Anayss of Approach n Modeng Broadband Traffc R.G. Garroppo, S. Gordano, S. Lucett, and M. Pagano Department of Informaton Engneerng, Unversty of Psa Va Dotsav - 566 Psa - Itay {r.garroppo, s.gordano, s.ucett, m.pagano}@et.unp.t Abstract. The (Crcuant Moduated Posson Process) modeng approach represents an appeang souton snce t provdes the ntegraton of traffc measurement and modeng. At the same tme, t mantans the Markovan hypothess that permts anaytca transent and steady-state anayses of queueng systems usng effcent agorthms. These reevant features of approach has drven us to anayze n more detas the fttng procedure when t s apped to actua broadband traffc. In the paper, nvestgatng the estmaton agorthm of mode parameters, we emphasze the dffcuty of n capturng the upper ta of margna dstrbuton of actua data, whch eads to an optmstc evauaton of network performance. As shown n the paper, a smpe reaton exsts between the number of sgnfcant egenvaues obtaned by the spectra decomposton and the peak rate that the structure s abe to capture. The reaton evdences the dffcutes of to mode actua traffc, characterzed by ong taed dstrbuton, as we as traffc data wth the we accepted hypothess of gaussan margna.. Introducton The approach for modeng arrvas process by means of a crcuant moduated Posson process, provdes a technque for ntegraton of traffc measurement and modeng [0], mantanng, at the same tme, the Markovan hypothess that permts anaytca transent and steady-state studes of queueng systems usng effcent agorthms [9]. The deveoped modeng theory has permtted to study the mpact of power spectrum, bspectrum, trspectrum, and margna dstrbuton of the nput process on queueng behavor and oss rate. These studes have hghghted the key roe payed on the queueng performance by the margna dstrbuton, especay n the ow frequences regon [8]. The technque for the constructon of a that matches margna dstrbuton and autocorreaton functon of the observed process has been presented n [,9], where the authors showed smuaton resuts wth measured traffc data to prove the goodness of ths approach. In ths paper, further anayss of fttng procedure w be presented, hghghtng a mtaton of the mentoned agorthm n matchng accuracy for the margna dstrbuton of observed rate process. Moreover, the presented study determnes the maxmum peak rate captured by the mode once the spectrum has been matched and emphaszes the necessty of a E. Gregor et a. (Eds.): NETWORKING 00, LNCS 345, pp. 340-35, 00. Sprnger-Verag Bern Hedeberg 00
Anayss of Approach n Modeng Broadband Traffc 34 structure contanng a arge number of effectve egenvaues to adequatey capture even the ght ta of a gaussan functon, usuay accepted as reastc for traffc dstrbuton n the core network []. The reevance of these consderatons s reated to the mpact of the margna dstrbuton ta of nput traffc on queueng behavour. Indeed, as shown n the numerca anayss Secton, optmstc performance are estmated when the peak rate s not matched. On the other hand, the actua traffc rate has a margna dstrbuton that n some cases exhbts a ta heaver than Gaussan [3,5]; under such condton, the modes resut nadequate to estmate reastc queueng performance. Lasty, some advce to overcome the exposed mtaton are brefy ntroduced.. Background on The fttng procedure of a mode many conssts of three steps [9], whch are brefy summarzed n ths secton. In the frst step, the autocorreaton functon of the observed rate process s estmated and then matched by a sum of exponentas (wth compex parameters λ k ) weghted by rea and strcty postve power coeffcents ψ k. Ths matchng s a non-near probem and cannot be soved drecty. An approxmate, but qute accurate, souton s obtaned by usng the Prony agorthm [6] to express the autocorreaton functon n terms of compex exponentas wth compex coeffcents, and then satsfyng the constrants on the ψ k s (whch must be rea and strcty postve) by matchng the power spectra densty (PSD) usng the nonnegatve east square (NNLS) method. The second step ams to desgn the transton frequences matrx Q of the underyng moduant contnuous tme Markov chan. In order to ft the PSD of the modeed process, the egenvaues of Q must contan a the λ k s obtaned n the prevous step; the use of a crcuant matrx permts to sove the nverse egenvaues probem. An effcent procedure to sove ths probem s the Index Search Agorthm (ISA), presented n []. The ast step s then the estmaton of a vector γ assocated to the Possonan generaton of arrvas n each state of the moduatng Markov chan, such that the mode matches the cumuatve dstrbuton functon (CDF), F(x), of the observed rate process. In more detas, the fttng procedure starts consderng that the autocorreaton functon of a mode wth N states s expressed by the foowng: R N ( λ τ ) τ = ψ + ( ) 0 ψ exp (.) = wth postve rea ψ 's. The Fourer transform of (.) can be expressed by: S ( ω ) = π ψ δ ( ω ) + ψ b ( ω ) 0 N = (.)
