A Multifractal Wavelet Model of Network Traffic

A Multifractal Wavelet Model of Network Traffic Proect Report for ENSC80 Jiyu Re School of Egieerig Sciece, Simo Fraser Uiversity Email: re@cs.sfu.ca Abstract I this paper, a ew Multifractal Wavelet Model (MWM) is studied ad simulatio results are preseted. We coclude that MWM has captured most of the properties of etwork traffic ad the computatioal complexity of MWM is oly O(N). The efficiecy of MWM makes it possible i the real-time applicatios..itroductio Network traffic modelig is oe of the key topics i the applicatio of etwork ad multimedia. Takig the multimedia dowload as a example, a well-desiged model will help make use of the etwork efficietly ad speed up the trasmissio of data. Oe of key issues of multimedia dowload is how to model the commuicatio chaels. This is because after we get the accurate iformatio, like distributio of the etwork delay, about chael, we may optimally distribute the multimedia data ad fetch them by optimal dowload strategy. The more accurate the iformatio is, the better we may fid the strategy. Here we cocer the accuracy of the etwork iformatio i a statistical sese. What we like to kow is how to predict the real-time delay of each lik over which the data is trasmittig, which has the close relatio with etwork traffic modelig. Lots of research that have bee doe is based o measuremet data upo a specific Iteret coectio [3][4]. They tried to use curretly existed models to match measured data, such as AR model, Berulli model, -state Markov chai model, ad k-th order Markov chai model. Although these methods give some solutios to predictio of etwork delay, obody has cocluded whether or ot these methods ca be geeralized to ay kids of Iteret situatios. Actually we kow the Iteret chael is so complex that these covetioal statistical methods really caot capture the rich properties of Iteret [3]. Noetheless, a buch of researchers i Rice Uiversity are doig the challegig obs with excellet isight []. By itroducig the cocepts of multifractal ito Iteret traffic model, they build a multifractal model that matches the traffic properties, such as Log-rage depedecy, self-similarity ad burstiess, very well. Wavelets are used as a atural tool to aalyze ad sythesize the model due to its iheret multiscalig property. Based o these ideas, a

Multifractal Wavelets Model (MWM) is preseted. Based o MWM, V. Ribeiro etc. also develop a ovel algorithm to estimate multifractal cross-trafiic [], which is equivalet to delay distributio [3]. The algorithm is adaptive, effective ad requires o a priori traffic statistics, which greatly proves the effectiveess ad efficiecy of MWM ad provides ew method to the applicatio i the multimedia area. I sectio, we give the tutorial about the fractal, multifractal ad wavelets, ad some geeral ideas how ad why these cocepts ca be related to etwork traffic model. I sectio 3, we provide the performace criterios for etwork traffic models. After we show how to extract the iformatio of these criterios from traffic data, we build the multifractal wavelet model followig these criterios. Sectio 4 presets the simulatio results ad gives the commets to MWM based o the criterios we provided. It ca be show that MWM capture most the properties of etwork traffic, such as self-similarity, log-rage depedece, burstiess ad multiscalig behaviors, with computatioal complexity of oly O(N). We also show a simple approach to improve the performace of this model by simulatio results. We closed this paper with coclusio ad future work.. Fractal, Multifractal Ad Wavelets. From Fractal To Multifractal By itroductio of idea of fractal, researchers have made a great progress i modelig etwork traffic compared with classical models. A carefully desiged fractal model ca to great extet capture the fractal properties of etwork traffic, such as self-similarity, burstiess ad Log-Rage Depedece... Fractal Ad Fractal Radom Process Fractal is a atural pheomeo that is of the self-simliarity o fier ad fier level behid the extremely irregular shape, such as coastlie, sowflake. A fractal obect is measured with a dimesio that is fractioal [5]. For etwork traffic, we caot capture its rich properties with those classical models dealig with oly time ad sample value. A fractal radom process has so irregular sample values that its graph has a effective dimesio that exceeds its topological dimesio of uity. Typically, the dimesio of a fractal radom process is betwee ad. [6]. This irregular property of fractal sigal model makes it possible to be used to model etwork traffic... Fractioal Browia Motio Fractioal Browia motio (fbm) has bee broadly used to model fractal radom process []. It is a geeralized versio of Browia motio or Wieer process by makig the icremets of

