WE present our preliminary work that develops a new

Transcription

1 ACM/IEEE PADS 28 1 Sochasic Process Models for Packe/Analyic-Based Nework Simulaions Rober G. Cole, George Riley, Derya Cansever and William Yurcik Index Terms Hybrid Simulaion Models, Even Driven Simulaions, Nework Models, Sochasic Queues and Diffusion Equaions. Absrac WE presen our preliminary work ha develops a new approach o hybrid packe/analyic nework simulaions for improved nework simulaion fideliy, scale, and simulaion efficiency. Much work in he lieraure addresses his opic, including [1] [11] [8] [12] [13] and ohers. Curren approaches rely upon models, which we refer o in his paper as Deerminisic Fluid Models [9] [12], o address he analyic modeling aspecs of hese hybrid simulaions. Insead we draw upon an exensive lieraure on sochasic models of queues and (evenually) neworks of queues o implemen a hybrid sochasic model/packe nework simulaion. We will refer o our approach as Sochasic Fluid Models hroughou his paper. We ouline our approach, presen es cases, and presen simulaion resuls comparing he measured queue merics from our approach for hybrid simulaion o hose of a deerminisic fluid model hybrid simulaion and a full packe level simulaion. We also discuss plans for fuure areas of research on his approach. I. Inroducion Consider a nework model as found in Figure 1. The nework model is comprised of nodes and communicaions links. Associaed wih each communicaions link is a queue. Traffic can arrive and depar he nework model a each node in he nework. Each node also carries inernal raffic which is forwarded hroughou he nework beween he raffic source and desinaion nodes. In hybrid simulaion models, some of he raffic is handled via analyic mehods and some of he raffic is handled via more CPU and memory inensive discree-even, packe-level handling. The more raffic modelled analyically, he more efficien he simulaion becomes (in he general case), and hus he simulaion can scale o larger neworks. Typically, here is a disincion drawn beween R. G. Cole is wih he Applied Physics Lab and he Deparmen of Compuer Science, Johns Hopkins Universiy, Balimore, MD. Phone: , rober.cole@jhuapl.edu G. Riley is wih he School of Elecrical and Compuer Engineering, Georgia Insiue of Technology, Alana, GA. Phone: , riley@ece.gaech.edu D. Cansever is he Program Direcor of Advanced Inerneworking, SI Inernaional, Inc. Phone: , derya.cansever@si-inl.com W. Yurcik is wih he Deparmen of Compuer Science, Universiy of Texas-Dallas, Dallas, TX. Phone: , wxy81@udallas.edu exernal arrival nework queue summed inernal load nework exernal deparure Fig. 1. An example nework of queues wih inernal and exernal arrival and deparure raffic. foreground raffic, which is of primary ineres, and background raffic, which exiss solely o provide compeiion o he foreground raffic being measured. In he hybrid approach discussed here, analyic models are used is o esimae he impac of he background raffic on he foreground raffic, which is handled explicily via discreeeven handling wih full packe level deail. In he remainder of his paper, we will assume ha he analyically modeled raffic represens, in some sense, background raffic and ha he explicily handled packe raffic is foreground raffic. We will use he erms background raffic and foreground raffic o disinguish beween he analyically modeled versus he explicily handled even packe raffic. Of course oher ways o divide up he analyically modeled and packe handled raffic are possible. There are wo ineresing aspecs of he analyic modeling in he conex of a nework of queues. One relaes o he allocaion and esimaion of he inermediae load generaed by he raffic a he nodes wihin he nework based upon he assumed rouing paerns, oal esimaed exernal raffic loads, finie queue sizes and associaed nework inernal packe losses. The oher aspec relaes o mehods of mixing, a a given queue, he analyically modeled background raffic wih he explicily handled foreground raffic. Our focus in his paper is he laer; hence we concenrae on mehods o mix he analyically modeled raffic wih he even-driven, packe-level raffic a a given queue. A fuure paper will concenrae on he nework equaions. The remainder of his paper is organized as follows: In he nex secion we discuss previous mehods for hybrid nework simulaion which rely upon deerminisic fluid model approximaions o queue dynamics. In Secion III we presen our approach based upon mehods in sochasic queue dynamics. In Secion IV we develop our hybrid equaions for a specific insance of a sochasic queue model, i.e., a Brownian Moion model based upon soluions o he Fokker-Planck Equaion. In Secion V we repor on our iniial simulaions invesigaions comparing hree es cases; one based upon pure even-driven packe-level sim-

