva o a cass arriva; he even no ype packes arrive in ime unis mus occur, which has probabiiy F. Then, he probabiiy ha a ransiion from a sae invoving an arriva of ype o anoher sae aso wih an arriva of ype is found by using he fac ha he nex arriva wi occur a ime wih densiy f T and wih he condiion ha no cass packes arrive in he mean ime, which happens wih probabiiy F. Hence, we can use he produc f T F o cacuae he compee ransiion probabiiy from a given sae o anoher, when boh saes have an arriva of ype. Aong he same ines, he densiy f ges muipied wih F T o make sure ha a ype packe is foowed by a ype packe and he ime uni he nex arriva is. Oher combinaions foow simiary. Ahough i does no denoe a densiy funcion, we use he noaion ft ij o denoe a produc of a densiy and a compemenary probabiiy when a cass i packe is foowed by a cass j packe. Tha is, he noaion used beow is ft = f FT, ft = ft F, ft = ft F, ft = f FT 3

14 which are based on he independence of he wo casses of raffic sreams. IV. /M/ WITH TWO CLAE: NON-PREEMPTIVE PRIORITY ERVICE We consider a mode of wo queues based on G/M/ by aking ino accoun wo casses of sef-simiar inpu raffic denoed by /M/, and anayze i on he basis of prioriy wih no preempion. Le he service ime disribuion have rae and for ype and ype packes, respecivey, and e ype packes have prioriy over ype packes. A. /M/ wih Two Casses The usua embedded Markov chain [68] formuaion of G / M / is based on he observaion of he queueing sysem a he ime of arriva insans, righ before an arriva. A such insans, he number in he sysem is he number of packes ha arriving packe sees in he queue pus packes in service, if any, excuding he arriving packe isef. We specify he saes and he ransiion probabiiy marix P of he Markov chain wih he sef-simiar mode for wo ypes of raffic. Le { X n : n } denoe he embedded Markov chain a he ime of arriva insans. As he service is based on prioriy, he ype of packe in service is imporan a each arriva insan of a given ype of packe o deermine he queueing ime. Therefore, we define he sae space as: = { i, i, a, s : a { a, a}, s { s, s, I}, i, i Z + } where a, a are abes o denoe he ype of he arriva, s, s are abes o denoe he ype of he packe in service, i, i are he number of packes in each queue incuding a possibe packe in service, and I denoes he ide sae in which no packe is eiher in service or being queued. ome of he saes in he sae space given above have zero probabiiy. For exampe, i,, a, is impossibe. s The paricuar noaion is chosen for simpiciy, ahough he impossibe saes coud be excuded from. Each possibe sae, he reachabe saes from each and he corresponding ransiion probabiiies wi be expained in he seque. 4

15 B. aes of he Embedded Markov Chain The saes of he Markov chain and he possibe ransiions wih respecive probabiiies can be enumeraed by considering each case. We wi anayze he saes wih non-empy queues and hose wih a eas one empy queue a he ime of an arriva, separaey. aes i, i, a, wih i i and s I : s, We can divide he saes and ransiions ino 6 groups because a, s can occur x=4 differen ways, and he nex sae p, q can be composed simiary in 4 differen ways as a p { a, a } and s q { s, s }. We anayze ony wo exampes in deai; he ohers foow simiary.,, Transiion from i, i, a, s j, j, a, s Consider he case where a ransiion occurs from an arriva of ype o an arriva of ype such ha he firs arriva has seen a ype packe in service, i packes of ype in he sysem equivaeny, oa of queue and he packe in service and i packes of ype in he sysem in his case ony queue. The ransiion occurs o j packes of ype and j packes of ype in he sysem wih a ype packe in service. This ransiion is shown in Fig. 4. Due o prioriy scheduing, an arriva of ype can see a ype packe in service in he nex sae ony if a ype packes incuding he one ha arrived in he previous sae are exhaused during he inerarriva ime. Tha is why j can ake ony he vaue and exacy i packes of ype are served. In conras, he number of packes served from queue +, say k, can be anywhere beween and i as a eas one ype packe is in he sysem, one being in service, when a new arriva occurs. The ransiion probabiiy is P{ X n + =, i k, a, s X n = i, i, a, s} = P { i + served from ype, k served from ype and a ype packe remains in service during T } where we use he fac ha he remaining service ime of a ype packe in service has he same exponenia disribuion Exp, due o he memory-ess propery of a Markovian service and we denoe he inerarriva ime beween a ype and ype arriva by T. Therefore, for k =, K, i 5

16 P{ X n + =, i k, a, s X n = i, i, a, s} = x f s f i + k x f + T dsdxd where Z m : sum of independen service imes of ype m packes, m=,, +. Noe ha m has an Erang disribuion wih parameers, as each service ime has an exponenia disribuion, and he sum + being m he sum of severa exponeniay disribued random variabes has a hypoexponenia disribuion. The numerica evauaion of hese densiy funcions is discussed in he foowing secion. Transiion from i, i, a, s j, j, a, s This is he case where a ransiion occurs from an arriva of ype o an arriva of ype such ha he firs arriva has seen a ype packe in service, i packes of ype in he sysem equivaeny, oa of queue and he packe in service and i packes of ype in he sysem in his case ony queue. The ransiion occurs o j packes of ype and j packes of ype in he sysem wih a ype packe in service. This ransiion is shown in Fig. 5. An arriva of ype sees a ype packe in service in he nex sae, which indicaes ha no ype packes has been served during his ransiion due o prioriy scheduing. In conras, he number of packes served from queue, say k, can be anywhere beween and i as a eas one ype packe is in he sysem, he one in service, when a new arriva occurs. The ransiion probabiiy is P{ X n + = i k +, i, a, s X n = i, i, a, s} = P{k served from ype, no packe served from ype and a ype packe remains in service duringt } = f s f k x f T x dsdxd The above wo ransiions are summarized beow. 6