Wieer process ormally distributed but o loger idepedet [7]. It preserves the self-similarity of Wieer process. The pdf of its icremet is [7] P( B(t)) = P(B(t + ε) B(t)) = πε H x u exp( )du 0 < H < () H ε We ca see that its icremet process, which is kow as fractioal Gaussia oise (fg), has ormal distributio with mea zero ad variace ε. Whe a=/, it is the Browia motio case. From pdf, we ca show the correlatio fuctios of fbm ad fg are H H H E[B(t )B(t )] = ( t + t t t ) () E[ B(t + τ) B(t)] = E[(B(t + ε + τ) B(t + τ))(b(t + ε) B(t))] = ( τ + ε H + τ ε H τ From these correlatio fucitos, we kow fbm is ot statioary, but fg is statioary, From the pdf of (), we see that H H B(t) = a B(at) a is costat (4) Where = is i the statistical sese, which meas B(t) ad B(at) is of the same distributio ad is statistical self-similar. Although fbm ad fg offers a simple ad good way to model ostatioary radom processes, ad some of authors has doe a good simulatio with Wavelet [8], but they have some limitatios i modelig etwork traffic [][6]: ) ). fbm ad fg are models based o Gaussia distributio, which meas there must exist some egative sigal. However, the data of etwork traffic should ot be less tha zeros. ). Network traffic will display short-term correlatio as well as well-kow Log-Rage depedece. It is icosistet with self-similarity fg ad fbm provide. 3). Ulike the fbm ad fg, which have a costat scalig value at differet momet orders, the scalig behaviour of momets of etwork traffic is ot homogeeous. H (3)..3 Multifractal Measure It is atural to thik about the questio ca we partitio etwork traffic ito multiple umber of fractal sets such that o each idividual partitio, the measure is homogeeous or uifractal? The aswer is yes ad this is also the motivatio of multifractal. From the motivatio, we kow multifractal refers to a measure method [5]. 3

. Multifractal Theory There are may special defiitios i multifractal theory [5]. We give some geeral ideas related to the modelig of etwork traffic... Local behavior---multifractal spectra Defie for a icreasig process Y at time t [] α( t) = lim α with k t k α k = log [Y] (6) k k [Y] = Y((k + ) ) Y(k ) k (5) = 0,, L (7) here α(t) meas the stregth of growth of Y. k is itervals i each partitio of process Y. The smaller the α (t), the faster Y grows at t. Defie f ( α) : = lim lim log #{ α k ( α ε, α + ) G 0 ε } (8) ε as (graied) multifractal spectrum, which is the frequecy measure of occurrece of a give stregth α. Here are some physical iterpretatios of α ad f G ( α ) : α >, meas Y has small icremet, α < meas Y has istat growth called burstiess if applied to etwork traffic. f G ( α ) = meas most of Y icremets is early equal to probability of a is higher tha b i process Y. α. f G (a) < f G (b) meas the If we kow the multifractal spectrum ( α ), we ca give a good iterpretatio to local f G behaviors of etwork traffic. The defiitio (8), however, is difficult to get with two limits i oe equatio. It will be show that we ca fid a easy way to compute partitio fuctio ad Legere trasform. f G ( α) with the help of.. Global behavior---partitio fuctio 4

Defie partitio fuctio as [] q T(q) : = lim log E[ ( k [Y]) ] (9) k = 0 where q is the momet of Y icremets at differet partitios. The T(q) could be iterpreted as the extet to which the measure deviates from a measure uiformly distributed o its support, which ca equivaletly be used to iterpret the global behavior of etwork traffic. Whe q>0, it describes the measure of relatively high icremet values, that is, bustiess. Whe q<0, it describes the measure cotaiig relatively small but ot-zero icremets[5]. Specially, whe q=0, T(q)=-; for a positive icremet process, whe q=, T(q)=0.[]..3 multifractal formalism---legere Trasform It ca be show that the multifractal spectrum ( α ) ad partitio fuctio T(q) are closely related by Legere trasform. The relatioship is as follows [][5]: f G ( α ) = qα T(q) ad f ( α) = q at T (q) = α (0) G With T(q) i had, by fidig the derivative of T(q), that is α, at every poit q, we ca plot the relatioship of f G ( α) versus α. (The details may see [][5]). This is illustrated i figure []. f G (a) Partitio Fuctio (b) Multifractal Spectrum Fig. Typical partitio fuctio ad Multifractal Spectrum.3 Wavelet Aalysis.3. Why Wavelet? It is atural to choose Wavelet as our aalysis tool to multifractal process. As well as low computatioal complexity, wavelets have iheret similar properties with multifractal sigal. 5