34 R.G. Garroppo et a. λ where b ( ω ) = F [ exp( λ τ )] =, and ( ) ω + λ b ω dω = ; hence, the ψ s represent the power assocated to each λ. The λ s are the egenvaues of the transton matrx, whch must ncude a the effectve ones that derve by the exponenta decomposton of R(τ), the autocorreaton functon of the measured rate process. Usng the Prony method, the estmated R(τ) can be wrtten as p k = 0 ( λ ) R( τ ) ψ P, k exp P, k τ. (.3) The presence of a constant term n R (τ) requres λ P, 0 to be mposed equa to zero, and consequenty ψ 0 = ψ P, 0 : ths s smpy obtaned appyng the Prony method to the autocovarance functon ( ) ( ) traffc rate), snce ( ) C τ = R τ γ (γ s the mean vaue of the observed R τ γ, and from (.3) ψ P, 0 = γ. After the NNLS matchng, the expresson (.3) remans substantay unchanged and can be rewrtten as p k = ( λ ) R( τ ) ψ P, 0 + ψ P, k exp P, k τ (.4) beng aware that p, the ψ P, k s and the λ P, k s may not be the same as those of (.3) (they surey w not be n the case of compex egenvaues). The order p of the exponenta decomposton may be much ess than the order N of the mode, and thus n the constructon of the transton matrx ony few λ s w be mposed equa to the λ P,k s. Indcatng wth the vector of ndces (of dmenson p) such that λ [ k ] = λp, k, the reaton ψ [ k ] = ψ P, k consequenty hods. On the other hand, n order to obtan R () τ R() τ, a the other ψ s w be mposed equa to zero. After havng determned the transton matrx Q (note that many soutons are possbe for each set of egenvaues, snce the order N of the matrx s hgher than the number p of desred egenvaues), the thrd step,.e. the desgn of the rate vector γ such that F ( x) F( x), nvoves the mnmsaton of the dstance between F ( x) and F ( x), whch s obtaned by usng the Neder-Mead Smpex Search method. Snce F ( x) s a pecewse step functon, whch jumps by /N at each vaue γ n γ, the task s to determne the optma vector γ whch mnmses the quantty N γ γ (.5) = 0 where γ s obtaned by the quantzaton of F ( x) n eves, whose amptude s /N.
Anayss of Approach n Modeng Broadband Traffc 343 Defnng β = ψ exp( jϑ ), =0,,,N-, the vector β [ β β β ] = 0,,..., N represents the Dscrete Fourer Transform of γ, and ts Inverse can be expanded as N π γ = γ + ψ exp j ϑ, for =0,,,,N- = N where the expresson of β has been substtuted. In order to obtan rea γ, β must exhbt the Hermtan property (.e. * N β, β = whch corresponds to ψ N = ψ and ϑn = ϑ ). Indeed, f β does not satsfy the Hermtan property, ts Inverse Fnte Fourer Transform γ cannot be rea. Under the condton of Hermtanty on β, the above reaton assumes the foowng expresson N = π γ = γ + ψ cos ϑ, for =0,,,,N- (.6) N that permts to estmate γ by appyng the Neder-Meade Smpex Search method to (.5) as a functon of ϑ. The Hermtan condtons on β are automatcay satsfed for those power coeffcents reated to conjugated compex pars of egenvaues, but cannot stand for rea ones, snce ony one ψ s assocated to each of them. To overcome ths probem, each rea egenvaue needs to be consdered twce. In order to mantan the same correaton structure (or equvaenty the same PSD), the correspondng power coeffcents w be assumed equa to haf of the orgna ψ s. 3. Anayss of Fttng Procedure The nvestgaton presented n ths work nvoves the ast step of the fttng procedure and evdences a reevant mtaton on the ta behavor of the margna dstrbuton of modes. Ths mtaton may consderaby affect the evauaton of queueng performance of actua traffc, eadng to an underestmaton of network resources needed to guarantee the target QoS expressed n terms of oss probabty. The frst observaton on the fttng procedure s that, puttng τ=0 n (.4), the varance σ of the rate process can be expressed as the sum of the ψ for =,,, N-. On the other hand, the maxmum theoretca rate achevabe by the mode s derved by (.6) puttng a the cosnes equa to +. In ths case, a second reaton nvovng ψ s can be smpy obtaned: N γ = ψ = γ. (3.) The maxmum rate devaton from the mean vaue s then mted by (3.), under N the constrant ψ = σ =. As we stated before, ony the p ψ s assocated to the effectve egenvaues are non zero. Among these p power coeffcents, some are
