Superposition of Multiplicative Multifractal Trac Streams. Los Angeles, CA fjbgao,

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1 Proceedngs ICC'000, June, 000, New Orleans, Lousana. Superposton of Multplcatve Multfractal Trac Streams Janbo Gao and Izhak Rubn Department of Electrcal Engneerng Unversty of Calforna, Los Angeles Los Angeles, CA emal: fbgao, Abstract: Source trac streams as well as aggregated trac ows often exhbt long-range-dependent (LRD) propertes. In ths paper, we model each trac stream component through the multplcatve multfractal countng process trac model. We prove that the superposton of a nte number of multplcatve multfractal trac streams results n another multfractal stream. Ths property makes the multfractal trac model a versatle tool n modelng trac streams n computer communcaton networks. There, a node s loaded by a trac ow resultng from the superposton of source streams and aggregated LRD (and other) streams. The structure and the burstness of the supermposed process s studed, and useful mathematcal relatons are obtaned. Introducton Recent analyss of hgh{qualty trac measurements have revealed the prevalence of long-range-dependent (LRD) features n trac processes loadng packet swtchng communcatons networks. Included are local area networks (LANs) [8], wde area networks (WANs) [9], varable-bt-rate (VBR) vdeo trac [,7], and world wde web (WWW) trac []. Wth the prevalence of LRD trac ows n data networks, the modelng of such trac has become mportant. It has been shown that a multplcatve multfractal trac model can yeld smlar queueng performance results smultaneously for a network operatng at low, medum, and hgh utlzaton levels when one compares the system-sze tal performance of a sngle server queueng system drven on one hand by a multplcatve multfractal trac model and on the other hand by the measured trac streams such as ows across local and metropoltan area networks, nvolvng a multtude of applcatons such as VBR vdeo and WWW orented servces [3-5]. Quanttatve understandng on why a multplcatve multfractal process makes a good LRD trac model has also been obtaned [6]. In communcaton networks, the trac process loadng nodal swtchng and transmsson processors s noted to be descrbed as the superposton of multple nput trac streams. In ths paper, we use the countng process model [4] to prove that superposton of multplcatve multfractal trac streams results n another multfractal stream. Ths property allows us to model LRD trac ows at derent network locatons usng a sngle multplcatve multfractal countng process model. In partcular, when there are a number of ndependent users each generatng a LRD source trac modeled by a multplcatve multfractal, n so far as the aggregated trac s concerned, one need only to smulate one multplcatve multfractal for the aggregated traf- c nstead of smulatng a bunch of multfractals for all the users. The remanng of the paper s organzed as follows. In Sec. we revew brey the countng multplcatve multfractal trac process model. In Sec. 3, we prove that the superposton of a nte number of multplcatve multfractals results n another multfractal. The burstness of the supermposed process s studed n Sec. 4. Conclusons are gven n Sec. 5. Multplcatve multfractal countng process trac model Consder a unt nterval. Assocate t wth a unt mass. Dvde the unt nterval nto two (say, left and rght) segments of equal length. Also partton the mass nto two fractons, r and? r, and assgn them to the left and rght segments, respectvely. The parameter r, called the multpler, s n general a random varable, governed by a probablty densty functon (pdf) P (r), 0 r. Each new subnterval and ts assocated weght (or mass) are further dvded nto two parts followng the same rule. Ths procedure s schematcally shown n Fg., where the multpler r s wrtten as r, wth ndcatng the stage number. Note the scale (.e., the nterval length) assocated wth stage s?. We assume that P (r) s symmetrc about r = =, and has successve moments ; ; :::. Hence, the weghts at