As we kow, wavelet aalysis acts as a mathematical microscope that allows us to zoom i (to fid details) or zoom out (to fid global behavior) a sigal. Whe the sigal exhibits self-similarity uder differet aggregatio levels, wavelets are aturally able to reveal it by its scalig ability. Actually the iteratig approximatios of wavelet aalysis at coarser ad coarser resolutios are a implicit way of aggregatig data, ad evaluatig wavelet or details coefficiets is ust a refied way of computig icremets [3]..3. Orthogoal Wavelet Decompositio The icreasig process Y(t) ca be decomposed by orthogoal scalig fuctio wavelet fuctio (t) as follows [] [8][9]: φ k (t) φ k ad J φ 0 k (t) + Wkϕ k (t) 0 k = J0 k Y (t) = U () k with W k = Y(t) ϕ (t) dt k called wavelet or detail coefficiets U k = Y(t) φ k (t) dt called scalig or approximatio coefficiets. Where represets the umber of scales of wavelet decompositio. A larger correspods to a higher resolutio. k is the umber of scalig coefficiets or wavelet coefficiet at the coarsest level. If k >, it meas that we do ot keepig decomposig the sigal till the scale with oly oe umber, ad also meas we have several sub-trees i had with the iterpretatio of biary tree to wavelet decompositio. Istead of evaluatig wavelet ad scalig coefficiets by the ier product i the defiitio (), it is possible to recursively compute them by cascaded discrete filter [9]. Takig Haar wavelet as a example [][9], the wavelet ad scalig coefficiets ca be computed by U k +,k +,k + = / (U + U ) ad Wk +,k +,k + = / (U U ) () We ca see it is easy to aalyze ad sythesize the sigal of iterest. 3. Aalysis ad Sythesis So far we have had some backgrouds for the motivatio of a Multifractal Wavelet Model (MWM). Now we will show how to lik wavelets ad multifractal together ad how to costruct 6

a good model for etwork traffic i details. 3. Criterios for a good etwork traffic model ). Positive Data: Back to limitatios of fbm ad fg we metioed before, the key of our model is to geerate positive data. ). Multifractal Scalig: Ca the model capture the multifractal scalig behaviors give the real trace to be multifractal? Here the scalig behaviors focus o scalig of momet as the sigal is aggregated with differet level. 3). Multifractal Spectra: It is kid of similar to last criterio but with the differet respective. We have discussed multifractal spectrum is oe of mai criterios for a multifractal process, by which we ca fid what is the probability of occurrece of burstiess of etwork traffic. 4). Log-Rage Depedece (LRD): It is oe of the most importat properties of etwork traffic. 5). Queuig Behavior: The queuig behavior of etwork traffic is importat because it will affect the etwork maagemet algorithm such as admissio cotrol that supports certai QoS demads []. 3. Trace Aalysis First of all, we eed to kow how to extract the performace criterios from the trace data. 3.. Log-Rage Depedece All processes with exact self-similarity exhibit LRD [3]. A process Y with property that its correlatio is osummable is said to exhibit log-rag depedece (LRD). LRD ca be equivaletly characterized i terms of behavior of the aggregated processes [] Y (m) [] = m km Y[i] i= (k )m + here m is the umber of data at a certai aggregatio level. If Y is self-similar sigal defied by (3) equatio (4), the (3) implies that Y[] = m H Y (m) [], is the idex of data, ad var(y[]) = m H var(y (m) []) Takig log at both sides of (4), we get (4) (m) log var(y []) = ( H) log m + C C = log (var(y[])) is costat (5) If Y is of strict secod-order scalig property, the log-log plot should be liear. By gettig the (m) slope p of the log var(y []) as a fuctio log m, we ca fid 7

