New Exploration of Packet-Pair Probing for Available Bandwidth Estimation and Traffic Characterization

Transcription

1 New Exploraton of Packet-Par Probng for Avalable Bandwdth Estmaton and Traffc Characterzaton Yu Cheng, Vkram Ravndran, Alberto Leon-Garca, Hsao-Hwa Chen Department of Electrcal and Computer Engneerng, Illnos Insttute of Technology, Chcago, IL, USA Emal: Department of Electrcal and Computer Engneerng, Unversty of Toronto, Toronto, ON, Canada Emals: {v.ravndran, Insttute of Communcatons Engneerng, Natonal Sun Yat-Sen Unversty, Kaohsung Cty, Tawan Emals: Abstract The Packet-par dsperson technques are the most common probng-based approach to measurng the bottleneck capacty of a path. In practce, the dsperson measurement, and therefore the bandwdth estmaton, could be serously dstorted by the cross traffc queung between or n front of the probe packet par. Almost all the exstng packet-par technques depend on heurstc flterng methods to fnd a fnal capacty estmate. In ths paper, we take a dfferent perspectve to explot the crosstraffc effect. We develop a queueng model to descrbe the output packet-par dspersons nterfered by the cross traffc, based on whch a new measurement technque to estmate the avalable bandwdth s derved. Another mportant contrbuton s that we for the frst tme reveal that the statstcs of the cross traffc, e.g. the margnal dstrbuton and the autocovarance functon of the arrval process, can also be nferred from the stochastc behavor of the output packet dspersons. Effcency of the proposed avalable bandwdth estmaton and traffc characterzaton technques are demonstrated by computer smulatons. I. INTRODUCTION The end-based network measurement has attracted wde attenton n recent years, by whch nternal network status s ndrectly nferred from end-to-end or edge-to-edge actve or passve traffc measurements 1. The end-based approach s more feasble than the router-based approach by requrng cooperaton of only the path end-ponts. Whle the end-based approach greatly facltates customers awareness of network status, t can also brng hgher flexblty and scalablty to the network management 2, 3. Packet dsperson technques 4 7 are the most popular end-based approach for bandwdth measurement. Two bandwdth metrcs that are commonly assocated wth a path are the capacty and the avalable bandwdth 7. Accordng to 7, we refer to the capacty lmtng lnk as narrow lnk and to the avalable bandwdth lmtng lnk as tght lnk, to avod the ambguty of the term bottleneck lnk whch has been wdely used for both metrcs. Among the packet dsperson technques, the pack-par dsperson technques are normally used for measurng the capacty of a path, and the packet-tran dsperson technques normally for measurng the avalable bandwdth of a path 8. Ths paper focuses on the packetpar technques. The basc dea of the packet-par technque s that when a back-to-back probe packet par of the same sze s launched nto a network, assumng no cross-traffc, the packet par wll be maxmally dspersed at the narrow lnk, and the dsperson wll go through the downstream lnks to reach the destnaton. If knowng the probe packet sze, the recever can then estmate the capacty of the path from the measured dsperson. Unfortunately, the cross traffc, referrng to the Internet traffc that shares the same frst-come-frst-serve (FCFS) buffer wth the probe packet par, exst n practce. The dsperson receved can be ether expanded by the cross-traffc packets queueng between the probe packet par at the narrow lnk, or compressed by the cross-traffc packets queueng n front of the probe par downstream of the narrow lnk. The expanson or compresson of the dsperson wll lead to under-estmaton or over-estmaton of the path capacty, respectvely. To overcome ths problem, the common approach s to generate a reasonable number of dsperson samples, maybe from packet pars wth dfferent szes, and use some heurstc flterng methods to fnd the fnal bandwdth estmaton 5, 6, 9, 1. In ths paper, we gve a new explotaton of the packetpar dsperson technques by developng a queueng model to descrbe the mpact of the cross traffc on the packet-par dsperson. The study s motvated by the objectve to buld a systematc methodology to nfer nternal network status from end-to-end measurements. Based on the queueng model, we fnd that the queueng behavor of the probe packet par can be dvded nto two stuatons. One s that the packet par are served n the same busy perod of the queue, where the packet par s termed as a unted par; the other s that the two probe packets fall nto dfferent busy perods, where the packet par s termed as a dvded par. The dspersons from the unted pars and dvded pars have dfferent dstrbutons. Whle the dvded-par dspersons are exploted to derve a smple and effcent avalable bandwdth measurement approach, the unted-par dspersons are exploted for traffc character-