17 Iniia ae Reachabe ae m =, Transiion Probabiiy s i, i, a,, i k, a,, =,, i s k K x f s f i + k x f + T dsdxd i, i, a, s i k +, i, a, s, k =,... i imiary, we can enumerae a 6 cases. f s f k x f T x dsdxd aes i, i, a, wih i or i equa o or s = I : s The saes when one queue is empy i.e. = i or i = or when boh queues are empy and he sysem is ide, i.e. i = i =, s = can be considered simiary. There are a oa of 8 such saes. The deais can be found in [69]. I C. Limiing Disribuion and Qo Parameers The seady sae disribuion π as seen by an arriva is obained by sovingπ P = π, where P is he ransiion marix of he Markov chain anayzed above. In pracice, he queue capaciy is imied in a rouer. o he Markov chain is finie and he seady sae disribuion exiss. Consider a finie sae sysem wih queue capaciy n. In a finie sysem, an arriva can occur a a fu queue described by he saes of he ype n, k, a, s m and k, n, a, s m. In hese cases, he queue is fu and he arriving packe is dropped. The ransiions for hese saes are he same as hose from a queue which has ony one vacan posiion ha is fied by he arriving packe, since in he aer, he arriving packe is queued, and he sae is now idenica o he fuqueue case. Thus, he ransiions for n, k, a, s m are he same as hose for n-, k, a, s m and simiary for k, n, a, s m. To he bes of our knowedge, no previous anayica expressions are avaiabe for he waiing ime of a G/M/ queue wih prioriy. Our anaysis reies on he imiing disribuion of he sae of he queue a he arriva insances, which can be compued using he anaysis given above for our sef-simiar raffic mode. In genera, he foowing anaysis is vaid for any G/M/ queueing sysem where he imiing disribuion π a he arriva insances can be compued. The expeced waiing ime for he high prioriy queue can be found as 7

18 W = J J j = j = j π j, j, a, s + J J j = j = j + π j, j, a, s where J and J are he respecive capaciies of each queue. This foows ceary from he fac ha an arriving packe of higher prioriy wi wai uni a packes of he same prioriy as we as he packe in service are served. Depending on he ype of he packe in service, we have he consiuen expressions in he sum. On he oher hand, we obain he expeced waiing ime for he ow prioriy queue by anayzing he evens ha consiue his deay. The amoun of work in he sysem a any ime is defined as he random sum of a service imes ha wi be required by he packes in he sysem a ha insan. The waiing ime of a ype packe can be wrien as: W = Z + Z + Z where Z is he amoun of work seen by he arriving packe in he sysem, Z is he amoun of work associaed wih high prioriy i.e.ype packes arriving during Z, Z 3 is he amoun of work associaed wih ype packes arriving during Z, and so on. As iusraed in Fig.6, he waiing ime of an arriving packe of ype is indeed given by he oa workoad buiding in fron of i. The arrows in he figure denoe he arriva imes of ype packes, and a he obique ines have 45 degrees ange wih he ime axis. In his figure he waiing ime is W = Z + Z + Z + Z as an 3 4 exampe. Le M j denoe he number of ype j arrivas over Z i, j=,,. Then W = Z + + M M + L where M j denoes he random sum of M j independen service imes of ype packes. Then, E [ W ] = E[ Z ] + E[ ] E[ M ] + E[ ] E[ M ] + L since he service imes and he arriva process are independen. For a saionary packe arriva process, we ge E M ] = E[ E[ M Z ]] = E[ c Z ] = c E[ Z [ j j j j j ] 8

19 9 due o menioned independence, where > c is a consan paricuar o he arriva process. Tha is, expecaion of he number of arrivas in any period of ime is proporiona o he engh of ha period because of saionariy in ime and ineariy of expecaion. In our saionary sef-simiar raffic inpu process, c is he expeced number of arrivas per uni ime which can be caed he arriva rae, given by he produc of he arriva rae of session arrivas, he arriva rae of packes over a session, and he expeced session engh [6]. Expiciy, / = δ λαδ b c. Hence, he expeced waiing ime reduces o + L + + = ] [ ] [ ] [ ] [ ] [ [ ] Z E c E Z E c E Z E W E ] [ ] [ ] [ ] [ ] [ W E c Z E Z E Z E c Z E + = = L in view of 7. Therefore, we ge,,,,,, π π W c s a j j j j s a j j j j W J j J j J j J j = = = = = which impies ha he raffic inensiy c mus be ess han. Anoher Qo parameer readiy avaiabe from his descripion of he sysem is he packe oss rae PLR due o a fu queue or equivaeny he sysem avaiabiiy. For each cass of raffic, his is he sum of he seady-sae probabiiies of saes where an arriva occurs for a fu queue: = = =,,,, J k m s m a k J PLR π V. NUMERICAL ANALYI In his secion, we presen a numerica exampe demonsraing he appicaion of he above anayica framework. We firs noe ha numericay soving he queueing sysem modeed in he previous secion amouns o cacuaing he

20 ransiion probabiiies of he corresponding Markov chain i.e. generaing he ransiion probabiiy marix P. The seady-sae disribuionπ hen can be obained by soving he ef-eigenvaue sysem π π = P. Consider he inegras given in ecion IV.B for finding he enries of P. Every ransiion probabiiy may be direcy or indirecy cacuaed from an inegra of he form: + x T dsdxd f x f s f mn k 8 where k=,, =,,J, =,,J and m, n =,. Here J and J are he respecive capaciies of queue and queue. The erm x f + in he inegra above is a hypo-exponenia disribuion. I is he densiy funcion of he service ime of packes of ype and packes of ype. I is he sum of wo Erang disribuions and is densiy funcion can be obained by convouion on he densiy funcions of he wo Erang disribuions, namey: + = = = x s x x s x s x ds e s x s e ds e s x e s ds s x f s f x f!!!! Noe ha if = =, hen we assume ha x x f δ = +, he Dirac-dea funcion. Thus, he generic ransiion probabiiy inegra 8, above reduces o:!! d dx ds e s x s e f e e x s x T x k mn k We firs noe ha for a sysem wih a finie queue capaciy, N = max J, J, he Markov chain formuaion eads o a sae space of size 4N + 4N + and hus we have a Markov marix, P, wih ON 4 eemens. However, here are ony ON disinc vaues of 8. Thus, a significan compuaiona saving can be obained by pre-compuing a ON vaues and fiing ou he O N 4 eemens of he Markov marix using hem. To obain he resus described beow, we se each queue o a capaciy of packes and packe arrivas occur according o he process described in secion III. For he higher prioriy cass we se he session arriva rae o λ =8s -, he in-session packe arriva rae o α = 5s - characerisic of VoIP raffic and he service rae o = 5s -. For he