H=0.5*P+ If 0.5<H<, Y exhibits LRD property [3]. If we deal with the real data with Haar wavelet trasform, by equatios of (), we kow scalig coefficiets at every level are ust the aggregatio of the process ad m is equal to (scale =0 represets the fiest scale). To obtai a more accurate estimatio of variace, we leave 5 umbers at the highest aggregatio level, that is, the coarsest wavelet scale (see the routie PlotVariaceTime.m ). Network traffic is ot strict secod-order scalig process, but we ca still use variace time plot as a tool to test its LRD []. 3.. Multifractal Scalig We ca get the iformatio of multifractal scalig behaviors of etwork traffic from partitio fuctio. For coveiece, we repeat its defiitio: T(q) : = lim log E[ k = 0 ( k q [Y]) ] = lim log With a larger umber of measured data, we assume the process we are dealig with is ergodic []. Uder this assumptio, we ca replace expectatio with average of time or sample. After haar wavelet trasform, T (q) at every scale (aggregatio level ) ca be show as [][3] S (q) T (q) = log k= 0 / W k q = log S (q) log q q T(q) = log S (q) = ( + log Wk ) (6) k= 0 If the process Y is of multifactal scalig behavior, a liear relatio betwee ad log (S (q)) should be expected ad the slope of these lies are the values of T(q) at poit q s. Here is the scale of wavelet trasform. A larger represets a fier scale. It should be aware that whe q<0, T(q) is used to measure the data that is very small but ozero. Here ozero is of physical meaig, that is, whe we deal with measured data, the resolutio of measure istrumets should be thought as zeros istead of real zeros. 3..3 Multifractal Spectrum As show i sectio..3, we ca get every poit of ( α ) by T (q) with the help of Legere Trasform f G ( α ) = q α T(q ) ad T ( q ) = α (7) f G 8

3.3 Buidig MWM Now we discuss how to build a multifractal model with Haar wavelets trasform. 3.3. Positive Data Guaratee Wavelet-domai modelig of positive process is complicated due to the fact that the wavelet coefficiets costraits required esurig a positive output is ot trivial. By equatios () with Haar wavelet trasform, we ca get / U = + (U W ) ad,k k + k / U = + + (U W ) (8),k k k As we kow, all scalig coefficiets U k of etwork traffic data will ot be less tha zero. Hece it is surprisigly easy to guaratee a positive data with Haar wavelet trasform. All we eed to do is let W U []. (9) k k To build a statistical model for W k, we may fid a radom variable A supported o the iterval [-,] ad compute every wavelet coefficiet by W = A k k U k plug (0) ito (8), we get U ( + A U ) U /,k = + k k ad ( A ) U /,k = + + k k () (0) 3.3. Modelig The LRD Property Sice we have applied a radom variable A ito model to guaratee a positive data, we ca use the degrees of freedom i the pdf of this radom variable to cotrol the correlatio ad LRD property of modelig data [][]. Because of its simplicity ad flexibility, we will use a symmetric beta distributio pdf supported o [-,] β(p,p) with f (a) = p ( + a) ( a) p β(p, p) p It ca be show f (a) exhibits differet shapes with the chage of values of q, which is illustrated i figure []. Its variace is give by var(a) = E[A ] = () p + We already kow the wavelet trasform geerates approximate decorrelated wavelet coefficiets 9

Fig Beta pdf with differet q for a LRD sigal, so we ca capture the correlatio structure of modelig data by properly settig the eergy of wavelet coefficiets at each scale []. The eergy of wavelet coefficiets is computed by their secod momet. With the symmetric structure of radom variable A ad wavelet coefficiets at every scale with respect to origi, we kow their meas are both equal to zero. Hece we ca fid eergy by calculatig the variace. To make our model begi with a little bit data, we should fix the eergy at the coarsest scale ad set ratios of eergy for other scales. From (0) ad (), we have E(W ) = E(A )E(U J0k ) var(w ) E(U ) J 0k J0k J0k = J0k p + 0 the E(U J0k ) p = 0 (3) var(wj0k ) Note that we caot replace E(U is ot equal to zero. The eergy ratios are calculated by J0 k ) with var(uj0k) sice the mea of U J0k (scalig coefficiets) E[W,k ] Var(W,k ) η = = (4) E[W ] Var(W ) k usig (),() ad (4), η E[A ] = E[A ]( + E[A k Var(A ) = ]) Var(A )( + Var(A = )) (/(p /(p + ) + ))( + /(p + )) Fially we get 0