2 zaton, that s, extractng the statstcs (e.g. the margnal dstrbuton and the autocovarance functon 11, 12) of the cross traffc. It s noteworthy that traffc characterzaton can be used to enhance QoS control on the customer sde and facltate resource management on the network sde through queueng analyss or effectve bandwdth technques To the best of our knowledge, no smlar dsperson technque for traffc characterzaton has been presented n the lterature. The remander of ths paper s organzed as follows. Secton II gves a revew of the related work. Secton III descrbes the queueng model governng the output dspersons under the cross traffc. Secton IV and Secton V present the proposed avalable bandwdth measurement and the traffc characterzaton technques, respectvely. Secton VI presents smulaton results to demonstrate the performance. Secton VII gves the concludng remarks. II. RELATED WORK It s ndcated n 15 that the dsperson expanson or compresson due to the cross-traffc leads to multmodal dstrbuton of the bandwdth measurements. The relatonshp between the network load, cross-traffc packet sze, probe packet sze and the locaton and strength of dfferent modes has been nvestgated n 7, based on whch a flterng methodology s proposed. Pasztor and Vetch proposed a queueng model n 16 to formalze and unfy the network dsperson effect, whch s the most related work to our study. Nevertheless, we step further n ths paper to decode the statstc nformaton of the cross traffc from the dsperson measurements. Moreover, n 16 the avalable bandwdth s estmated by unted-par measurement (accordng to our termnologes), whle n ths paper we estmate the avalable bandwdth by dvded-par measurement. Wth FCFS forwardng, the avalable bandwdth s only meanngful over an averagng tmescale 8. Therefore, the packet-tran dsperson technques 6, 17, nstead of the packet-par dsperson, are normally used to measure the avalable bandwdth. Recently, Dovrols, Ramanathan, and Moore showed that the dsperson of long packet trans does not measure the avalable bandwdth of a path; nstead, t measures a dfferent throughput metrc referred to as the asymptotc dsperson rate (ADR) 7, 18. The SLoPS technque 8 detects the avalable bandwdth by sendng probe packet streams wth varable testng rates, based on the observaton that the one-way delay of a packet stream ncrease obvously when the streams rate s hgher than the avalable bandwdth. The TOPP technque 19 shows that such a delay prncple can be combned wth the packet-par dsperson measurement to estmate the avalable bandwdth. Note that the avalable bandwdth estmaton technques proposed n ths paper also use packet-par dsperson measurement, but based on queueng analyss. III. PACKET-PAIR DISPERSION In ths secton, we descrbe a queueng model to formalze the packet-par dsperson behavor. We model a store-and- 1 g 1 τ 1 g 1 τ 1 1 τ 1 W 1 g 2 g W 1 2 τ (a) 1 W 2 τ 1 τ 1 2 τ (b) W g τ t t Hop -1 Hop Hop +1 Hop -1 Hop Hop +1 Fg. 1. Two types of dsperson behavors: (a) the second packet reaches queue before the frst leaves, (b) the second packet reaches queue after the frst has departed. forward router on a path as a FCFS queue whch s served at a fxed rate of C bts per second (or equvalently bytes per second), the capacty of the lnk connected to the router. A network path consstng of lnks from 1 to n s modeled as the concatenaton of n queues, where each queue has servng capacty (1 n). We defne a probe packet par as two consecutve probe packets that are transmtted along a path between the source and the destnaton. The frst packet n the par has a sze L bts, and the second packet has a sze L. If L = L, then the packet-par s called symmetrc, otherwse t s called asymmetrc. We refer to other packets n the network other than the probe traffc as cross traffc. A. Dsperson Governng Equatons The packet-par dsperson at a gven pont s defned as the the tme when the last bt of the frst packet arrves at the pont to the tme when the last bt of the second packet arrves. The dsperson of a packet par leavng queue s denoted as, also termed as the output dsperson of queue. We assume that the propagaton delay along a lnk s constant, and therefore the output dsperson from a queue s equal to the nput dsperson at the next downstream queue. The packet-par dsperson behavor at a queue can fall n two cases 7, as llustrated n Fg. 1. In one case the second probe packet arrves before the frst probe packet leaves the queue, and n the other case the second probe packet arrves after the frst probe packet has left. For queue, the two cases of the output dsperson are descrbed by the followng governng equatons 7: = where σ (j) { σ f 1 σ 1 + σ σ otherwse = τ (j) + W (j) and τ (j) = L(j) for j = 1, 2. W s the s the queueng delay of the frst probe packet, and W queueng delay of the second probe packet due to the cross traffc nserted between the probe packet par.