21 ower prioriy cass we se he session arriva rae o λ =5s -, he in-session packe arriva rae o α = 8s - and he service rae o =. In he foowing secions, we invesigae he effecs of varying he Hurs parameer.5 < H < on he deay and packe oss rae Qo parameers. For numerica accuracy, we have performed some evauaion experimens o verify ha we obain a sochasic marix. Whie performing a numerica check of he Markov ransiion marix, we have found ha he sum of he ransiion probabiiies of each row of he marix is, giving evidence ha he marix P is indeed sochasic. In fac, he recommended defau queue size by Cisco for prioriy queueing impemenaion, paricuary for rea ime appicaions such as voice is [7]. Ahough he compuaion of P seems o be somewha cosy, i is cerainy possibe o sove a sysem wih packes in a reasonabe amoun of ime. To show he pracicabiiy of he approach, here we give some iming informaion. Compuing a compee row of P for he smaer vaued saes ike 3, 4, a, s akes around 6 sec and for higher vaued saes such as 8, 8, a, s akes abou -5 min in MATLAB, which can be performed ceary in parae. The running ime for a 3-packe sysem is ess han minues and for a sysem wih queues packes each, compuing P akes up o 3 hours depending on he vaue of H in MATLAB wihou any opimizaion. This ime coud be reduced remendousy if direcy coded for exampe in a anguage such as C by eiminaing he overhead ime caused by he oos of MATLAB. On he oher hand, much effor has been dedicaed o sove for he saionary disribuion of arge Markov chains over he recen years. The curren sae of he ar enabes soving a Markov chain wih a biion saes using ieraive mehods [7]. VI. IMULATION REULT In his secion, we expain he simuaion resus and presen a comparison wih he numerica anaysis, which serves o vaidae he anayica modeing. Firs of a, we provide some comprehensive deais abou simuaion framework foowed by accuracy consideraions and comparison of simuaion and numerica resus. A. imuaion Framework A comprehensive discree-even simuaor for queueing sysems was bui o undersand and evauae he Qo behaviour of sef-simiar raffic. The simuaion engine is highy moduar by design aowing free cusomizaion of he raffic generaor and he scheduing ogic. This aows for he ready evauaion of any scheduing discipine under any specific kind of inpu raffic.

22 The key eemen for he scheduer ogic is he cheduer cass. Here we used he empae mehod design paern [7]. This aows any scheduing agorihm o be oosey couped bu easiy inegraed, overriding he exising program skeeon. Prioriycheduer was acuay impemened o anayse he corresponding Qo behaviour. A raffic generaor was aso wrien, which impemens he raffic mode described in ecion III. This generaor may aso be readiy over-ridden by anoher raffic mode. A number of oher associaed casses were wrien o faciiae program funcion and accuracy. These incude: imuaion. This cass served as he simuaion engine moving ime forward and updaing he even is ec. RandomNumber. A cass for generaing random number wih specific disribuions incuding: uniform, exponenia, Poisson, Compound-Poisson and Pareo. Packe. A cass used o sore he sysem sae as encounered by each packe. Addiionay, a speciais numerica agorihm [73] was impemened for compuing he variance o comba he numerica insabiiy in he aggregaion of he Qo saisics. The Qo resus from he simuaion sudies aong wih heir corresponding heoreica vaues are presened in he nex subsecions. B. Accuracy Consideraions Geing accurae resus from simuaing he raffic mode discussed in ecion III requires aenion. The numbers of packes are direcy simuaed raher han he iner-arriva ime disribuions. We discuss he reaed issues here. One issue arises from he infinie pas assumpion of he raffic mode in ecion III. This assumpion is necessary o guaranee saionariy. In simuaion however, we are forced o repace wih a sufficieny arge negaive number say T <. In [6], he expeced error difference in he number of packes generaed over a given inerva is anayzed, due o he runcaion of he infinie pas o T. For raffic generaed on [, ] we have: δ δ λb α T δ δ δ + O T NB: The expeced error is arger for he highy sef-simiar version of he raffic mode. As H approaches from beow, δ approaches from above and he expeced error becomes very arge. δ

23 Noe aso ha for a given consan runcaion poin T, he error increases ineary as we increase he inerva under consideraion [, ]. This woud indicae ha a shorer simuaion inerva is desirabe in erms of raffic mode accuracy. However, he heoreica vaues obained from he anayica modeing represen he Qo parameers of he sysem whie in seady-sae. I is ikey ha he queueing sysem does no reach seady-sae for sma vaues of. Thus, for he idea choice of, here is a rade-off beween raffic mode accuracy and reaching he sysem seadysae. This rade-off was addressed by quaiaive observaions in his work. Furher invesigaion ino he accuracy of he simuaion is ikey o be of ineres. Anoher issue arises from he difficuy of simuaing heavy-aied disribuions in genera and he Pareo disribuion in paricuar. ession duraions in our raffic mode are governed by a Pareo disribuion. Thus being abe o accuraey generae Pareo disribued numbers is imporan o he accuracy of he simuaion sudy. Figure 7 shows expeced heoreica mean vaue of he Pareo disribuion versus he vaues acuay obained from random number generaion experimens. 95% confidence inervas are aso shown. For each of H=.55,.75 and.95 we show he heoreica mean and 5 poins showing he experimena mean. Each Experimena Mean poin in Fig. 7 represens he saisica aggregae mean for approximaey he same number of random number generaions RNG as in he simuaion resus presened foowing > 5, and so he anaysis here has direc reevance o he resus. We can ceary see he exremey high variance in he daa as H approaches. In fac, for H=.95 severa poins are no shown because hey were we-off he graph. This is a direc consequence of he infinie variance of he Pareo disribuion. The probem is paricuary acue for H cose o as he ai of he disribuion is heavies and we are more ikey o see exremey arge vaues generaed by he RNG. As he above discussion shows i is very difficu o obain accurae resus from a simuaor generaing random numbers from a very heavy-aied disribuion. The ai is heavies for H cose o and generaing accurae simuaion resus proves o be paricuary difficu in ha range. Gross e a. sudy a reaed issue in deai in [7] and concude ha care mus be aken in simuaions invoving Pareo disribuions as hey can ead o arge errors due o he heavy ai. I shoud aso be noed hough, ha he buk of empirica evidence [, 8-9, 74-75] suggess ha H ~ [.7,.85] is he region of ineres in nework raffic. Ergo i is his range of vaues of H ha are of primary ineres in he foowing resus and no he vaues very cose o jus discussed. 3