2 2 ACM/IEEE PADS 28 ulaions, one based upon hybrid deerminisic fluid/packe simulaion and one based upon our hybrid sochasic process/packe simulaion. In Secion VI we discuss conclusions and fuure invesigaions. II. Deerminisic Models Mos curren approaches o performance speedup for nework simulaions involving hybrid even-analyic simulaion rely on analyic models based upon deerminisic Fluid Flow Approximaions (FFAs), e.g., [7] [4] [1] [11] [8] [13]. We refer o hese approaches as Deerminisic Models because he evoluion of he queue dynamics modeling he background analyic raffic is assumed o be a deerminisic process capured in he form of differenial equaions. The fluid dynamics is derived from he inegraion of hese equaions. In some cases he differenial equaions are solvable, e.g., fixed arrival rae models, and numerical inegraion of he differenial equaions is no necessary. Boh cases address variable mean arrival raes; one hrough ime dependen variables in he differenial equaions [9] [13] and one hrough discree even fluid models where rae changes are propagaed hroughou he nework via evens [4] [11] [7]. Two approaches o using deerminisic analyic models for hybrid simulaions are found in he lieraure. One approach, used in [13], divides he nework ino a hierarchy consising of a high speed core and a lower speed edge. In he core, all raffic is modeled analyically and in he edge all raffic is modeled via packe-level, even-driven simulaion. Packe raffic raversing he core is convered o fluid load on he core and is hen convered back o packe-level raffic a he far edge. Anoher approach, used in [4] [11] and [8], is o rea some of he raffic hroughou he nework analyically. Then develop mehods o explicily mix boh analyic and packe raffic a each muliplexing poin hroughou he nework. Our ineress are in his laer approach. A. Deerminisic/Packe Mixing Equaions Here, we develop he simples, leas assuming mehod o mix deerminisic raffic wih packe raffic. There exis mehods o improve he fideliy of our example deerminisic model, e.g., [4] [11] [8], bu hese improvemens require a priori knowledge of he behavior of he foreground raffic, which in general is no known. This choice is made here for several reasons. Firs, we wish o make an equivalen comparison o our sochasic models presened below in order o access is abiliy o negae a reliance on a priori raffic knowledge. Second, our main focus in his paper is o access aspecs of he viabiliy of he sochasic modeling and mixing approach agains full packe level simulaions. Assume ha λ d is he average deerminisically modeled raffic rae ino he queue over a ime averaging inerval δ. We assume ha δ is large compared o a ypical packe ransmission ime bu small compared o he oal simulaed ime. Noe ha we assume ha he parameers describing he arrival processes, boh deerminisic and sochasic, in his paper o be fixed. Le µ be he server ω max work ω ( ) i ω ( ) + i i i+1 i+2 i+3 packe arrivals slope = λ µ ω i ( work carried by packe i ) d ime Fig. 2. An illusraion of he hybrid deerminisic modeling approach for mixing raffic a a queue. rae a he queue, i.e., he inverse of he link bandwidh. Le w() be he oal workload in he queue a ime, and w ( i ) and w + ( i ) be he oal workload in he queue jus prior o and jus following (respecively) a disinguished ime even, e.g., he arrival of packe i a ime i. Le w i be he work carried by he ih packe o he queue. Finally, le w max be he maximum amoun of work (or backlog) in he queue due o is finie size. Figure 2 gives a picorial represenaion of he mehod used o mix analyically modeled background raffic wih explicily handled packe raffic. Individual packe arrival evens are indicaed along he boom axis of he figure. These packes carry wih hem a given workload which is a funcion of heir packe size and he link bandwidh serving he queue in quesion. If he packe is allowed o ener he queue, hen he workload in queue jumps from w ( i ) o w + ( i ) where he difference is he workload, w i, carried by he packe. In beween packe arrivals, he workload evolves deerminisically. As menioned, we assume he queue dynamics evolve a a consan rae beween packe arrivals (as long as he workload does no exceed he maximum w max or drops below he minimum of zero) given by he difference λ d µ which is fixed over he averaging ime inerval δ. Then he explici handling of he packe raffic and he process for updaing he impac of he background raffic are as follows: If, during he curren δ ime averaging period, λ d < µ, i.e., he background raffic does no overflow he server rae. For each packe arrival bringing w i work o he queue, do he following asks: Incremen he packe arrival couner i a he queue. Updae he queue backlog jus prior o he packe arrival ime i, as w ( i ) max[w + ( i 1 )+(λ d µ)( i i 1 ), ] (1) If w i < w max w ( i ), hen accep he packe, updae queue workload as w + ( i ) = w ( i ) + w i and schedule he packe deparure even a ime service i = w ( i )/µ + i (2) Else, discard he packe and se w + ( i ) = w ( i ). Else λ d µ. Here, he background fluid (which is assumed consan) is coninually overflowing he queue.