3 B. Cross-Traffc Effect In the dsperson governng equatons, the cross traffc affects the output dsperson through the queueng delay W (j). To explctly descrbe the cross traffc effect, we classfy the packet-par probng nto two cases. The frst case s defned as unted-par probng, where the two probe packets fall n the same busy perod of the queueng process. In ths case, the server s always busy before the second packet arrves, whle the frst probe packet may depart before or after the second probe packet arrves. Let A(t) denote the cross-traffc arrval process (n bytes) to the queue, the output dsperson n ths case s = A( 1) + L. The second case s defned as dvded-par probng, where the two probe packets fall n dfferent busy perods of the queueng process. In ths case, the frst probe packet defntely leaves before the second probe packet arrves, as shown n Fg. 1(b). The output dsperson n ths case s = 1 + L L + W W. (3) In summary, the output dsperson under the cross-traffc effect can be expressed as = { A( 1) + L 1 + L L + W W IV. AVAILABLE BANDWIDTH ESTIMATION for unted-pars for dvded-pars. (4) Accordng to the dsperson governng equaton (4), the unted-par measurement s normally combned wth the exstng flterng technques to estmate the capacty of the narrow lnk along a path 7. In ths secton, we propose a novel dvded-par measurement method derved from (4) to estmate the avalable bandwdth of a path. The avalable bandwdth s estmated through the utlzaton measurement. In a FCFS work-conservng queue, the utlzaton factor u s defned as the proporton of tme that the lnk s busy. Assume the queue s nfnte and no packet loss happens, the utlzaton of lnk- can be expressed as u = r (5) A where s the lnk capacty, and r = lm (t) t t s the average rate of the traffc arrval process A (t) to queue-. Accordng to 7, the avalable bandwdth at lnk s defned as A = (1 u ), and the avalable bandwdth of a path havng H hops s A = mn (1 u ) = A tght-lnk. (6) =1...H Wth symmetrc unted-par probng, t s not dffcult to derve from (4) that u = E L / 1. In ths method, the lnk capacty should be measured frst for utlzaton estmaton; the capacty estmaton error wll then mpact the utlzaton measurement. Such a dsadvantage can be avoded by dvded-par measurement. If we use symmetrc dvded-pars, from (4), we know that = 1 + W W. (7) As the two probe packets are n dfferent busy perods, the amount of queued traffc they see (and therefore the two delaytmes W and W ) are ndependent and have the same dstrbuton, assumng a statonary queueng process. In order to fnd the utlzaton, we frst nvestgate the probablty of W = W. Consderng the possble two cases, we have P W = W =P W = W = + P W = W. It s far more lkely for the watng tmes to be equal due to that both probe packets found the queue empty (a common occurrence, especally n low-utlzaton cases) than due to that the two non-zero watng tmes are equal. Therefore, t s reasonable to get the approxmaton P W = W P = P =, W = P W W = W =. (8) Furthermore, accordng to the defnton of the utlzaton factor, we have 1, P W = = P W = P lnk s dle = 1 u. Combnng (7), (8), and (9), we obtan P = 1 = P W = W (1 u ) 2 (1) where an estmator of u can then be derved as (9) û 1 P = 1. (11) It s expected that the estmator of (11) has a good performance when the utlzaton s not very hgh, whch wll be demonstrated through smulatons n Secton VI. The estmator can also be appled to a mult-queue path. Wthout loss of generalty, we denote lnk-1 as the tght lnk of a path wth H hops. The relatonshp of (1) can