24 Fig. 8 shows a comparison of he numerica and simuaion resus for he packe oss rae. The resus appear o vaidae he modeing. We noe he significan increase in he Packe Loss Rae of he ower prioriy queue as he degree of sef-simiariy increases. VII. TET BED IMPLEMENTATION ON CICO 84 ERIE MODULAR ROUTER In his secion, we describe he inerim resus of he IP Qo ess running non-preempive prioriy scheduing on a Cisco Moduar Rouer 84 and presen a comparison wih he numerica and simuaion resus given in he previous secions. A. Tes Bed Descripion A Cisco 84 Moduar Rouer wih Cisco Qo feaures running Cisco IO.4 was conneced o wo Linux worksaions hrough dedicaed Mbps Eherne inks as shown in Fig. 9. We impemened a raffic generaor on he ender worksaion, which simuaneousy generaed wo differen sef-simiar raffic sreams over UDP. We impemened wo sinks INK and INK on he Receiver worksaion o receive he wo differen casses of raffic on differen pors. B. Cisco 84 Rouer Configuraion wih Prioriy Queueing We impemened Prioriy Queueing in a Cisco Moduar Rouer 84 o provide differenia reamen o he differen casses of sef-simiar raffic. Prioriy Queueing s mos disincive feaure is is scheduer. I suppors a maximum of four queues: High, Medium, Norma and Low. If he High queue aways has a packe waiing, he scheduer wi aways serve he packes from his queue. On he oher hand, if he High queue does no have a packe waiing, bu he Medium queue does, one packe is aken from he Medium queue and hen he process sars over a he High queue. The ow queue ony ges service if he High, Medium, and Norma queues do no have any packes waiing [7]. Any number of queues ou of four can be configured on an inerface; he scheduer simpy serves hese configured queues and skips ohers. As we have wo kinds of raffic, we ony configured wo queues; High and Medium a he oupu inerface Fa/. As shown in Fig. 9, here are wo inerfaces Fa/ inpu inerface and Fa/ oupu inerface. We need o cassify differen kinds of raffic a he inpu inerface and assign hem o he proper queue a he oupu inerface on he basis of desinaion por number. We briefy cover he configuraion seps here: 4

25 We defined he prioriy is, cassified he raffic a inpu inerface Fa/ and assigned hem o he proper queue a he oupu inerface Fa/ by execuing he foowing commands: prioriy-is prooco ip high udp 63 prioriy-is prooco ip medium udp 63 Nex we specified he maximum size of each queue a he oupu inerface: prioriy-is queue-imi 6 8 Finay we assigned he prioriy is o he oupu inerface Fa/ by execuing he foowing command. prioriy-group C. Time ynchronizaion beween ending and Receiving Machine In order o obain an accurae measure of he one-way deay hrough he nework, he cocks on he sending and receiving machines had o be synchronized. Nework Time Prooco NTP [76] was used for his purpose, as i mees our accuracy requiremens and here are numerous readiy avaiabe impemenaions. To have accurae ime synchronizaion beween he sending and receiving machine s cocks and no o inerrup wih he sef-simiar raffic passing hrough he rouer, we used dedicaed Eherne pors over a cross-over cabe for he NTP connecion. We assigned an IP address o he sending machine s eherne card and an IP address o he receiving machine s eherne card as deaied in Figure 9. An NTP primary server, or sraum, was conneced o a high precision reference cock and equipped wih NTP sofware. Oher compuers sraum s, equipped wih simiar sofware auomaicay queried he primary server o synchronize heir sysem cocks. We made he sending machine as he NTP primary server in our nework. The NTP primary server was conneced o a high precision reference cock au.poo.np.org o synchronize is sysem s cock. Then we execued he foowing command on he receiving machine which was acing as NTP cien in he nework and aso equipped wih NTP sofware o synchronize is sysem cock wih he primary NTP server: npdae u Furher, o achieve rea ime synchronizaion beween he sender and receiver s cocks, a sma program was wrien, o enabe NTP o run as a background process. We execued he foowing command on he rouer which is aso he 5

26 NTP cien in our nework in he goba configuraion mode o synchronize is sysem cock wih he NTP primary server: np server D. Measuremen of Queueing Deay for Muipe Casses of ef-imiar Traffic A packes in a nework experience deay from when he packe is firs ransmied o when i arrives a is desinaion. Fig. 9 shows he differen kinds of deay a packe experiences from source o desinaion. We expain hem here, briefy: eriaizaion Deay: is he ime i akes o encode he bis of a packe on o he physica inerface and can be cacuaed by dividing he number of bis sen by ink speed. Propagaion Deay: is he ime i akes a singe bi o ge from one end of he ink o he oher and can be cacuaed by using he formua: inkengh 8. m / s 3 Processing Deay: refers o he ime aken by he rouer o examine he packe a he inpu inerface and pacing i in he oupu queue on he oupu inerface 4 Queueing Deay: consiss of ime spen in he queues inside he rouer ypicay jus in oupu queues in a rouer. 5 Transmission Deay: is he deay ha he scheduer akes o pu he packe from oupu queue on o he ink; i is same as seriaizaion deay [7]. In our deay cacuaions, we can ignore he processing deay inside he inpu inerface of he rouer and a he receiving machine as his is in order of few microseconds, severa orders of magniude smaer han he expeced deay. The propagaion deay hrough he nework is aso negigibe and herefore ignored. Compensaing for he seriaizaion deay a he sending machine and ransmission deay a he oupu inerface of he rouer, we found he foowing queueing deay for he wo differen casses of sef-simiar raffic in our es bed experimens Refer o Tabe. Fig. shows he mean deay, in which he es bed resus have been poed wih 95% confidence inerva agains numerica and simuaion resus. We see he significan derimena impac of increasing he Hurs parameer he degree of sef-simiariy on he Qo offered. We aso noe he characerisics of a prioriy queueing sysem: as he oad increases, we see a significan 6