2 the stage N, fw n (N); n = ; :::; N g, can be expressed as w n (N) = u u u N, where u l, l = ; :::; N, are ndependent dentcally dstrbuted (..d) random varables havng pdf P (r). When w n (N) s nterpreted as the loadng to a network (representng the total count of message unts) n a tme slot of length?n T, where T s the total tme perod one s nterested n, then ths process becomes a countng trac process model. The multfractalty of the multplcatve P process refers to the fact that M q () = E( N (w n= n(n)) q ) (q), wth =?N, (q) =? ln( q )= ln [6]. Fgure : Schematc llustratng the constructon rule of a multplcatve multfractal. 3 Superposton of multplcatve multfractals Consder the superposton of (an arbtrary) k ndependent multplcatve multfractal trac streams. Let these multfractal trac streams be denoted as MF,..., MF k. Ther multpler dstrbutons are P () (r); :::; P (k) (r). These dstrbutons are assumed to be symmetrc about =, and have successve moments q (), = ; :::; k, q = ; ;. The superm- trac stream s denoted by MF (sk) P, MF (sk) = Pposed k k = MF, wth 0 < ; :::; k <, = =. A weght w (sk) (N) of MF (sk) at P the stage N can then be expressed as w (sk) k (N) = = w () (N) = P k = u () u() at stage N, and u () N, where w() (N) s a weght of MF, = ; :::; N are..d random varables governed by pdf P () (r), for = ; :::; k. We rst prove the followng smple propertes for the weghts of MF (sk) at stage N. P () E(w (sk) (N)) = E(w (sk) k (N)) = = E(w () (N)) =?N ; = ; :::; N : () Snce MF,..., MF k are ndependent, we have V ar(w (sk) (N)) = V ar(w (sk) (N)) = P k = V ar(w() (N)); = ; :::; N : We can now estmate the Hurst parameter for MF (sk) by consderng the varance-tme relaton. Let W (m) = (w (sk) + + m?m+ w(sk) (m) m )=m, W = (w () + + m?m+ w() m )=m, P = ; :::; k; then we have V ar(w (m) K ) = = (m) V ar(w ); where V ar(w (m) ) = ( () )N (4 () )?k??n, = ; :::; k [6]. Snce each term n V ar(w (m) ) s an exponental term, for reasonably large N, only the one correspondng to the maxmum of (), = ; :::; k, domnates. Droppng other terms, we then obtan a powerlaw relaton between V ar(w (m) ) and m. We thus nd the Hurst parameter H (sk) for MF (sk) H (sk) mn(h () ; ; H (k) ); where H (), = ; ; k, are the Hurst parameters assocated wth MF (), = ; ; k. We thus observe a sharp derence between the superposton of multfractal trac and superposton of other LRD trac models such as fractonal Brownan moton processes or the heavy-taled ON/OFF models. In the latter stuatons, the supermposed process assumes a value for ts Hurst parameter whch s the maxmum of the source trac streams [0]. () E[(w (sk) P P (N)) q k ] = E[( = u () u() N )q ] = l[ (sk) q ] N q [+ 0 b =0 y N ]; where l, (sk) q, fb ; y ; y < ; = 0; :::; q 0 g, and q 0 are sutable constants. Proof: P P k E[( = u () u() N )q ] q! = E[ q (!qk! u () u() N )q ( k u (k) u (k) N )qk ] P = q! q!qk! [q (() q ) N ] [ qk k ((k) q k ) N ], where P k q = = q. Let (sk) q = max ( () q q ;:::;q k (k) q k ): () Assume among all the terms () q q (k) k, there are m terms that attan the above maxmal value. Group those terms together. P We can then wrte E[(w (sk) (N)) q ] = l[ (sk) q ] N q [ + 0 b =0 y N ], wth l = l(m), (sk) q, fb = b (m); y ; y < ; = 0; :::; q 0 g, and q 0 = q 0 (m) are sutable constants. Ths concludes the proof. From the above result, we mmedately have E[(w (sk) ) q ] N q ; N! : We are now ready to prove the followng theorem. Theorem : MF (sk) s (asymptotcally n N) a multfractal. The proof s qute straghtforward f one notces that M q () = N E[(w (sk) (N)) q ] (q), wth =?N, (q) =? ln( (sk) q )= ln, when N!.