then be extended to H P = 1 (1 u 1 ) 2 (1 u ) 2. (12) If the utlzaton of other lnks s much lower than that of H the tght lnk,.e. =2 (1 u ) 2 1, the estmator of (11) can then well approxmate u 1. In a more strct sense, t s not dffcult to see that the estmaton from (11) gves a upper bound of u 1 n the mult-queue case. To the best our knowledge, no measurement technque smlar to the proposed dvded-par estmaton has yet been presented n the lterature. =2 1 If the nput process s a Posson process, the n (9) can be replaced by a =, accordng to the PASTA theory (.e. Posson arrvals see tme average).

4 V. TRAFFIC CHARACTERIZATION In ths secton, we show that the packet-par dsperson measurements can be used to estmate the statstcal characterstcs, e.g. the margnal dstrbuton and the autocovarance functon of the cross-traffc packet arrval process. To facltate the estmaton, we use the ndependent and dentcally dstrbuted (d) packet-sze assumpton and the fact that Internet packet sze follows a known mult-modal dstrbuton. A. Internet Packet Sze Dstrbuton Internet measurements have shown that packets of a few szes tend to make up the vast majorty of Internet packets As an example, we refer to the measurements performed by the Cooperatve Assocaton for Internet Data Analyss (CAIDA) on the traffc passng through the NASA Ames Internet Exchange (AIX) between February 21st and 27th, 2 2. The cumulatve dstrbuton functon of the packet szes s re-plotted n Fg. 2. From the fgure, t can be seen that the vast majorty of packets were of a few szes,.e. 4 bytes, 576 bytes, and 15 bytes, whch s attrbuted to the behavor of the Transport Control Protocol (TCP). As CAIDA does not provde the hstogram, we produce an approxmate numercal probablty mass functon (PMF) from Fg. 2: k {, 1,..., 39} k = 4 3 P S = k = k {41, 42,..., 575} (13) 7 k = k {577,..., 1499} 7 k = 15 B. Cross-Traffc Margnal Dstrbuton Estmaton In the followng, we develop a method to estmate the margnal dstrbuton of packet arrval process and the margnal packet-sze dstrbuton, based on the probablty generatng functon (PGF). Consder a unted-par wth dsperson s sent to probe a queue. Cross-traffc can be completely descrbed by two random varables, whch are the random varable N( ) to denote the number of packets that arrve n a gven nterval, and the random varable S to denote the packet sze. The total amount of cross-traffc A( ) can then be expressed as A( ) = N( ) n= S n, where S n are d random varables. Let us defne the PGFs of N( ) and X as G N( ) (z) = Ez N( ) = P N( ) = kz k (14) G S (z) = Ez S = P S = kz k. (15) Accordng to the ndependence assumptons gven at the begnnng of ths Secton, t s the well-known result 24 that G A( ) (z) = Ez A( ) = E Ez A( ) N( ) = G N( ) (G S (z)). (16) Fg. 2. Packet sze dstrbuton at NASA AIX, February 21 27,2 Wth unted-par measurement, the total cross-traffc volume can be obtaned from the dspersons. Agan, usng 1 and to represent the nput and output dspersons assocated wth the queue under consderaton, respectvely, we have A( 1 ) = L. (17) By sendng a reasonable number of probe packet pars, the hstogram of A( ) (we drop the subscrpton of 1 for the convenence of expresson), and therefore the correspondng PGF G A( ) (z), can be calculated. Usng q k ( ) to denote the hstogram probablty P A( ) = k, the PGF G A( ) (z) s then calculated as G A( ) (z) = q k ( )z k. (18) Gven G A( ) (z), our objectve s to estmate the margnal dstrbuton P N( ) = n, brefly denoted as p n ( ), for n =, 1,...,. One fact we are to use s