27 increase in he deay for he ower prioriy queue. The sigh difference beween es bed and numerica resus is ikey due o congesion a he NIC of he Receiver worksaion, paricuary when sef-simiariy increases. VIII. APPLICATION OF THE MODEL Here we briefy presen he prime appicaions of he mode. Wih he remendous growh in daa raffic, he eecommunicaion indusry is evoving is core neworks owards IP echnoogy. An a-ip Differv mode is widey considered o be he mos promising archiecure for guaraneed Qo provisioning in NG wireess neworks. This is argey due o is scaabiiy, mobiiy suppor and he abiiy o iner-nework heerogeneous radio access neworks [77]. To ranspor UMT services hrough IP neworks wihou oosing end-o-end Qo provisioning, an accurae and consisen Qo mapping is required. According o 3GPP, UMT-o-IP Qo mapping is performed by a ransaion funcion in he GGN rouer ha cassifies each UMT packe fow and maps i o a suiabe IP Qo cass [78]. Being abe o accuraey mode he end-o-end behaviour of differen casses of IP raffic conversaiona, sreaming, ineracive and background passing hrough a Differv domain is essenia o he guaraneed deivery of various Qo parameers. evera queueing oos have been deveoped ha can be impemened in IP rouers wihin differen Qo domains incuding Prioriy Queueing PQ, Cusom Queueing CQ, Weighed Fair Queueing WFQ, Cass Based Weighed Fair Queueing CBWFQ and Low-Laency Queueing LLQ [7]. This paper specificay considers he Qo behaviour of PQ. Work on he oher oos is ongoing. Our mode is direcy appicabe o he probem of deermining he end-o-end queueing behavior of IP raffic hrough boh Wired and wireess IP domains. Modeing accuracy is mos crucia hough, in resource-consrained environmens such as wireess neworks. For exampe, our mode is direcy abe o anayze he behavior of differen Qo casses of UMT raffic which have been proven saisicay sef-simiar and ong-range dependen passing hrough a Differv domain, in which he rouers impemen prioriy queueing. The mode enabes igher bounds on acua behaviour so ha over-provisioning can be minimized. I aso enabes ransaions of raffic behaviour beween differen kinds of Qo domains so ha i is possibe o map reservaions made in differen domains o provide session coninuiy. We have joiny considered raffic engineering and Qo issues. The fundamena hemes of his sudy span raffic modeing, sochasic anaysis and nework design. I aso provides significan insigh and guidance for he design of NG-IP based neworks. 7

28 IX. CONCLUION AND FUTURE WORK In his paper, we have conribued o he accurae modeing of wireess IP raffic behavior, by presening a nove anayica mode based on a G/M/ queueing sysem under differen casses of sef-simiar inpu raffic. We have anayzed i on he basis of non-preempive prioriy and derived expici expressions for he expeced waiing ime and packe oss rae for muipe casses. The accuracy of he mode is demonsraed by comparing he numerica souion of he anayica modeing o simuaion experimens and he acua es-bed resus. The presen sudy can be used as a guide for he efficien aocaion of buffer space and bandwidh for individua raffic casses wih he aim of guaraneeing he Qo required by differen appicaions whie minimizing excessive aocaion. Furher, he mode represens an imporan sep owards he overa aim of undersanding reaisic under sef-simiar raffic end-o-end Qo behaviour in erms of Qo parameers such as deay, jier and hroughpu of muipe raffic casses passing hrough heerogeneous wireess IP domains Inerv, Differv and MPL. Our fuure work wi anayze he Qo performance of differen domains impemened wih differen queueing discipines such as CQ, LLQ and CBWFQ. We pan o deveop various modes for prioriy, poing and he combinaion of poing and prioriy sysems and use ieraive mehods o sove he Markov chains. ACKNOWLEDGEMENT The deaied review and commens of an anonymous referee are greay appreciaed. The auhors woud aso ike o hank Khaid Hameed and Adee Baig for heir grea hep in he es bed impemenaion and Rashid Mehmood for his vauabe commens on he numerica issues of a Markov chain. REFERENCE [] W. Leand, M. Taqqu, W. Wiinger and D. Wison, On he sef-simiar naure of Eherne raffic exended version, IEEE/ACM Transacions on Neworking, vo.. no., pp. -5, Feb [] V. Paxon, Empiricay derived anayica modes of wide-area TCP connecions, IEEE/ACM Transacions on Neworking, vo., pp , Aug. 994 [3] V. Paxon and. Foyd, Wide-area raffic: he faiure of Poisson modeing, in Proc. ACM IGCOMM 94, London, U.K., Aug. 994, pp [4] P. Danzig,. Jamin, R. Caceres, D. Mize and D. Esrin, An empirica workoad mode for deriving wide-area TCP/IP neworks simuaions, Inerneworking: Research and Experience, 3, pp. -6, March, 99. [5] H. Fower and W. Leand, Loca area nework raffic characerisics, wih impicaions for broadband nework congesion managemen, IEEE JAC, 97, pp , epember, 99 8