3 COLE, e al: STOCHASTIC PROCESS MODELS FOR PACKET/ANALYTIC-BASED NETWORK SIMULATIONS 3 Hence, we have ha w() = w max and no packe raffic is allowed ino he queue. Therefore, he model has he foreground raffic experiencing 1% packe loss. As menioned, here are ways o overcome his effec, see, e.g., [4], bu hey rely on having an esimae of he foreground raffic rae. Wihou a priori knowledge of he foreground raffic behavior, he deerminisic model will deny he foreground raffic access o he buffer. This siuaion can be addressed by esimaing foreground raffic load based on previous behavior, as has been done elsewhere. possible sochasic rajecories ω max work ω( + i ) mean drif ω( i+1) i i+1 packe arrivals a packe loss esimae ω( + i+1 ) σ m = λ µ diffusional densiy ime s III. Sochasic Models In his secion we presen our approach o hybrid nework simulaions which relies upon sochasic models of he background raffic and explici handling of packe evens for he foreground raffic. As discussed previously, wo aspecs of his level of inegraion need consideraion, i.e., mehods o mix he analyic background raffic wih he explici even handling of he foreground raffic a a common queue, and mehods o esimae he arrival processes of he background raffic a he inermediae queues in he nework based upon nework rouing and exernal arrival processes. We concenrae on he former in his paper. A. Sochasic/Packe Mixing Equaions Here we describe our mehod o mix he sochasic modeled background raffic and he packe raffic a a given queue wihin he nework. In some sense, our approach defines a naural exension o he deerminisic/packe hybrid approach described above. However, our approach eliminaes he need for ad-hoc mehods and assumpions o mixing analyic and packe raffic a common queues, does no require a priori knowledge of foreground packe behavior and applies independen of wheher he overall load on he server is under or exceeds is capaciy. As above, assume ha λ s is he average background raffic (now reaed as a sochasic process, hence he subscrip s ) rae ino he queue over a ime averaging inerval δ. This will represen he average rae of raffic associaed wih he analyically modeled raffic. However, we also characerize he variaion in he background arrival process hrough is (CoV) funcion, C v [1]. The CoV is defined as C v = σ s /λ 1 s where σ s is he sandard deviaion of he iner-arrival imes for he incoming sochasic background raffic. Here, λ 1 s is he mean iner-arrival ime for he incoming sochasic background raffic. We assume he characerizaion of he background raffic is known, i.e., (λ s,c v ), a each node (queue) in he nework. Le µ be he server rae a he queue, i.e., he inverse of he link bandwidh. Le w() be he oal workload in he queue a ime, and w ( i ) and w + ( i ) be he oal workload in he queue jus prior o and jus following (respecively) a disinguished ime even, e.g., he arrival of packe i. Le w i be he work carried by he ih packe o he queue. Finally, le w max be he maximum amoun of work (or backlog) in he queue due o is finie size. Fig. 3. An illusraion of he hybrid sochasic modeling approach for mixing raffic a a queue. The algorihm o deermine a specific w ( i ) given a value of w + ( i 1 ) is based upon he knowledge of he emporal evoluion of he Cumulaive Disribuion Funcion (CDF) of he sochasic process. The CDF is defined as F (w, w, ) = P r[x < w X = w ] and saisfies d w F (w, w, ) = 1 and lim o F (w, w, ) = δ(w w ), he Dirac Dela Funcion. Our mehod applies for any chosen model of he CDF of he sochasic processes. Here we ouline he mixing algorihm and in Secion IV we give an example of one paricular sochasic process, e.g., a Brownian Moion process from heavy raffic approximaions of queues [3]. We use a CDF o deermine probabilisically he evoluion of w ( i ) given w + ( i 1 ). Figure 3 gives a picorial represenaion of he mehod used for mixing analyically modeled sochasic raffic wih explicily handled packe raffic. Individual packe arrival evens are indicaed along he boom axis of he figure. These packes carry wih hem a given workload which is a funcion of heir packe size and he link bandwidh serving he queue in quesion. If he packe is allowed o ener he queue, hen he workload in queue jumps from w ( i ) o w + ( i ) where he difference is he workload carried by he packe. In beween packe arrivals, he workload evolves according o a given sochasic process based upon load parameers which characerize he associaed arrival process, i.e., λ s and he coefficien of variaion, C v. Given he sochasic process and is Cumulaive Probabiliy Disribuion, we can deermine an appropriae w ( i+1 ) given w + ( i ). We accomplish his by sampling agains he given CDF for each packe inerarrival. The explici handling of he packe raffic and he process for updaing he impac of he sochasic raffic are as follows: For each packe arrival, do he following asks: Incremen he packe arrival couner i a he queue. Updae he queue backlog jus prior o he packe arrival ime i, by sampling agains he Cumulaive Probabiliy Disribuion Funcion, F (w, i w + ( i 1 ), i 1 ). Specifically, given a Disribuion Funcion F (X) = P r[x < w], and a random variable Y uniformly disribued beween and 1, hen we sample agains he disribuion by he formula w = F 1 (Y ). This yields a specific value for

4 4 ACM/IEEE PADS 28 w ( i ). If w i < w max w ( i ), hen accep he packe, updae queue workload as w + ( i ) = w ( i ) + w i and schedule he packe deparure even a ime service i = w ( i )/µ + i (3) Else, discard he packe and se w + ( i ) = w ( i ). The approach oulined above leads o a few observaions. Firs, his approach is a naural exension o he mehods previously discussed. Second, his approach explicily accouns for background raffic loss and even packe loss for all arrival load condiions, no jus for background loads less han he line rae. Specifically, he impac of background raffic on he packe raffic is handled a each packe arrival, while he impac of packe raffic on background raffic (and hence background raffic loss) is handled by reseing he iniial queue sae following each packe arrival. Third, unlike for deerminisic background models where i is ofen necessary o know a priori he foreground load in order o ge good esimaes of foreground packe loss, his approach requires no such informaion or esimaion. Insead, he foreground packe loss is a naural resul of he flucuaions in he background raffic and he curren load placed on he sysem by he foreground raffic. The above procedure provides a beer esimaion of aspecs of he foreground and background losses and heir cross ineracions. IV. An Example Sochasic Model Our approach o analyic/packe level mixing a a given queue wihin he nework was described independenly of he underlying sochasic process describing he background raffic. Hence, i is possible o rely upon a number of poenial sochasic processes. We describe one here and rely upon i in our simulaion resuls in Secion V. Furher, i is cerainly possible o use several sochasic processes o model he background raffic based upon which model bes represens he resuls as a funcion of sysem load and oher aspecs of he sysem model. This is a opic for fuure research. The sochasic model we invesigae here plays a prominen role in he work on heavy raffic limis in queuing models and is based upon he Brownian Moion sochasic process [3]. A. Brownian Moion Model For our iniial invesigaions, we assume ha he sochasic process is well approximaed by a Brownian Moion model which is sricly appropriae for heavy raffic limis, see, e.g., [3]. Given his, and assuming a GI/G/1/K queueing sysem, we hen have ha he Cumulaive Disribuion Funcion, F (w i, i w + ( i 1 ), i 1 ) saisfies he Fokker-Planck Equaion [3] wih appropriae boundary condiions o be discussed below, i.e., F = m F w σ2 2 F w 2 (4) where m = λ s µ, σ 2 = λ s C 2 v + µ C 2 v,µ, F (w, = ) = H(w w ), and H(x) is he Heavy-side Funcion. For he purpose of his paper, we assume ha all packes have he same size, hence we model a GI/D/1/K queueing sysem and have ha he CoV for he service process is zero. This assumpion implies ha σ 2 = λ s C 2 v. The Fokker-Planck equaion has an analyic soluion derived in [3] F (w, w, = ) = α Φ( w w m σ ) + β e 2mw/σ2 Φ( w w m σ ) (5) F (w, = ) = H(w w ) (6) Φ( ±w w m σ ) = 1 ±w w m σ 2π e x2 /2 dx (7) The mulipliers, α and β, are deermined by Boundary Condiions (BCs) of he sysem. The mos naural BCs found in he lieraure are and lim F (w, w, = ) = (8) w lim F (w, w, = ) = 1 (9) w