that the Internet packet sze dstrbuton takes only a few, say m, domnant values as we have dscussed. Then, G S (z) reduces to G S (z) = a 1 z l1 + a 2 z l2 + + a m z lm (19) where l 1, l 2,..., l m, arranged as l 1 < l 2, < l m, are the m domnant packet szes and a 1, a 2,..., a m are the correspondng probabltes. Defne the probablty vector p = (p, p 1,..., p lm ), where p lk = a k (k = 1,... m) and other values n the vector are equal to. If use p (n) = (p (n), p(n) 1,..., p(n) nl m ) to denote the probablty vector of the random varable S (n) = n =1 S, and the convoluton operaton, we have For the PGF, we have p (n) = p p p (wth n fold). (2) nl m G S (n)(z) = (G S (z)) n = p (n) k zk. (21)

5 Based on the result of (21), (16) can be unwrapped as G A( ) (z) = G N( ) (G S (z)) = (G S (z)) n p n ( ) = n= n= nl m p (n) k zk p n ( ) = n= p (n) k p n( ) z k. (22) Comparng (18) and (22), we can get q k ( ) = p (n) k p n( ). (23) n= It s possble to solve p n ( ) from (23). However, t s mpractcal to estmate p n ( ) up to n. In practce, the cross-traffc nput to a queue s lmted by the upstream lnk capacty. Lettng L mn denote the mnmum packet sze allowed, we can obtan an upper bound of N( ) as and (23) reduces to N( ) N max = 1 /L mn (24) q k ( ) = N max n= p (n) k p n( ). (25) Wth vectors p (n) (n =, 1,..., Nmax) avalable from (2), defne p (k) = (p () k, p k,..., p(n max ) k ). Assume that we can fnd Nmax + 1 ponts of k,.e. k, k 1,..., k N max, so that the ( Nmax + 1 ) ( Nmax + 1 ) matrx P, wth p (k) for =, 1,..., N max ( ) as the row vectors, s nvertble. Moreover, from the hstogram of A( ), we can have the values of (q k ( ), q k1 ( ),..., q kn ( )), denoted as vector max q( ). Use the vector p( ) to denote the cross-traffc margnal dstrbuton (p ( ), p 1 ( ),..., p N max ( )), whch can be estmated as p( ) T = P 1 q( ) T. (26) In the above, we have dscussed the cross-traffc margnal dstrbuton estmaton n the sngle-queue context. In a multqueue path, f there s only one bottleneck, narrow as well as tght, and the downstream lnks only ncur gnorable delay, the estmaton of (26) s expected to have an acceptable performance. C. Packet Sze Dstrbuton Estmaton We can also estmate the packet sze dstrbuton based on (16), f the packet arrval margnal dstrbuton s avalable from traffc modelng or from hstorcal data measurements. The estmaton s straghtforward as G S (z) = G 1 N( ) (G A( )(z)). (27) For example, let s assume that the self-smlar Internet traffc can be well modeled by an M/G/ nput process 11. The Margnal dstrbuton s then Posson wth average rate λ packets/second, gvng G N( ) = e λ (z 1). Usng ths result, G S (z) = λ ln(g A( )(z)). (28) Wth the PGF known, the probablty vector p can be obtaned wth an nverse fast Fourer transform (IFFT) operaton. VI. PERFORMANCE EVALUATION In ths secton, we present some computer smulaton results to llustrate the performance of the proposed traffc characterzaton technques. We develop a sngle-queue smulator usng the event-drven smulaton technque n the MATLAB language. In all the smulaton examples, we use the d packetsze assumpton, and the cross-traffc packet sze dstrbuton follows the CAIDA PMF gven n (13). A. Utlzaton estmaton In ths example, we examne the performance of the utlzaton estmaton technque based on dvded-par measurement. We smulate a 1 Mbps queue. The packet-pars are symmetrc wth sze 15 bytes, and nter-par nterval s.5s. Posson cross traffc s sent for four dfferent target utlzatons (,,, and.7) and fve dfferent nput dspersons (.16,.32,.64,.128 