3 We can prove that the moments of the multpler dstrbuton of MF (sk) are (asymptotcally n N) stagendependent. Hence, one expects MF (sk) to be for most practcal stuatons a multplcatve multfractal. Indeed, we have observed n our numercal smulatons that MF (sk) s asymptotcally a multplcatve process. To llustrate ths feature, we present an example nvolvng the superposton of multplcatve multfractal trac streams, MF and MF, wth the largest stage number beng 8. Let = = =, and the multpler dstrbutons for MF and MF be truncated Gaussan: P (r) e?(r?=) ; 0 r, wth = 50 and 00 for MF and MF, respectvely. To estmate the transformaton between w (s) (N + ) and w (s) (N), we use on one hand, by MF (e), and on the other hand, by MF (s), and compare the system sze tal dstrbutons. Fg. 3 shows ths comparson. We observe that the system-sze tal dstrbutons are almost dentcal for the two queueng systems for all the four utlzaton levels, = 0:3, 0.5, 0.7 and 0.9. Hence, MF (s) s ndeed equvalent to MF (e) n terms of queueng performance. r (N) = w(s)? (N + ) ; = ; :::; N ; () w (s) (N) where w (s) (N) and w (s)? = ; :::; N, are the weghts of MF (s) at stages N and N +, respectvely. We compute the dstrbuton P N (r) from ts hstogram based on fr (N); = ; :::; N g. We then plot P N (r) vs. r for derent stages N. Fg. shows P N (r) vs. r curves for N = 3; :::; 7. Fgure 3: System-sze tal dstrbutons obtaned when M F (s) (dashed lnes) and M F e (sold lnes) are used to drve dentcal sngle server queueng systems. The utlzaton levels are ndcated on the gure. 4 Burstness of the supermposed process Fgure : Multpler dstrbutons P (r) vs. r curves for the stage numbers N = 3; :::7. We observe that those curves neatly collapse together, ndcatng that P N (r) s qute ndependent of the stage number N for reasonably large values of N. Next we model MF (s) by a sngle deal multplcatve multfractal, MF (e). For ths purpose, we smply use the dstrbuton P N (r) as calculated for reasonably large values of N as the multpler dstrbuton for MF (e). From Fg. we nd that P N (r) can be well tted by a truncated Gaussan dstrbuton wth = 67. We then generate MF (e) tll stage N = 8. We drve a sngle server queueng system, In ths secton, we study the burstness of the supermposed multfractal trac processes. A trac process A s sad to be more bursty than a trac process B f a sngle server queueng system yelds a longer system sze tal dstrbuton when the process A s used to drve the queueng system. For ease of exposton, we shall only consder superposton of two trac streams. Furthermore, we assume that = = =, each source trac stream has 8 countng states, and all source trac streams have the same mean. We consder three cases: () Superposton of two homogeneous sources. That s, MF and MF are two derent realzatons of a multplcatve multfractal process wth a gven multpler dstrbuton. () Superposton of two heterogeneous sources. That s, MF and MF have derent multpler dstrbutons fp (r)g. () Superposton of a multplcatve multfractal process wth a Posson process. We shall show that n case (), MF (s) s always less bursty than ether MF or MF. In case (), f we assume MF to be more bursty than MF, then MF (s) s always less bursty than MF, but can be more bursty than

4 MF. In case (), MF (s) s less bursty than the multfractal trac component, whle more bursty than the Posson process. The latter can actually be substtuted by a determnstc process. Note the latter to be a partcular multplcatve multfractal process wth P (r) = (r? =). A good startng pont for the study of cases () and () s to employ the varance-tme relaton obtaned n the last secton. Wthout loss of generalty, we assume that MF s at least as bursty as MF. When the multpler dstrbutons fp (r)g for MF and MF are of the same functonal form (but wth derent parameters), we conclude the followng nequalty, V ar(w (m) ) V ar(w (m) ) [3,4]. We then have V ar(w (m) ) < V ar(w (m) ). Ths nequalty motvates our observaton that MF (s) s less bursty than MF. Ths s vered to be the case by the followng numercal examples (as well as other cases examned by us and not presented here). wth (p; d) = (0:66; 0:3). The system-sze tal dstrbutons for queueng systems operatng at utlzaton level = 0:9 when MF, MF, and MF (s) are used to drve them are shown n Fg. 4. We observe that MF (s) s clearly less bursty than ether MF or MF. For case () nvolvng the superposton of two heterogeneous sources, let P (r) be gven by truncated Gaussan wth = 50 for MF and = 00 for MF. We generate one realzaton for MF, and realzatons for MF, and consder superposton of the realzaton of MF wth those of MF. The systemsze tal dstrbutons for queueng systems operatng at utlzaton level = 0:7 when MF, MF, and MF (s) are used to drve them are shown n Fg. 5. Fgure 5: System-sze tal dstrbutons obtaned when M F (s), M F and M F are used to drve dentcal sngle