29 [6] R. Gusea, A measuremen sudy of diskess worksaion raffic on he Eherne,, IEEE Transacions on Communicaions, 389, pp , epember 99 [7] R. Jain and. Rouhier, Packe Trains Measuremens and a new mode for compuer nework raffic, IEEE JAC, 46, pp , epember 986 [8] M. Crovea and A. Besavros, Expaining Word Wide Web Traffic ef-imiariy, Tech. Rep. TR-95-5, Boson Universiy, C Dep, Boson, MA 5, Aug. 995 [9] M. W. Garre and W. Wiinger, Anaysis, Modeing and Generaion of ef-imiar VBR Video Traffic, ACM Compuer Communicaion Review, vo. 4, Oc. 994, IGCOMM 94 ymposium [] W. Wiinger e a, aisica anaysis of CCN/7 raffic daa from working CC subneworks, IEEE. Journa on eeced Areas of Communicaion, vo., no. 3, pp , Apr. 994 [] M. Crovea and A. Besavros, ef-imiariy in Word Wide Web Traffic: Evidence and Possibe Causes, in ACM igmerics, May 996 [] J.C Boo and M. Grossgauser, On he Reevance of Long-Range Dependence in Nework Traffic, Compuer Communicaion Review, vo. 6, no. 4, pp. 5-4, Ocober 996. [3] Z. L. Zhang, V. Ribeiro,. Moon and C. Dio, ma-time caing behavior of inerne backbone raffic: An Empirica udy, in IEEE INFOCOM, march 3 [4] M.. Taqqu, ef-imiar processes. In. Koz and N. Johnson, ediors, Encycopedia of aisica ciences, vo. 8, pp Wiey, New York, 988. [5] W. Wiinger, M. Taqqu and A. Erramii, A bibiographica guide o sef-simiar raffic and performance modeing for modern high speed neworks, In F. P. Key,. Zachary and I. Ziedins, ediors, ochasic Neworks: Theory and Appicaions, pp , Caredon Press, Oxford, 996 [6] H. Homa and A. Taskaa, WCDMA for UMT, Radio Access for Third Generaion Mobie Communicaions, nd Ediion, John Wiey & ons, Ld., pp. -5 [7] J. Yang and I. Kriaras, Migraion o a-ip based UMT neworks, IEEE s Inernaiona Conference on 3G Mobie Communicaion Technoogies, 7-9 March,, pp. 9-3 [8] W. aings, Inegraed ervices Archiecure: The Nex-Generaion Inerne, Inernaiona Journa of Nework Managemen, 9, 999, pp [9]. Bake e a., An Archiecure for Differeniaed ervices, IETF RFC 475, Dec. 998 [] Rosen E. e a., Muiprooco Labe wiching MPL Archiecure, RFC 33, Jan. [] K. Venken, J. De Vriend and D. De Veeschauwer, Designing a Differv-capabe IP-backbone for he UTRAN, IEEE nd Inernaiona Conference on 3G Mobie Communicaion Technoogies, 6-8 March, pp []. Maniais, C. Grecas and I. Venieris, End-o-End Qo Issues Over Nex Generaion Mobie Inerne, IEEE ymposium on Communicaion and Vehicuar Technoogy,, VCT-, 9 Oc,, pp [3] P. Newman, Neiion Inc. In earch of he A-IP Mobie Nework, IEEE Communicaion Magazine, vo. 4, issue, Dec. 4, pp

30 [4] G. Aranii, F. Caabro, A. Iera, A. Moinaro and. Puiano, Differeniaed ervices Qo Issues in Nex Generaion Radio Access Nework: a New Managemen Poicy for Expedied Forwarding Per-Hop Behavior, IEEE Vehicuar Technoogy Conference, VTC 4-Fa, vo. 4, 6-9 ep. 4, pp [5]. Uskea, A IP Archiecures for Ceuar Neworks, nd Inernaiona Conference on 3G Mobie Communicaion Technoogies, 6-8 March, pp [6] Jeong-Hyun Park, Wireess Inerne Access for Mobie ubscribers Based on GPR/UMT Nework IEEE Communicaion Magazine, vo. 4, issue 4, Apri, pp [7] K. Danie Wong and Vijay K. Varma, upporing Rea-Time IP Muimedia ervices in UMT, IEEE Communicaion Magazine, vo. 4, issue, Nov. 3, pp [8] 3GPP, Universa Mobie Teecommunicaion ysem UMT; Qo Conceps and Archiecure, T3.7V6, March 4 [9] R. Chakravory, J. Carwrigh and I. Pra, Pracica Experience wih TCP over GPR, in IEEE GobeCom, Nov. [3] D. chwab and R. Bun, Characerizing he use of a Campus Wireess Nework, in IEEE INFOCOM, March 4 [3] X. Meng,. Wong, Y. Yuan and.lu, Characerizing Fows in Large Wireess Daa Neworks, in ACM Mobicom, ep 4 [3] A. Baachandran, G. M. Voeker, P. Bah and P. Venka Rangan, Characerizing user behavior and nework performance in a pubic Wireess LAN, igmerics Performance Evauaion. Review, vo. 3. no.,, pp [33] T. Henderson, D. Koz and I Abyzov, The changing usage of a maure campus-wide wireess nework, in Proc. ACM Mobicom, ACM Press, pp. 87-, epember, 4 [34] J. Ridoux, A. Nucci and D. Veich, eeing he difference in IP raffic: Wireess versus Wireine, in Proc. of IEEE InfoCom 6 [35] Y. Zhou and H. ehu, Performance of shared oupu queueing in ATM swiches under sef-simiar raffic, in Proc. of Appied Teecommunicaion ymposium, Washingon, D.C., UA, Apri 6-, [36] A. Erramii, O. Narayan and W. Wiinger, Experimena queueing anaysis wih ong-range dependen packe raffic, IEEE/ACM Transacions on Neworking, vo. 4, no., pp. 9-3, Apri 996 [37] M. Zukerman e a, Anayica Performance Evauaion of a Two Cass Differv ink, IEEE IC, 5-8 Nov., vo., pp [38]. Kasahara, Inerne raffic modeing: A Markovian approach o sef-simiar raffic and predicion of oss probabiiy for finie queues, IEICE Transacions on Communicaions: pecia Issue on Inerne Technoogy, vo. E84-B, no. 8, pp. 34-4, Augus [39] H. Yousefi zadeh, A neura-based echnique for esimaing sef-simiar raffic average queueing deay, IEEE Communicaions Leers, 6, pp. 49-4, [4] J. M. Chung, Z. Quan, Impac of ef-imiariy on Performance Evauaion in Differv Neworks, IEEE MWCA, 4-7 Aug., vo., pp [4] B. Tsybakov and N. D. Georganas, ef-imiar raffic and upper bounds o buffer overfow in ATM queue, Performance Evauaion, 36, 998, pp