η = ( p + ) / (5) p To summarize, at every level, we fid p by η ad p, the by usig the radom umber geerator of beta fuctio with parameter p, we ca geerate some radom umbers betwee [-,] whose quatity is equal to the quatity of wavelet coefficiets at th scale. Whe fittig the radom umbers ito equatio (), at the fiest scale we get a series of scalig coefficiets, which are ust the realizatio of our model. We ca see the algorithm oly cost the time of O(N) to sythesize a N-poit data. 3.3.3 Modelig The Root Scalig Coefficiets The theory of cascaded filters tells us the procedure we use to sythesize the data should begi from coarsest approximatio (scalig coefficiets) ad add the details (wavelet coefficiets) recursively. Therefore the ob we still eed to do thus far is to model the root scalig coefficiets. If there are eough scales i wavelet trasform, the scalig coefficiets at coarsest scale ca be see as the sum of a large umber of idepedet radom variable. By Cetral Limt Theory we ca expect the root scalig coefficiets approximately follow Gaussia distributio [], thus we ca model them oly through their mea ad variace. For gettig a better statistical iformatio about the mea ad variace, we have to leave eough coefficiets at the coarsest scale. However, egative data are uavoidable because of Gaussia distributio, which will rui our whole model. Here the bottom lie to this assumptio is that the mea greatly outweighs the variace so that the probability of a egative value is very low. We model the root scalig coefficiets as follows: cotrol the umber of scales of wavelet trasform such that there are eough scales used ad eough data left at the coarsest scale (6 scales used ad 5 scalig coefficiets left i my simulatio); compute the mea ad variace of these data; geerate the Gaussia radom umbers with parameter mea ad variace whose quatity is equal to the quatity of scalig coefficiets; check ad make sure all the data is positive; 4. Simulatio ad Results 4. Real Trace Data The real trace we used is paug89 that recorded the first millio arrivals (about 34.8 secods) of the daylog trace started at :5 a.m., 9 August 989 o the "purple cable (the ickame of a Etheret cable at the Bellcore Morristow Research ad Egieerig facility,

buildig MRE-)". The actual accuracy is roughly 0 microsecods [0]. We chose iterarrival times as our traiig data because iterarrival times, beig cotiuous-valued, are most atural for our Multifractal Wavelet Model. Besides, ulike the data of packet iter-arrival times, byte-per-time, the aalysis of iterarrival times avoids the problem of choosig the time iterval []. We extract the iterarrival times from paug89 ad plot them i figure 3 grouped by 00, 0 ad packet. I fact it is a process of zoomig i. Fig 3 Iterarrival Times Of Real Trace The data plotted of the ext subfigure correspod the last 0% of the last subfigure. That meas, from left to right, (:000000) grouped by 00 packets, (900000:000000) grouped by 0 packets ad (990000:000000) grouped by packet are plotted, respectively. From the figures we get that the iterarrival times should fall ito the category of fractal process because it is irregular ad self-similar o matter how fie scale we see it. 4. Test Multifractal Scalig Behavior of Real Trace Usig equatio (6) we plot the relatio of ~ log S (q) i figure 4 (a). We ca see they are of liear relatios. We also plot agaist log (S (q))) i figure 4 (b) to show the liear relatio more clearly. ( From figure 4, we ca see the excellet scalig property at most scales (aggregatio levels) that shows the real trace paug89 is multifractal [][]. We ote that the two figures i figure 4 are kid of differet from the figure 8 of paper []. It is because the y axis i figure 4(a) is actually log (S (q)) that is cosistet with equatio (6). It does ot affect the fial result of partitio fuctio ad multifractal spectra which will be see soo.

(a) (b) Fig 4 Multifractal Scalig Behavior. (a) scale versus log (S (q)) (b) scale versus Delta( log (S (q)) Scale : from to 9 ad = is the coarsest scale. I both figures, from top to bottom, each lie represets differet momets q =[.5,.0,.5,.0, 0.8, 0.4, 0, -0.3, -0.5, -0.8, -, -.3]. I figure (a), the right triagles represet computed values ad the gree lies through the right triagles are the fitted lies i the sese of least square. 4.3 Geeratig MWM Data With β Radom Variable Based upo the results preseted i sectio 3.3 Buildig MWM (refer to the routie GerateMwmData.m ), we ca geerate oe set of data by MWM. We repeat the routies twice due to the u-determiistic property of the algorithm. We deote the two realizatios of MWM as Geerated Trace ad Geerated Trace ad paug89 as Real Trace. 4.4 Test Model By Criterios 4.4. Positive Data We plot geerated traces grouped by 00, 0, ad packet i figure 5 with the same approach as the real trace. It is clear that all the data geerated by MWM is positive. It also has the irregular shape similar to real trace plotted i figure. 4.4. Log Rage Depedece We plot the relatio of variace ad time By equatio (5) to test the LRD property of real trace ad geerated traces( refer the routie PlotVariaceTime.m ) i figure 6. From the relatio of variace ad time of real trace, we fid iterarrival times are ot strict secod-order scalig. However, we ca coclude the geerated traces are of LRD property. We 3