344 R.G. Garroppo et a. reated to rea λ, hence each of them needs to be spt nto two terms wth haved magntude. Therefore, the resutng set of coupes (λ, ψ ) after ths operaton conssts of q eements, wth p q << N. In the remanng of the paper, we w refer to (λ, ψ ) as eements of ths set; consequenty f λ R λ N- =λ and ψ N = ψ, wth ψ equa to haf of the orgna power coeffcent obtaned by NNLS agorthm. Thus reaton (3.) can be rewrtten as q γ γ = ψ [ k ] (3.) k= where s now a vector of ndces of dmenson q. In most actua cases, the measured peak rate s qute hgher than the mean vaue and then, n order to capture the ong taed behavor of the rate dstrbuton, the sum n equaton (3.) shoud be as arge as possbe. The probem of maxmzng (3.) wth q the constrant = ψ [ ] = σ k k can be easy soved usng the Lagrange-Mutpers method, eadng to the souton Consequenty ψ = q. (3.3) [ k ] σ max ψ { γ γ } = ψ [ k ] q k= q σ (3.4) hods. Ths equaton represents the ntrnsc mtaton of n terms of maxmum achevabe rate as a functon of the number q of effectve egenvaues, n the hypothess of eveny dstrbuted power coeffcents. Consderng that the dstrbuton of the amptudes of the domnant ψ s w hardy be ke (3.3), the γ vaue obtaned from the above reaton represents ony an upper bound, actuay dffcut to reach. However, (3.4) gves an ndcaton on the mnmum number of exponentas requred to reach a target peak rate, fxed the varance and the mean vaue of the observed rate process. As an exampe, suppose that the observed process presents a gaussan margna dstrbuton wth mean x and varance σ ; hence, the CDF s x x x F ( x) = Q σ where Q ( y ) = exp π y dx (3.5) Usng a 500 state to mode ths rate process, the CDF resuts dvded n ntervas whose heghts are /500= 0-3 (the hgh number of states has been chosen n order to obtan a fne quantzaton of CDF). The maxmum eve of the quantzed CDF s then mted to the vaue 0.998, correspondng to ( x x) σ equa to 3.09. The
Anayss of Approach n Modeng Broadband Traffc 345 comparson of ths reaton wth (3.4) eads to q=3.09 9.55. Therefore, the orgna autocovarance shoud be decomposed n, at east, 0 exponentas; more key they w not be suffcent, snce the assumpton that a power coeffcents ψ s are of equa magntude s not easy verfed. Indeed, a more reastc scenaro s that few ψ s (around 6 or 8, as supported by the anayss n the Numerca Secton) w be domnant wth respect to the others and consequenty the reproduced peak rate w be such that γ γ = 6 σ, n correspondence of whom the orgna CDF w assume the vaue Q ( 6) 0. 993. Hence, the ta of the mode w be shorter than the one of the observed rate process. The reevance of ths drawback can be ponted out envsagng that, especay for traffc whose power spectrum s concentrated n the ower regon of the frequences [7] (ths assumpton s supported by the sef smar nature hghghted by the recent modeng resuts based on the anayss of acqured traffc data []), the ta of the margna dstrbuton has a deep mpact on the network resources requred to guarantee a target oss probabty. q Note that from (.6) γ γ MIN = ψ [ k] can aso be derved, whch, together k= wth (3.), mpes the foowng genera reaton q γ γ = ψ [ k]. (3.6) k= 4. Numerca Resuts To test the reevance of the presented anayss, we consder two sets of smuatons: the frst one s carred out appyng the fttng procedure on synthetc data wth a we defned and known spectra decomposton, whereas the second one refers to actua traffc data. In the frst smuaton scenaro, we have generated two traces havng the same mean vaue, varance and power spectrum, dfferng ony by ther probabty densty functons (one s gaussan and the other one s tranguar). The vaues of the mean and the varance of the data (γ =4500 ces per second, σ =0 6 cps ) have been chosen n order to obtan a neggbe probabty of havng negatve vaues n the gaussan rate trace. In order to bud the modes of the two traces, we have frst decomposed the autocovarance functons nto a sum of compex exponentas. Then, appyng the NNLS agorthm, ony the three coupes of egenvaues shown n Tabe.(a) (here and n the remanng of the paper, the ** remnds that the rea egenvaue s consdered twce and that the correspondng ψ has been aready haved n magntude, accordng to the procedure descrbed n the prevous sectons) have turned out to be assocated to power coeffcents ψ dfferent from zero (note that ψ equas the mposed varance σ ).