31 [4] A. Adas and A. Mukherjee, On Resource Managemen and Qo guaranees for ong-range dependan raffic, In Proc IEEE INFOCOM, 995, pp [43] M. Paruekar and A. Makowski, Tai Probabiiies for a Muipexer wih sef-simiar inpu, In proc IEEE INFOCOM, 996, pp [44] I. Norros, A orage Mode wih sef-simiar inpu, Queueing ysem, 6, 994, pp [45] C. F. Chou e a, Low Laency and efficien packe scheduing for sreaming appicaions, IEEE ICC, -4 June, 4, vo. 4, pp [46] A. Kos and B. Kepec, Performance of VoIP appicaions in a simpe Differeniaed ervices nework archiecure, IEEE EUROCON, 4-7 Juy,, vo., pp. 4-7 [47] J. M. Chung and H. M. oo, Anaysis of non preempive prioriy queueing of MPL neworks wih Buk arrivas, IEEE MWCA, 4-7 Aug., vo. 3. pp [48] ai. Kanhere and Harish ehu, Fair, Efficien and Low-Laency Packe cheduing using Nesed Defici Round Robin, Proceedings of he IEEE Workshop on High Performance wiching and Rouing HPR, May [49] N. F. MIR and A. Chien, imuaion of Voice over MPL communicaion neworks, IEEE ICC, 5-8 Nov., vo., pp [5] Y. Koucheryavy, A. Krednze,. Lopain and J. Harju, Performance esimaion of UMT reease 5 IM-subsysem eemens, 4 h Inernaiona Workshop on Mobie and Wireess Communicaion Neworks, IEEE MWCN, pp , 9-, epember, [5] Z. hao and U. Madhow, A Qo framework for heavy-aied raffic over he wireess Inerne, in proc. of MILCOM, vo., pp. -5, 7- Oc. [5] I. Norros, The managemen of arge fows of connecioness raffic on he basis of sef-simiar modeing, IEEE Inernaiona Conference on Communicaions, vo., pp , 8- June, 995 [53] A. Kemn, C. Lindemann and M. Lohmann, Traffic modeing and characerizaion for UMT neworks, IEEE Gobecom, vo. 3, pp Nov. [54] M. Jiang, M. Nikoic,. Hardy and L. Trajkovic, Impac of ef-imiariy on Wireess Daa Nework Performance, IEEE ICC,, vo., pp [55] D. aehe, K. Leibniz and P. Tran-Gia, ource raffic modeing of wireess appicaions, Research Repor eries No. 6, Insiue for Informaik, Universiy of Wurzburg, Germany, June [56] D. aehe, K. Leibniz and K. Tsipois, Qo of Inerne wih GPR, Research Repor eries No. 83, Insiue of Compuer cience, Universiy of Wurzburg, Germany, January [57] J. Ridoux, A. Nucci and D. Veich, Characerizaion of Wireess Traffic based on emi-experimens, Technica Repor- LIP6, December 5 [58] Z. ahinogu and. Tekinay, On Muimedia Neworks: ef-imiar Traffic and Nework Performance, IEEE Communicaion Magazine, vo. 37, issue, Jan. 999, pp [59] I. Norros, On he use of Fraciona Brownian Moion in heory of connecioness neworks, IEEE Journa on eeced Areas in Communicaions, vo. 3. no. 6, Augus 995, pp

32 [6] P. Benko, G. Maicsko and A. Veres, A Large-scae, passive anaysis of end-o-end TCP Performances over GPR, in IEEE INFOCOM, March 4 [6] M. Cagar, A Long-Range Dependan Workoad Mode for Packe Daa Traffic, Mahemaics of Operaions Research, 9, 4, pp. 9-5 [6] H. P. chwefe, L. Lipsky, Impac of aggregaed sef-simiar ON/OFF raffic on deay in saionary queueing modes exended version, Performance Evauaion, 43,, pp. 3- [63] M. Ifikhar, B. Landfed and M. Cagar, An anayica mode based on G/M/ wih sef-simiar inpu o provide end-o-end Qo in 3G neworks, in proc. of Mobiwac 6 IEEE/ACM MWIM 6, pp. 8-89, Torremoinos, Maaga, pain Ocober nd, 6 [64] G. Fay, F. Roueff, P. ouier, "Esimaion of he memory parameer of he infinie-source Poisson process", Bernoui, 3, 7, [65] I. Kaj, Limiing fraca random processes in heavy-aied sysems, In Fracas in Engineering, New Trends in Theory and Appicaions, Eds.J. Levy-Lehe, E. Luon, pringer-verag London, 5, pp [66] I. Kaj and M.. Taqqu, Convergence o fraciona Brownian moion and o he Teecom process: he inegra represenaion approach. In Braziian Probabiiy choo, h Anniversary Voume, Eds. M.E. Vares, V. idora Vicius, 7. [67]. M. Ross, Inroducion o Probabiiy Modes,, Academic Press, an Diego. [68] E. Cinar, Inroducion o ochasic Processes, 975, pp. 78 [69] M. Ifikhar and B. Landfed, An Anayica Mode Based on G/M/ wih ef-imiar Traffic Inpu and Non-Preempive Prioriy ervice Discipine, Technica Repor, ANRG, choo of IT, Universiy of ydney, Nov. 7. [7] W. Odom and M. J. Cavanaugh, IP Teephony ef-udy Cisco DQo Exam Cerificaion Guide, Cisco Press, 4, pp [7] R. Mehmood. Disk-based echniques for efficien souion of arge Markov chains, PhD Thesis, choo of Compuer cience, Universiy of Birmingham, Birmingham, UK. Ocober 4. [7] D. Gross, J. hore, M. Fischer and D. Masi, Difficuies in imuaing Queues wih Pareo ervice, Proceedings of he Winer imuaion Conference, [73] D. Kunh, The ar of Compuer Programming, vo., emi numerica Agorihms, 3 rd ediion, pp. 3. Boson, Addison- Wesey [74] Kihong Park, Gi Tae Kim and Mark E. Crovea, On he reaionship beween fie sizes, ranspor proocos and sef-simiar nework raffic, in proc. of he Inernaiona Conference on Nework Proocos, pp. 7-8, Oc. 996 [75] T. Tuan and K. Park, Muipe ime scae congesion conro for sef-simiar nework raffic, Performance Evauaion, 999 [76] D. L. Mis, impe nework ime prooco NTP version 4 for IPv4, IPv6 and OI, RFC 3, IETF, Oc hp:// [77] Y. Cheng, H, Jiang, W, Zhuang, Z. Niu and C. Lin, Efficien Resource Aocaion for China s 3G/4G Wireess Neworks, IEEE Communicaion Magazine, vo. 43, issue, Jan 5, pp [78] R. Ben Ai, Y Lemieux and. Pierre, UMT-o-IP Qo Mapping for Voice and Video Teephony ervices, IEEE Nework, vo. 9, issue, March/Apri 5, pp