(a) Geerated Trace (b) Geerated Trace Fig 5 Two Realizatios Of MWM, grouped by 00, 0 ad packets. (a) Geerated Trace (b) Geerated Trace Fig 6 Variace-Time Plot. m is umber of the aggregated data, m= meas the fiest scale. ca see the geerated traces capture much of the correlatio structure except at coarser aggregatio levels (m=~6). All H is betwee [0.5,] that meas they exhibit LRD property. 4

4.4.3 Multifractal Scalig The relatio of ~ log S (q) ad ~ log (S (q))), for geerated traces, are show i ( figure 7. We ca see that the geerated traces exhibit similar multifractal scalig behavior to real trace except the momet q strogly egative, which ca bee see more clearly by partitio fuctios T(q) i figure 8. The values of T(q) are calculated with the slope of these lies got from fittig ~ log S (q) to be liear i least square sese. (a) Geerated Trace (b) Geerated Trace Fig 7 multifractal scalig Behavior of geerated traces Scale : from to 9 ad = is the coarsest scale. I both figures, from top to bottom, each lie represets differet momets q =[.5,.0,.5,.0, 0.8, 0.4, 0, -0.3, -0.5, -0.8, -]. The right triagles represet computed values ad the gree lies through the right triagles are the fitted lies i the sese of least square. 5

(a) Geerated Trace (b) Geerated Trace Fig 8 Partitio Fuctios Of Real Trace Ad Geerated Traces. The mageta lie is fitted lie to real trace, ad cya lie is fitted lie to geerated trace. We ca see geerated traces match real trace very well whe momet q is from 0.5 to. Why does it happe? As we have discussed i sectio 3.., whe q<0 i partitio fuctio, we are tryig to measure the data that is very small but ozero that is of physical meaig, I real trace, we treat 0-5 as its zero due to the resolutio of measure istrumets, but for geerated trace, we treat real zero as its zero because it is geerated by Matlab. I fact we fid the miimum order of geerated trace is 0 -, which is the reaso why more small values are foud i geerated trace, ad why the partitio fuctios of geerated traces deviate more from uiform distributio whe q<0. The mismatch whe q> is because we did ot modelig the scalig behavior of higher momets of real trace whe buildig our MWM model. It will be improved whe add these costraits ito our model. 4.4.4 Mutifractal Spectra Multifratal spectrum has similar iterpretatio with partitio fuctio because we compute them by partitio fuctios with the help of Legere trasform. By equatio (7), fidig the taget of T(q) at every poit, we plot the multifractal spectra of geerated traces ad real trace i figure 9. From figure 9, we see that the spectra of geerated traces closely match real trace whe f( α ) ear to, which meas the geerated trace capture much of the most ofte occurred evets. The similarity o the left of two spectra ( α <) idicates geerated traces also capture the property of bustiess of real trace. The divergece of the spectra of geerated traces from real trace o the right idicates that the frequecy of observig large α is kid of too high. I multifractal spectrum, larger α meas more small icremet, which correspods to the part of q<0 i partitio fuctio, the we have the same iterpretatio as before. 6

(a) Geerated Trace (b) Geerated Trace Fig 9 Multifractal Spectra Of Real Trace Ad Geerated Traces. 4.5 A Simple Approach To Improve The Performace Of MWM I [] authors discussed about how to improve the mismatches metioed above. They replace the beta distributio with poit mass distributio with two degrees of freedom i those scales of mismatch. Rather tha the complex method, we preset a simple solutio. The divergece of spectra of geerated traces from real trace is maily because the data resolutio of geerated traces is smaller tha the oe of real trace. If we trucate all geerated data to the resolutio of real trace, we get a set of data with the same resolutio as real trace. We used the same criterios to test the trucated data ad the simulatio results are show i figures 0. From figure 0, we ca see after the data of geerated trace are trucated to the same resolutio as real trace, the LRD property has almost o chage, but the multifractal scalig, partitio fuctio ad multifractal spectrum have the improvemet to some extet. Especially whe q<0 i partitio fuctio ad o the right of spectrum, the divergece from real trace is greatly reduced. Therefore, it ca be used as a simple way to improve the performace of MWM by trucatig the geerated data to the same resolutio as the real trace before we use geerated data to the applicatio. 7