346 R.G. Garroppo et a. Tabe. Egenvaues wth non-zero power coeffcent dervng from the NNLS agorthm: (a) Synthetc Gaussan trace; (b) trace Vdeoconference ; (c) trace October89 Accordng to (3.6), the maxmum devaton achevabe usng ths set of egenvaues s equa to γ γ = ψ 367 cps. (4.) The obtaned vaue s qute cose to the mt 6 σ 450 cps, snce the power coeffcent magntudes have a quas-unform dstrbuton. Two modes have been but from these egenvaues to match the two dfferent CDFs. The γ s for the two modes have resuted equa to 6660 cps and 6500 cps for the tranguar and the gaussan hypothess, respectvey. These mts do not consder the exponenta and I.I.D. generaton of arrvas (Possonan mcrodynamcs) n each state of the mode, whch affects the upper tas of margna dstrbutons, as we as the peak rates of generated traces. In partcuar, the atter ones are hgher than the respectve γ s, (see the thrd coumn of Tabe, whch summarzes the man statstcs of the anayzed data traffc and of the reated traces generated by the correspondng modes). The resuts of the fttngs n terms of PSD and compementary probabty (CP) of the generated traces n the gaussan hypothess are shown n Fg.. The exceent matchng of the PSD s not accompaned by an equvaenty good fttng of the dstrbuton functon. Ths msmatch s not easy reveaed by the CDF s comparson, hence a CP pot s aways requred to apprecate possbe dfferences n the upper tas. In the tranguar case, we observed good fttng resuts for both PSD and CP pots, but we do not report the reatve fgures here for sake of smpcty. In order to estmate the errors n the evauaton of queueng performance ntroduced by the msmatchng of the CDF ta, we have anayzed the resuts obtaned by means of dscrete-event smuatons. The smuatons have been carred out feedng a FCFS G/D//K queueng system wth the four traces; n the remanng of the paper we w ndcate wth µ the servant constant ce rate. The sze K of the buffer has been fxed equa to 50 ces, correspondng to a maxmum deay ntroduced by the queue, varabe wth the normazed offered oad, around 50msec. The traffc s competey contaned n the LF and the MF regon of the queue [7]. Therefore, no further fterng of traffc s possbe wthout osng a porton of ts spectrum,.e. MF components, whch woud affect the queueng performance evauaton. The resuts are shown n the ast two coumns of Tabe 4 and emphasze as the reduced peak of the trace reated to the gaussan case eads to a sghty optmstc resources aocaton (neary 6%). It s
Anayss of Approach n Modeng Broadband Traffc 347 mportant to hghght that the desgns have been repeated usng dfferent vaues for N, rangng from 00 to 500, but no apprecabe dfference has been notced. In the second set of smuatons, two rea traffc traces have been consdered: the two data sets are reated to a vdeoconference servce and a LAN traffc trace. The descrpton of the characterstcs of the vdeoconference traffc data are descrbed n [4], whereas the second trace s the we known October89 trace coected at Becore Labs. In both anayses, the data (.e. the estmates of the rate processes) refer to the equvaent number of ATM ces (cacuated as the number of transmtted bytes dvded by 48) per second observed n non overapped tme ntervas of ength Tu. In the frst case, we assume Tu equa to the frame perod,.e. 40 ms, whereas for the second traffc trace the vaue of 00 ms has been chosen. The partcuar shape of the autocovarance functon of the vdeoconference trace has ed to a spectra decomposton characterzed by the egenvaues reported n Tabe.