33 Biographies of he Auhors Mohsin Ifikhar Mohsin Ifikhar received his Bc Eecrica Engineering from Universiy of Engineering and Technoogy Lahore, Pakisan in 999 and M.Eng.c. in Teecommunicaions from UNW Ausraia in. Curreny he is a PhD candidae in Advanced Neworks Research Group choo of IT, Universiy of ydney. During his PhD candidaure, he has pubished severa papers in inernaiona conferences and journas and has been awarded severa prizes incuding iemens Prize for soving an indusry probem 6, Neworks and ysems Prize in research projec work, schoo of IT, 7. He has been receny awarded Endeavour Posgraduae Feowship o pursue a 6 monhs Posdocora Research in 8 a Deparmen of Mahemaica cience King Fahd Universiy, audi Arabia. He is he member of IEEE, ACM and IET. His research ineress incude Qo, IP/Wireess IP raffic modeing, Markov Chains, ef-imiar raffic modeing, Nework Cacuus, Queueing Theory and Poing modes. Tejeshwar ingh T. ingh received his BE ofware Engineering/Bc Mahemaics from Universiy of ydney in 7. He is curreny working in he Windows Neworking eam a Microsof. His research ineress incude Qo, Markov Chains, Numerica ouion of Markov Chain and ef-imiar raffic modeing. 33

34 Dr. Bjorn Landfed Dr. Landfed sared his sudies a he Roya Insiue of Technoogy in weden. Afer receiving a Bc equiv, he coninued sudying a The Universiy of New ouh Waes where he received his PhD in. In parae wih his sudies in weden he was running a mobie compuing consuancy company and afer his sudies he joined Ericsson Research in ockhom as a enior Researcher where he worked on mobiiy managemen and Qo issues. In, Dr. Landfed ook up a posiion as a CICO enior ecurer in Inerne Technoogies a he Universiy of ydney wih he choos of Eecrica and Informaion Engineering and he choo of Informaion Technoogies. Dr Landfed has been awarded 8 paens in he U and gobay. He has pubished more han 5 pubicaions in inernaiona conferences, journas and books and been awarded many compeiive grans such as ARC discovery and inkage grans. Dr. Landfed is aso a research associae of Naiona ICT Ausraia NICTA. Curreny, he is serving on he edioria boards of inernaiona journas and as a program member of many inernaiona conferences and is supervising 8 Ph.D sudens. Dr. Landfed s research ineress incude; mobiiy managemen, Qo, performanceenhancing middeware, wireess sysems and service provisioning. Dr. Mine Cagar M. Cagar received her B. and M. degrees in Indusria Engineering from Midde Eas Technica Universiy and Biken Univerisy, respecivey. he received a Ph.D degree in aisics and Operaions Research from Princeon Universiy in 997. he worked as a pos-docora research scienis a Becore in Morrisown in Nework Design and 34

35 Traffic Research Group during he is curreny an associae professor in Deparmen of Mahemaics a Koc Universiy, Turkey, which she joined in 999. Her curren research ineress incude sochasic modeing in eecommunicaion neworks; in paricuar raffic modeing, epidemic agorihm and queueing. 35

36 Figures Fig. : A impified UMT Nework Archiecure r s Fig. : Iusraion of he raffic process. Horizona segmens represen he sessions, heir enghs are deermined by r, arriva imes s are he projecions of he diamonds o he horizona axis and he packe arrivas are indicaed by verica segmens over he sessions. 36

37 r B A C s Fig. 3: Various regions where he arriva imes and he engh of sessions fa. The obique ines make a 45º degree ange wih he s-axis. The session enghs in A are arge enough ha hey expire afer. In conras, he expiraion imes are before for hose sessions in B. Fig. 4: An Exampe of Markov Chain Transiion from i, i, a, s j, j, a, s 37

38 Fig. 5: An Exampe of Markov Chain Transiion from i, i, a, s j, j, a, s Work Z 3 Z Z Z 4 ime Fig.6: Waiing ime of a ype packe in erms of Zj s. 38

39 The Pareo Disribuion: Theoreica Mean vs. Experimena Mean Hurs Parameer Mean Vaue Theoreica Mean Experimena Mean Fig.7: hows he heoreica mean of he Pareo disribuion vs. ha acuay obained hrough he random number generaor for H=.55,.75,.95. Packe Loss Rae Packe Loss Rae: Numerica vs. imuaion Hurs Parameer Numerica Resus High Prioriy Queue Numerica Resus Low Prioriy Queue imuaion: High Prioriy Queue imuaion: Low Prioriy Queue Fig.8: Packe Loss Rae: Numerica vs. imuaion Resus 39

40 Fig. 9: Tes Bed eup Mean Deay Deay ms Hurs Parameer Numerica High Prioriy Queue Numerica Low Prioriy Queue imuaion High Prioriy Queue imuaion Low Prioriy Queue Tes Bed High Prioriy Queue Tes Bed Low Prioriy Queue Fig. : Mean Deay vs Hurs Parameer 4