w max where he firs BC addresses he lower limi of zero on he work in he sysem and he laer BC ensures ha he finie size queue limis he work in he sysem o he maximum work allowed. Solving for α and β using hese BCs, we ge and α 1 = Φ(w max,+ ) e 2mwmax/σ2 Φ(w max, ) (1) where we have used he abbreviaions β = α (11) Φ(w max,± ) = Φ( ±w max w m σ ) (12) We also invesigae he use of alernaive BCs. Consider he sysem saring ou wih a very large iniial value for he work, w. If we are ineresed in he prediced work in he sysem afer a very shor ime inerval, we would expec ha a lower limi on he work in queue would be given by w µ. Hence, an alernaive se of BCs, which we refer o BCs wih a moving lower BC, is and lim F (w, w, = ) = (13) w max[,w µ] lim F (w, w, = ) = 1 (14) w w max Solving for α and β using hese BCs, we ge α 1 = Φ(w max,+ ) Φ(w max,+)φ(w max, )e 2m(wmax wmin)/σ2 Φ(w min, ) (15)

5 COLE, e al: STOCHASTIC PROCESS MODELS FOR PACKET/ANALYTIC-BASED NETWORK SIMULATIONS 5 F(w,) Time = 1. (sec) Time = 2. (sec) Time = 1. (sec) Time =.25 (sec) Time =.5 (sec) Time =.1 (sec) F(w,) Time = 1. (sec) Time = 2. (sec) Time = 1. (sec) Time =.25 (sec) Time =.5 (sec) Time =.1 (sec) Sources Sinks Foreground Buffer C1 R1 S1 Background Mbps 5 Kbps.2.2 Fig. 5. The reference connecion used for he simulaion runs. 1 2 Work (Kbis) 1 2 Work (Kbis) Fig. 4. Evoluion of he CDF for wo cases of load equal o.98 (lef) and 1.2 (righ). and β = αφ(w min,+)e 2mwmin/σ2 Φ(w min, (16) where w min () = max[,w µ] and we have used he furher abbreviaions Φ(w min,± ) = Φ( ±w min w m σ ) (17) These ses of equaions for he CDF compleely deermine he ime evoluion of he sochasic sysem. The CDF wih BCs a w = and w = w max are well sudied and are discussed in [5] and [6]. In Figure 4 we presen some resuls for he shape of he CDF for he moving lower BCs, for wo server uilizaion of.98 and 1.2 and various imes. For hese examples we used a buffer size of 5 packes of 56 Byes each, an iniial queue workload of 1792 Byes represening a buffer a 8% occupancy and a server rae of 5 Kbps. We see ha for boh he cases, where loads are less han and greaer han server capaciy, here is a finie probabiliy ha he buffer is no fully occupied. This fac allows he packe level raffic access o he buffer even in server overload condiions, unlike he deerminisic case. V. Resuls In his secion we presen he resuls for our invesigaions ino he accuracy of he sochasic models in predicing queuing behavior in a single queue sysem. We compare he resuls of his mehod o he comparable even driven simulaion model of he equivalen queuing sysem and of he equivalen deerminisic background raffic model as described above. We use he Georgia Tech Simulaion ool (GTNeS) [2] for our simulaion sudies of he base case (packe level deail only) as well as he hybrid cases. For our simulaion resuls, we developed wo new modules wihin GTNeS. The wo modules developed are a) A Hybrid Inerface Module (HIM) which handled he mixing of he analyic background raffic wih he foreground packe raffic, and b) A new raffic generaion applicaion, he Hyper-Exponenial Arrival (HEA) module, which generaes applicaion level UDP packes according o a Hyper-Exponenial Process [1]. This process models a source which alernaes beween a high rae and a low rae raffic source, where each rae period is exponenially disribued. Hence, by seing he raes equal, we reproduce a simple and relaively smooh Poisson raffic paern. Our use of he Hyper-Exponenial Process allows us o model bursy raffic sources, which are well known o exis in he Inerne, by seing he raes o differen values. Our reference opology used for he simulaion runs is shown in Figure 5. Here node C1 is he source for wo differen applicaions; one which is a 5Kbps Poisson packe sream represening our foreground UDP raffic case and one represening our Hyper-Exponenial background UDP packe sream which can run a differen raes and bursiness facors. Boh applicaion raffic, foreground and background, ravel o node R1 over a 1 Mbps link and hen ono node S1 over a 5 Kbps link. The link beween R1 and S1 represens he queueing poin, or boleneck in our simulaions. For each reference case sudied, we ran hree differen simulaion models; firs is he base case