(b). In ths rea scenaro, we have obtaned ony few egenvaues wth non-zero power coeffcents, enforcng the hypothess ntroduced n the prevous secton regardng the number of domnant ψ s. In ths case the maxmum achevabe devaton s equa to γ γ = ψ 5950 cps and the mean vaue s approxmatey 4350 cps. Hence, we shoud expect a peak rate of about 0300 cps, aganst a measured peak rate of about 000 cps. The peak rate error s very hgh (neary 43%) due to the very unfavorabe condton on the power coeffcent dstrbuton. Indeed, the power spectra decomposton of the consdered trace presents ony a snge coupe of domnant effectve rea egenvaues and, at the same tme, ts margna dstrbuton exhbts an upper ta behavor heaver than gaussan [4]. The modeng resuts are shown n Fg..(a) and.(b), whch represent the PSD and the compementary probabty matchng, respectvey. In partcuar, Fg..(b) shows ceary that the obtaned peak rate s much ower than the observed 000 cps, as expected. Ths arge dfference n terms of peak rate eads to a very optmstc evauaton of the queueng behavor, as ponted out by the smuatons resuts, see Tabe, coumns 4 and 5, whch contan the servant ce rate requred n order to reach a ce oss probabty of 0-4 and 0-5. The ast anayss refers to the frst 000 s of the above mentoned LAN traffc trace; we do not consder the entre data set snce a shft of the mean vaue has been notced out of ths tme perod. The reevant egenvaues obtaned after the NNLS matchng of the spectrum are reported n Tabe.(c), and correspond to a theoretca maxmum devaton γ γ 554 cps. Addng the estmated mean (about 606 cps) to ths γ = max γ of vaue, the mt 540 cps s obtaned, wth respect to a mode ( ) 45 cps and of a data set peak of 7800 cps. Fg. 3.(a) and 3.(b) present the matchng of PSD and compementary probabty respectvey: the former shows the good fttng of the consdered second order statstc, whereas the atter confrms the mtaton of the mode n capturng the upper ta behavor of margna dstrbuton. The ast two coumns of Tabe ceary evdence the entty of the reatve error n resources aocaton when the modes does not capture the upper ta of margna dstrbuton (and partcuary the peak ce rate).
348 R.G. Garroppo et a..5e+06 Gaussan Gaussan 0. Power Spectra Densty [cps^/(rad/s)] e+06 500000 Compementary Probabty 0.0 0.00 0.000 0 0 3 6 9 5 e-05 0 000 000 3000 4000 5000 6000 7000 8000 9000 W [rad/s] Rate [cps] (a) (b) Fg.. Comparson of (a) Power Spectra Densty and (b) Compementary Probabty of Synthetc Traffc Data Gaussan Case 3e+07 Vdeoconference Vdeoconference.5e+07 0. Power Spectra Densty [cps^/(rad/s)] e+07.5e+07 e+07 Compementary Probabty 0.0 0.00 5e+06 0.000 0 0 5 0 5 0 e-05 0 5000 0000 5000 0000 5000 (a) W [rad/sec] Rate [cps] (b) Fg.. Comparson of (a) Power Spectra Densty and (b) Compementary Probabty of Vdeoconference Trace.4e+07 October89 October89.e+07 0. Power Spectra Densty [cps^/(rad/s)] e+07 8e+06 6e+06 4e+06 Compementary Probabty 0.0 0.00 0.000 e+06 0 0 4 6 8 e-05 0 000 4000 6000 8000 0000 000 4000 6000 8000 (a) W [rad/s] Rate [cps] (b) Fg. 3. Comparson of (a) Power Spectra Densty and (b) Compementary Probabty of October89 Trace
Anayss of Approach n Modeng Broadband Traffc 349 Tabe. Reevant statstcs of anaysed traces and correspondng modes Mean [cps] Varance [cps ] Peak [cps] µ (P oss =0-4 ) [cps] µ (P oss =0-5 ) [cps] Tranguar 4493.00e+6 7000 5930 6033 ( mode) (4506) (.00e+6) (7500) 605 (+.6%) 670 (+.3%) Gaussan 4490.00e+6 8344 60 6390 ( mode) (4503) (0.99e+6) (7330) 5855 (-4.