server queueng systems. The utlzaton level s 0.7. Fgure 4: System-sze tal dstrbutons obtaned when M F (s), M F and M F are used to drve dentcal sngle server queueng systems. The utlzaton level s 0.9. We examne rst case () nvolvng the superposton of two homogeneous sources. We consder two realzatons of a multplcatve process wth ts P (r) functon gven by P (r) = q + p(r? =) =? d r = + d 0 otherwse We observe that whle MF (s) s always less bursty than MF, t can be both more bursty or less bursty than MF. Closer examnaton reveals that when MF (s) s more bursty than MF, the superposton of MF and MF s n phase,.e., at the rst few stages (correspondng to long tme scales), large weghts of MF are added to large weghts of MF. When MF (s) s less bursty than MF, the superposton s out of phase,.e., large weghts of MF at the rst several stages are added to small weghts of MF. Fnally, we examne case () nvolvng the superposton of a multplcatve process and a Posson process. Snce a Posson trac process s much much less bursty than a multplcatve multfractal trac process, we expect that t can be eectvely approxmated by a determnstc process. Let our MF be the same as that dscussed n case (). Denote the superposton of MF and a Posson process by MF (+p), and the superposton of MF and a determnstc process by MF (+d). We then drve a sngle server queueng system by MF (+p) and MF (+d), and compute the system sze tal dstrbutons under derent utlzaton levels. The results are shown n Fg. 6. Clearly we see that MF (+p) s equv-

5 alent to MF (+d). In other words, a Posson process can be substtuted by the determnstc process. We have studed the superposton of multplcatve multfractal countng processes. We have proved that the superposton of an arbtrary nte number of multplcatve multfractal trac streams results (asymptotcally) n another multfractal. The Hurst parameter for the supermposed process s the same as the correspondng one for the source trac stream that has the largest second moment of the multpler dstrbuton. Furthermore, we nd n numercal smulatons that the supermposed process s typcally asymptotcally a multplcatve process, and can be modeled by a sngle deal multplcatve multfractal. These propertes ensure that trac streams representng LRD source trac as well as LRD aggregated trac n a communcatons network can be characterzed by a sngle convenent model. In partcular, these results shed lght on why aggregated LAN and WAN trac streams can be eectvely represented as an deal multplcatve multfractal trac stream [3,4]. We have also examned the burstness of the supermposed trac streams by measurng the tal dstrbutons they nduce when appled to a sngle server queueng system. By examnng a wde range of LRD trac cases, we demonstrate that the supermposed process s less bursty than the most bursty trac component. We also note that when a Posson process s supermposed wth a bursty multplcatve multfractal trac (as a nonneglgble component), the Posson component can be eectvely replaced by a determnstc process n dervng a supermposed process that provdes the same queueng tal features as those exhbted by the orgnal process. Acknowledgment Ths work s supported by UC MICRO/SBC Pacc Bell research grant 98-3 and by ARO grant DAAGIJ Fgure 6: System-sze tal dstrbutons obtaned when M F (+p) (sold curves) and M F (+d) (dashed lnes) are used to drve a queueng system. 5 Conclusons REFERENCES [] J. Beran, R. Sherman, M.S. Taqqu, and W. Wllnger, 995: Long-range-dependence n varable-btrate vdeo trac. IEEE Trans. on Commun., { 579. [] M.E. Crovella and A. Bestavros, Self-smlarty n World Wde Web Trac: Evdence and Possble Causes. IEEE/ACM Trans. on Networkng, 5, 835{ 846. [3] J.B. Gao and I. Rubn: Multplcatve Multfractal Modelng of Long-Range-Dependent Trac. Proceedngs ICC'99, Vancouver, Canada, June, 999. [4] J.B. Gao and I. Rubn: Multfractal modelng of countng processes of Long-Range Dependent network trac, Proceedngs SCS Advanced Smulaton Technologes Conference,San Dego, CA, 999. [5] J.B. Gao and I. Rubn: Multfractal Analyss and Modelng of VBR Vdeo Trac, Electron. Lett., n press. [6] J.B. Gao and I. Rubn, 000: Statstcal propertes of multplcatve multfractal processes n modelng telecommuncatons trac streams. Electron. Lett., 36, 0. [7] M.W. Garret and W. Wlllnger, Analyss, modelng and generaton of self-smlar VBR vdeo trac. In Proc. ACM SIGCOMM, London, England, 994. [8] W.E. Leland, M.S. Taqqu, W. Wllnger, and D.V. Wlson, 994: On the self-smlar nature of Ethernet trac (extended verson). IEEE/ACM Trans. on Networkng,, {5. [9] V. Paxson and S. Floyd, 995: Wde Area Trac- The falure of Posson modelng. IEEE/ACM Trans. on Networkng, 3 6{44. [0] W. Wllnger, M.S. Taqqu, M.S. Sherman, and D.V. Wlson, 997: Self-smlarty through hghvarablty: Statstcal analyss of ethernet LAN trac at the source level. IEEE/ACM Trans. on Networkng, 5 7{86.