which is a full even-driven packe level simulaion of boh he foreground and he background raffic, second is he deerminisic hybrid case where he foreground raffic is reaed a he packe level while he background raffic is reaed analyically as a fixed rae fluid, and hird is he sochasic hybrid case where he foreground raffic is reaed a he packe level while he background raffic is reaed analyically as a sochasic fluid wih fixed mean and variance. To invesigae impacs of differen foreground packe flows on he hybrid sysems, we also ran hese hree cases when he foreground raffic is a TCP-Reno sream from nodes C1 o S1 hrough R1, where we se he slow-sar hreshold o 2 KByes. For each of he above cases, we varied he mean and he CoV of he background Hyper-Exponenial UDP sream o cover a range of raffic loads and loss raes. Table V liss he various mean background raes and example parameers represening he exreme CoV invesigaed for each mean rae. For all he resuls presened, we averaged he meric values over en independen simulaion runs wih differen random generaor seedings for each. Each simulaion run consised of an iniializaion period of 5 seconds where jus he background raffic processes run (in order o allow he sysem o equalibriae) and hen we iniiaed he foreground process. We coninued each simulaion run for an addiional 1 seconds. We firs presen our resuls for he case of a UDP foreground packe-based sream. For his foreground process we measure he packe level delay and loss. In Figure 6 we show he resuls for mean delay esimaes for he hree cases of base, deerminisic and sochasic raffic for a background sream wih a CoV equal o uniy. The raes simulaed in he plos are 3, 35, 4, 45 and 5 Kbps. Noe, however, ha hese are he applicaion-level raes and due o packe overhead of 48 byes per packe, he

6 6 ACM/IEEE PADS 28 TABLE I Parameers for he Hyper-Exponenial background process. 1.8 Base De Soc_mvBC λ 1 s Cv H r H L r l (Kbps) (msec) (Kbps) (msec) (Kbps) Loss Probabiliy respecive line raes are 328, 383, 438, 492 and 53 Kbps. Here we see mean delay of he foreground sream increasing as he mean rae of he background raffic increases for all hree cases. However, he sochasic case does a beer job a racking he delays recorded for he base simulaion case, while he deerminisic case underesimaes he delays a lower loads and overesimaes he delay a higher loads Base De Soc_mvBC Background Rae (Kbps) Fig. 6. Resuls for he UDP foreground delay versus background raffic rae. Figure 7 presens resuls on he foreground packe loss probabiliy for he differen background raffic raes where he CoV is uniy. The loss meric is a much more sensiive and challenging es of models due o is dependence upon he high perceniles of he work-in-queue probabiliy disribuions. We see exremely low packe losses for all cases for raes less han 4 Kbps, as expeced. Above 4 Kbps, where he link is approaching a load near uniy, he loss begin o increase. The losses for he deerminisic model rapidly approach uniy, because i has no mechanism o allow packe raffic enry ino he buffer when he fluid inpu rae equals or exceeds he service rae of he sysem. While he resuls for he sochasic model are quie close o he base case loss resuls. As menioned previously, he sochasic model explicily accouns for he flucuaions in Background Rae (Kbps) Fig. 7. Resuls for he UDP foreground loss probabiliy versus background raffic rae. he background raffic and his is wha allows he packe raffic lossy enry ino he buffer. In he plos in Figure 8 we invesigae he impac of increased CoV in he background sream on he foreground packe merics. Figure 8 presen he resuls for mean raes of 3, 4, 43 and 45 Kbps in he background sreams. We see ha he mean delay increases as he CoV increases for lower loads, while for higher loads he mean delay acually decreases for increasing CoV. The deerminisic model does no accoun for variaions in he dynamics of he background sream so i is incapable of racking hese effecs. We show resuls for he moving lower BC discussed above for he sochasic Brownian Moion Model. The sochasic model capures he rends in he delay versus load and CoV. However, he rends are no as pronounced as in he base simulaion case. This resul will be addressed in fuure