%) 600 (-5.9%) October89 606 7.48e+6 7600 5360 60 ( mode) (6030) (6.03e+6) (780) 0330 (-3.7%) 0590 (-34.3%) Vdeoconference 4350 6.5e+6 0975 6400 850 ( mode) (440) (6.0e+6) (00) 9670 (-4.0%) 980 (-47.0%) 0000 8000 6000 Vdeoconference Vdeoconference October89 Ocotber89 Mu [cps] 4000 000 October89 trace 0000 8000 6000 Vdeoconference trace 4000 - - -3-4 -5-6 Log0(Ce Loss Probabty) Fg. 4. Performance Comparson In the cases of the vdeoconference and LAN traces, a more compete anayss of modeng performance s shown n Fgure 4, where the target CLP s potted versus the servant ce rate needed to guarantee t. The anayss of Tabe ponts out two mportant resuts: the frst one concerns the crtca behavor of the mode even n the gaussan case, whch mpes an optmstc resource aocaton (4 to 6%). The second resut hghghts as actua traffc exhbts a sower decay of margna dstrbuton wth respect to the gaussan hypothess eadng to a ess sutabe envronment for the approach, evdenced by the arge errors suffered (n terms of peak rate, matchng of the upper ta of margna dstrbuton and consequenty of the network resources needed to guarantee a target ce oss probabty). 5. Improvement Proposas to Overcome the Drawback We have observed that, under the same condtons on q, the unform dstrbuton of ψ s magntudes s the ony one that guarantees the maxmum mode peak γ. Thus, t s desrabe to have a spectra decomposton wth power coeffcents exhbtng ths feature. To the am of comng cose to the unform dstrbuton and ncreasng the parameter q, a straght souton can be to spt each domnant exponenta n the sum of two or more terms. The j-th term of the exponenta decomposton s represented by
350 R.G. Garroppo et a. the parameters λ j and ψ j, as descrbed n Secton. Our suggeston s to obtan an equvaent contrbuton to the autocorreaton functon usng a number B of power coeffcents (assocated to the gven exponenta). Ths procedure resuts n havng B terms λ j,,ψ j, for each coupe λ j, ψ j, where λ j = λ and, j ψ j, = ψ j B, =,,,B. Consequenty, the autocorreaton functon remans unchanged, whereas the contrbuton of the decomposed coupe λ j, ψ j to the peak rate becomes B B ψ j ψ j, = = B ψ,.e. B tmes the orgna one (.e. ψ j j ). = = B Unfortunatey, ths souton aso presents a drawback, whose reevance needs further nvestgaton: the ncreased number of egenvaues to be assgned to Q can cause the ncrement of the mnmum number of the mode states N that permts the souton of the ISA probem. As a consequence, an hgher N means that the mode parameters dervaton takes a onger tme, thus mtng the use of approach n a rea tme performance estmator. Another possbe approach to overcome the presented mtaton s to ncrease the vaue of T. In such a way, the peaks of the traffc data w be reduced, hence makng easer to capture them by the mode. Unfortunatey, aso ths procedure presents some drawbacks. Frst of a, a mt exsts on the maxmum vaue of the tme quantum T. Indeed, usng hgher T can cause the oss of a sgnfcant porton of nformaton assocated to the traffc data [7], eadng to naccurate queueng performance evauaton. Furthermore, ncreasng T reduces the varance σ of the resutng trace. Consequenty, the constrant on the ψ s sum ( ψ =σ ) produces a set of power coeffcents of reduced magntude, vanshng the advantage ganed by the trace peak reducton. Further study s then needed to hghght the trend of the peak rate decay wth respect to the tme quantum T n dfferent smuaton scenaros, and to derve a reaton wth the ta behavor. 6. Concusons The paper presents an anayss of the agorthm for the measurement-based parameters estmaton of modes, rasng some warnngs to be aware of n the use of ths modeng approach. In partcuar, the man resut of the paper s the anaytca dervaton of an ntrnsc mtaton of the fttng procedure n the modeng of traffc characterzed by ong-taed margna dstrbuton. Furthermore, the anayss shows that even n the gaussan hypothess, a structure contanng a arge number of effectve egenvaues s necessary to adequatey ft the CDF. In genera, the mtaton manfests when the dfference between the peak and the mean rate of the traffc data set exceeds few tmes ts standard devaton. In ths condton the mode cannot match wth suffcent accuracy the behavor of the CDF upper ta, eadng to optmstc predcton of the network resources needed to guarantee the target QoS (n terms of ce oss probabty).