and.256) second, to examne the goodness of the utlzaton estmator gven n (11) over a wde range of scenaros. Each estmate s obtaned from 1 trals, and batch method s appled to fnd the 99% confdence nterval. The estmaton results are gven n Fg. 3(a) 3(d). Correspondng to each estmaton pont, there s a actual utlzaton pont, whch s the accurate measurement of the true average utlzaton that occurs n the queue over the smulaton run. The actual utlzaton s obtaned by dvdng the total tme durng whch the queue s servcng a cross-traffc packet (probepacket servce tme s not counted) by the total tme of the smulaton run. From Fg. 3, we can see n all the utlzaton scenaros, the estmator has a robust performance where the actual utlzaton falls n the confdent estmaton range, except when nput dsperson s too small,.e. the case of.16 seconds. There the estmator sgnfcantly overestmate the actual utlzaton. The reason s that the small nput dsperson leads to unted packet pars, where the assumpton that the watng tmes of the two packets are ndependent breaks down, whch contrbutes to the estmaton devaton. B. Packet Sze Dstrbuton Estmaton In ths example, we llustrate the performance of the packet sze dstrbuton estmator gven n (27). A 1 Mbps queue s smulated. We consder Posson arrval process wth packet sze dstrbuton followng the CAIDA PMF. Four scenaros wth the combnaton of (u, ) = (,.6), (,.12), (.99,.6), and (.99,.12), are smulated. Probe packet sze s set correspondngly to guarantee the unte-par measurement, and nter-par nterval s.5s. In all the scenaros, a seres of 1 probe pars are sent for samplng, and Nmax = 12 s used consderng that N max p k ( ) 1.. The packet sze dstrbuton s estmated accordng to the followng procedure. Measure the output dspersons and convert the dspersons to traffc loads n bytes accordng to (17). Generate

6 Utlzaton estmate vs. nput dsperson for queue wth capacty 1Mbps Utlzaton Estmate p=(1-sqrt(pnput dsp.=output dsp.)) Actual utlzaton Utlzaton estmate (99% confdence nterval) Input dsperson (n seconds) (a) Utlzaton estmate vs. nput dsperson for queue wth capacty 1Mbps Utlzaton Estmate p=(1-sqrt(pnput dsp.=output dsp.)) Actual utlzaton Utlzaton estmate (99% confdence nterval) Input dsperson (n seconds) (c) Utlzaton estmate vs. nput dsperson for queue wth capacty 1Mbps Utlzaton Estmate p=(1-sqrt(pnput dsp.=output dsp.)) Actual utlzaton Utlzaton estmate (99% confdence nterval) Input dsperson (n seconds) Utlzaton (b) Utlzaton estmate vs. nput dsperson for queue wth capacty 1Mbps Estmate p=(1-sqrt(pnput dsp.=output dsp.)) Actual utlzaton Utlzaton estmate (99% confdence nterval) Input dsperson (n seconds) (d) Fg. 3. Utlzaton estmaton based on dvded-par measurement of a 1 Mbps queue wth target utlzaton (a), (b), (c), and (d).7. TABLE I ESTIMATED CROSS-TRAFFIC PACKET SIZE DISTRIBUTION, USING IFFT METHOD Scenaro Smulated Estmated probabltes of three man modes u True probabltes 7 7 the hstogram of A( ), and then take the (15N max + 1)-pont FFT of the hstogram to obtan G A (z). Calculate G S (z) = λ ln(g A(z)). Take the (15N max + 1)-pont IFFT of G S (z) to get the packet sze dstrbuton estmate ˆP {S = k}. The estmated PMFs for the four scenaros are plotted n Fgures 4(a) to 4(d). Although the maxmum packet sze s 15 bytes, the FFT and IFFT operatons wll spread the estmaton errors to all the ponts. In Fg. 4, ˆP {S = k} for k =, 1,..., 3 are plotted to llustrate the error spreadng. In all the four scenaros, the estmated dstrbuton correctly dentfes the three man packet