sudies. We now presen our resuls for he case where he foreground raffic is a TCP packe sream. Due o TCP s adapive windowing policy, he foreground raffic in hese simulaion runs will aemp o uilize he remaining bandwidh on he link beween R1 and S1. Hence, he foreground loads vary for differen cases of background raffic loads. We firs show a se of resuls comparing he TCP goodpu versus rae for he various simulaion cases. These resuls are shown in Figure 9. We see ha he goodpu generally decreases wih increasing background rae. Furher, he deerminisic model underesimaes he TCP goodpu, due o is over esimaion of he foreground packe losses (from previous resuls above). While he sochasic models do a more reasonable job in racking TCP hroughpu. Furher, he esimaes of he sochasic models improve for higher loads as expeced. We show he resuls for boh BCs discussed above. Tha he TCP goodpu as modeled by he sochasic hybrid case racks well he base case is a ribue o he models abiliy o esimae packe loss probabiliy, as TCP goodpu is highly non-linear wih respec o packe loss. We show he resuls for background sreams wih increasing Coefficiens of Variaion, where we are presening he average goodpu of he TCP foreground process. Figure 1 show he resuls for a mean background raffic rae of 3, 4, 45 and 5 Kbps. We observe ha he TCP

7 COLE, e al: STOCHASTIC PROCESS MODELS FOR PACKET/ANALYTIC-BASED NETWORK SIMULATIONS Base(3Kbps) De(3Kbps) Soc_mvBC(3Kbps) Base(4Kbps) De(4Kbps) Soc_mvBC(4Kbps) Base(43Kbps) De(43Kbps) Soc_mvBC(43Kbps) Base(45Kbps) De(45Kbps) Soc_mvBC(45Kbps) Fig. 8. Resuls for he UDP foreground delay versus CoV for various background raffic raes. goodpu for he base case shows a weak dependence on he CoV for lower loads and a greaer, increasing dependence on CoV a higher loads. Furher, he analyic models resul in an underesimaion of he TCP goodpu for all cases. We presen he resuls for boh sochasic models, i.e., he models resuling from differen lower BCs. The sochasic models capure he qualiaive ends shown in he base case. However, quaniaively hey underesimae he TCP goodpu. As before, we see ha he deerminisic model resuls are independen of he CoV of he background raffic sream. 1 Base De Soc mvbc Soc Background Rae (Kbps) Fig. 9. Resuls for he TCP foreground goodpu versus background raffic rae. VI. Conclusions and Fuure Sudies We have presened he iniial resuls of an invesigaion ino he use of sochasic queueing models for he purpose of developing hybrid analyic and packe level, even driven nework simulaion ools. Our ulimae objecive is o improve he scalabiliy of nework simulaions, decrease heir run-imes and mainain high fideliy of heir resuls. There are numerous aspecs of his opic which require invesigaions, including packe/analyic model mixing a nework queues and esimaions of sochasic model parameers a inerior queues in he nework. In his paper, we have only begun invesigaing he iniial quesion of how o mix sochasic analyic fluids wih packe level evens and performed some iniial invesigaions ino he fideliy of hese mehods. However, we are encouraged by hese iniial resuls. We believe he use of sochasic background models lead o a naural, unforced means of mixing packe and analyic raffic in common queues. This is no he case when relying upon deerminisic models which force designers o make ad-hoc assumpions and decisions wih respec o how o mix packe and analyic raffic and which ofen have o rely upon a priori knowledge of foreground raffic behavior. There remains much work o furher explore and develop a high fideliy hybrid sochasic and packe-level nework simulaion capabiliy. Fuure work includes: Invesigae he use of improved sochasic models and approximaions from he field of Heavy Traffic resuls in queuing heory o improve he fideliy of he predicions. Exend our mehods o neworks of queues. Furher explore he fideliy of hese mehods in he presence of oher background raffic ypes and loads and heir impac of oher foreground applicaion raffic. Invesigae he benefis of using hybrid sochasic/packe simulaion echniques when modeling processes wih long ailed disribuions. VII. Acknowledgemens We would like o hank he reviewers for heir deailed and insighful commens and in poining ou Reference [7].