Anayss of Approach n Modeng Broadband Traffc 35 Dscrete-event smuatons drven by actua traffc data and synthetc traces, generated accordng to the correspondng modes, have confrmed the resuts of the proposed anaytca study. Moreover, the anayss of the smuaton resuts shows the practca reevance of the mtaton n a snge server queueng system, a reevant case study n performance comparson. In partcuar, the anayzed cases hghght the optmstc resources aocaton produced by the fttng errors on the CDF. The presented mtaton reduces the fed of appcabty of the modeng approach, hence further studes are needed to overcome ths drawback. To ths am, some advce are presented as possbe soutons to be nvestgated. References. R. G. Adde, M. Zuckerman, T.D. Neame, "Broadband Traffc Modeng: Smpe Soutons to Hard Probems" IEEE Communcatons Magazne, August 998, pp.88-95. H. Che, San-q L, Fast Agorthms for Measurement-Based Traffc Modeng, IEEE Journa on Seected Areas n Communcatons, June 998, pp. 6-65 3. R. G. Garroppo, S. Gordano, S. Porcare, G. Procss, "Testng α-stabe processes n modeng broadband teetraffc" Proc. of IEEE ICC 000, New Oreans, Lousana, USA, 8- June, 000 4. R. G. Garroppo, S. Gordano, M. Pagano, "Stochastc Features of VBR Vdeo Traffc and Queueng Workng Condtons: a Smuaton Study usng Chaotc Map Generator" n Proc. of IFIP Broadband Communcatons 99, Hong Kong, November 999 5. A. Karasards, D. Hatznakos, "A Non-Gaussan Sef-Smarty Processes for Broadband Heavy-Traffc Modeng", n Proc. of GLOBECOM 98, pp. 995-3000, Sdney, 998 6. Steven M. Kay, Modern Spectra Estmaton: Theory & Appcaton, Prentce-Ha, 988. 7. Y. Km, San-q L, Tmescaes of Interest n Traffc Measurement for Lnk Bandwdth Aocaton Desgn, Proc. IEEE, Infocom '96, March 996, pp. 738-748 8. San-q L, Cha-Ln Hwang, Queue Response to Input Correaton Functons: Contnuous Spectra Anayss, ACM/IEEE Transactons on Networkng, December 993, pp. 678-69 9. San-q L, Cha-Ln Hwang, On the Convergence of Traffc Measurement and Queueng Anayss: A Statstca-Matchng and Queueng (SMAQ) Too, IEEE/ACM Transactons on Networkng, February 997, pp. 95-0 0. San-q L, S. Park, D. Arfer, "SMAQ: A Measurement-Based Too for Traffc Modeng and Queueng Anayss. Part I: Desgn Methodooges and Software Archtecture", IEEE Communcatons Magazne, August 998, pp. 56-65. W.Wnger, M.S. Taqqu, A. Erram, "A bbographca gude to sef-smar traffc and performance modeng for modern hgh speed networks", n F.P. Key, S. Zachary and I. Zedns eds., Stochastc networks: Theory and Appcatons n Teecommuncaton Networks, Vo. 4 of Roya Statstca Socety Lecture Notes Seres, pp. 9-04, Oxford Unversty Press, Oxford, 996