szes,.e. 4, 576, and 15 bytes, and the estmated probabltes are qute close to the true CAIDA values. The estmate values of the three domnant probabltes are presented n Table I. In the fgures, we can observe that estmaton errors exst n all the scenaros and spread to ponts larger than 15. Also, the FFT and IFFT operatons lead to negatve estmates for some ponts. When the nput dsperson or the utlzaton ncreases, the estmaton error also ncrease. For example, the estmaton n Fg. 4(d) wth (u, ) = (.99,.12) s much noser than that n Fg. 4(a) wth (u, ) = (,.6). The estmaton error s due to the nsuffcency of measurement samples. In the case of (u, ) = (,.6), the packet arrval rate s.8443 packets/dsperson, correspondngly, 6 p k(.6) 1.. In the case of (u, ) = (.99,.12), the packet arrval rate s packets/dsperson, correspondngly, 12 p k(.12) 1.. Wth the same number of samples, many more modes need to be estmated n the latter case to obtan the hstogram of A( ), therefore the accuracy decreases. We have done experments to do estmaton wth more probng pars, the nose n the case of (u, ) = (.99,.12) can be obvously suppressed. C. Packet Arrval Process Characterzaton In ths paper, we omt the performance evaluaton of the margnal dstrbuton estmaton technque proposed n secton V-B, due to the paper length lmt. In stead, we gve a detaled nvestgaton n 25 of the traffc characterzaton technque usng packet-par probng. In 25, we dscuss how to construct a nvertble matrx P for margnal dstrbuton

7 Estmated probablty Estmated probablty.4.4 Packet sze n bytes (a) Packet sze n bytes (c) Estmated probablty Estmated probablty Packet sze n bytes.4 (b) Packet sze n bytes Fg. 4. Packet sze dstrbuton estmaton n four scenaros, wth (u, ) set as (a) (,.6), (b) (,.12), (c) (.99,.6), and (d) (.99,.12). estmaton, extend the estmaton technque to estmate the autocorrelaton functon of a self-smlar process, and present smulaton results to demonstrate the effcency of the traffc characterzaton for both Posson arrvals and self-smlar arrvals. Moreover, we dscuss how the edge-based traffc characterzaton s n fact a black box system that can be appled for end-to-end QoS provsonng along a path. VII. CONCLUSIONS In ths paper, we propose new packet-par dsperson technques for network measurement. As the dsperson measurement s taken at end systems or network edges, t consderably strengthen the customers capablty for QoS control and brng hgher flexblty and scalablty to network management. We develop a queueng model to descrbe the stochastc relatonshp among the nput dsperson, the cross traffc, and the output dsperson. Based on the queung model, new avalable bandwdth estmaton and traffc characterzaton technques have been derved. Partculary, traffc characterzaton,.e. estmatng the margnal dstrbuton and autocovarance functon of the traffc arrval process based on dsperson measurement, s a novel technque for the frst tme beng presented. Whle bandwdth estmaton manly focus on the control of the average throughput, traffc characterzaton can facltate the control of other QoS metrcs, e.g. the delay and loss rate. Performance of the proposed bandwdth estmaton and traffc characterzaton technques have been demonstrated through computer smulatons. We have dscussed the proposed dsperson-technques manly for a sngle-queue or sngle-bottleneck path. Developng effcent mult-queue measurement technques n a systematc way s stll an open problem. (d) REFERENCES 1 M. Coates, A. O. Hero III, R. 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