8 8 ACM/IEEE PADS Base(3Kbps) De(3Kbps) Soc_mvBC(3Kbps) Soc(3Kbps) Base(4Kbps) De(4Kbps) Soc_mvBC(4Kbps) Soc(4Kbps) Base(45Kbps) De(45Kbps) Soc_mvBC(45Kbps) Soc(45Kbps) Queueing Neworks, I. Nonequilibrium Disribuions and Compuer Modeling, Journal of he Associaion for Compuing Machinery, Vol. 21, No. 3, [7] Liu, B., Guo, Y., Kurose, J., Towsley, D. and W. Gong, Fluid Simulaion of Large Scale Neworks: Issues and Tradeoffs, PDPTA, [8] Liu, J., Parallel Simulaion of Hybrid Nework Traffic Models, IEEE ACM PADS 7, San Diego, CA, USA, June 27. [9] Mira, D., Sochasic heory of a fluid model of producers and consumers coupled by a buffer, Adv. in Appl. Prob., Vol. 2, [1] Nicol, D. and G. Yan, Simulaion of Nework Traffic a Coarse Time-scales, IEEE ACM PADS 5, Monerrey, CA, USA, June 25. [11] Liljensam, M., Nicol, D., Yuan, Y. and J. Liu, RINSE: he realime Ineracive Nework Simulaion Environmen for Nework Securiy Exercises, IEEE ACM PADS 5, Monerrey, CA, USA, June 25. [12] Misra, V., Gong, W.B. and D. Towsley, Fluid-based Analysis of a Nework of AQM Rouers Supporing TCP Flows Wih an Applicaion o RED, Proceedings of ACM SIGCOMM 2, 2. [13] Gu,Y., Liu, Y. and D. Towsley, On Inegraing Fluid Models wih Packe Simulaion, IEEE INFOCOM 4, Base(5Kbps) De(5Kbps) Soc_mvBC(5Kbps) Soc(5Kbps) Fig. 1. Resuls for he TCP foreground goodpu versus Cv for various background raffic raes. References [1] Cooper, R., Inroducion o Queueing Theory, Norh-Holland, Oxford, [2] Riley, G., The Georgia Tech Simulaion Tool, hp://maniacs.ece.gaech.edu/ 28. [3] Heyman, D., A Diffusion Model Approximaion for he GI/G/1 Queue in Heavy Traffic, Bell Labs Technical Journal, Vol. 54, No. 9, November [4] Kiddle, C., Simmonds, R., Williamson, C. and B. Unger, Hybrid Packe/Fluid Flow Nework Simulaion, 17h Workshop on Parallel and Disribued Simulaion, June 23. [5] Kobayashi, H., Applicaions of he Diffusion Approximaion o Queueing Neworks, I. Equilibrium Queue Disribuions, Journal of he Associaion for Compuing Machinery, Vol. 21, No. 2, [6] Kobayashi, H., Applicaions of he Diffusion Approximaion o