(a) multifracral scalig behavior (b) Variace Time Plot (c) Partitio fuctio (d) multifractal spectra Fig 0 Simulatio Results For Trucated Data Of Geerated Trace 8

5. Coclusio Based o performace criterios, we develop a Multifractal Wavelet Model (MWM) ad show its effectiveess ad efficiecy by simulatio results. The Multifractal Wavelet Model ca capture most of the properties of etwork traffic, such as self-similarity, burstiess Log-Rage depedece (LRD) ad multiscalig behavior, which greatly overweighs other classical stochastic models. By usig the Haar wavelet trasform, the total cost of buildig the model is O(N) for a N-poit output, which is more suitable to be used to olie applicatios, such as some ogoig research distributed storage ad retrieval for multimedia [4]. 6. Future Work As it has bee show i sectio 3., queuig behavior is oe of importat performace criterios of etwork traffic models. It should be added ito our simulatio to show if MWM works well. From the mismatches of multifractal spectra ad partitio fuctios betwee geerated traces ad real trace (figure 8 ad figure 9), we see we eed to add more costraits to cotrol the behavior of geerated trace whe momets are beyod 0.5 ad. As we kow, etwork traffic behaves differetly i differet time ad places. It is ecessary to test the effectiveess of MWM with traffic trace of owadays because the trace we used i this paper is kid of old eve though it is typical. The low computatioal complexity of MWM provides some hope to real time applicatios, such as the olie etwork delay predictio based o etwork traffic model, which is what we are cotiue workig o. 7. Refereces [] R. Riedi, M.S. Crouse, V. Ribero, ad R.G. Baraiuk, A Multifractal Wavelet Model with Applicatio to Network Traffic, IEEE Tras. Ifo. Theory, Special issue o multiscale statistical sigal aalysis ad its applicatios, vol. 45, pp. 99-08, April 999. [] V. Ribeiro, M. Coates, R. Riedi, S. Sarvotham, B. Hedricks, ad R. Baraiuk, Multifractal Cross-traffic Estimatio, i Proc. ITC Specialist Semiar o IP Traffic Measuremet, Modelig, ad Maagemet, Moterey, CA, Sept. 000, pp. 5--5-0. [3] P. Abry, R. Baraiuk, P. Fladri, R. Riedi, ad D. Veitch, Multiscale Nature of Network Traffic, IEEE Sigal Processig Magazie, 8-46, May 00 [4] Jacques Vaisey, Aup Basu, Distributed Storage ad Retrieval for Multimedia 9

Commuicatios, A NESC Strategic Grat Applicatio, usubmitted. [5] David Harte, Multifractals, Theory ad applicatio, Chapma& Hall/CRC, 00 [6] Gregory Worell, sigal processig with Fractals: a Wavelet-based Approach, Pretice Hall, 996 [7] Keeth Falcoer, Fractal Geometry: Mathematical Foudatios ad Applicatios Joh Wille& Sos, 990 [8] Patrick Fladri, Wavelet Aalysis ad Sythesis of Fractioal Browia Motio, IEEE Trasactios o Iformatio Theory, Vol.38, No., March 99 [9] A. Abbste, C.M. Decusatis ad P. K. Das, Wavelets ad Subbads: Fudametals ad Applicatio, Birkhauser, 00 [0] http://ita.ee.lbl.gov/html/cotrib/bc-readme.txt [] R. Riedi ad J. L. vehel, Multifractal properties of TCP traffic: a umerical study, Tech. Rep. 39, INRIA Rocquecourt, Frace, Feb. 997[olie]. Available www.iria.fr/rrrt/publicatioseg.html [] Matthew S. Co\rouse, R. H. Riedi, V.J. Ribeiro ad R.G. Baraiuk, A Multifractal Wavelet Model For Positive Process, 998, IEEE available: www.dsp.rice.edu [3] T.Mirfakhrai, S. Payadeh, A Delay Predictio Approach for Teleoperatio over the Iteret, Proceedig of the 00 IEEE Iteratioal Coferece o Robotics& Automatio, Washigto, May, 00 [4] Q.P. Wag, D.L. Ta, Nig Xi ad Y.C. Wag, The Cotrol Orieted QoS: Aalysis ad Predictio, Preceedig of the 00 IEEE Iteratioal Coferece O Robotics& Automatio, Seoul, Korea, May, -6, 00 0