On the Conflict of Truncated Random Variable vs. Heavy-Tail and Long Range Dependence in Computer and Network Simulation

Transcription

1 Journal of Computatonal Informaton Systems 7:5 (0) Avalable at On the Conflct of Truncated Random Varable vs. Heavy-Tal and Long Range Dependence n Computer and etwork Smulaton Wanyang DAI Department of Mathematcs and State Key Laboratory of ovel Software Technology, anjng Unversty, anjng 0093, Chna Abstract Heavy-taled dstrbuton (HTD) and long-range dependence (LRD) have appeared n the studes of telecommuncaton, fnance, and etc. However, the truncaton of a heavy-taled random varable s nevtable n a computer and network smulaton due to the hardware and software lmtatons such as numberng format of bnary dgts and pseudo random number generators. Thus a natural ssue arses: how well the approxmaton of a concerned objectve n a smulaton wll be f a truncated random varable s used to replace the orgnal heavy-taled one? As ponted out n ths paper, the dfference of the means (or varances) between the truncated verson and the orgnal one wll stll be nfnte f the orgnal random varable s heavly taled wth nfnte mean and/or varance, whch ndcates that the truncated approxmaton may be not consstent wth what should be f the HTD/LRD assumptons are mposed. So, based on ths observaton and our prevously establshed approxmaton theory va reflectng Gaussan process (RGP) wth or wthout LRD, we can provde more reasonable nterpretatons to some well-known large-scale computer and statstcal experments n characterzng the superposton of nternet traffc sources, whch clarfy that the fndngs n these smulatons should be more close to the realty but not to the mathematcal assumptons about HTD/LRD, whch are mposed n ther models and analyss. Hence our fndngs smooth, to some extent, the gaps between HTD/LRD theory and real-world smulatons. More mportantly, our fndngs also ndcate that t s more sutable to use the truncated verson of a heavy-taled random varable n a smulaton and related analyss n order to be more consstent wth realty and to avod more complcated dscussons va HTD/LRD theory. Keywords: Hgh-speed etwork; Source Traffc; Characterzaton; Queueng System; Computer and etwork Smulaton; Pseudo Random umber Generator; umberng Format of Bnary Dgts; Truncated Random Varable; Heavy-taled Dstrbuton; Long-range Dependence; Reflectng Gaussan Process. Introducton HTD/LRD have been used n many areas to characterze source traffcs n telecommuncaton (see, e.g., Refs. [5~6], [], [3], [9], and etc.), prce jumps and correlatons n fnance (see, e.g., Refs. [], []), and etc. Partcularly, n the early 990s, the authors such as Refs. [] and [9] dsclosed one of the most nterestng fndngs n hgh-speed networks,.e., source traffc data were shown to possess LRD. Snce then, stochastc modelng assocated wth LRD data has become an actve area of research. For examples, n the early 000s, a number of scentsts n Bell Labs focused on traffc data and related queueng modelng based on statstcal multplexng. Ther achevements (see, e.g., Refs. [5~6], and etc.) through large-scale computer and statstcal experments ndcate that the packet nterarrval tmes and packet szes tend to..d. Correspondng author. Emal addresses: nan5lu8@netra.nju.edu.cn (Wanyang DAI) / Copyrght 0 Bnary Informaton Press May, 0

2 489 W. Da. /Journal of Computatonal Informaton Systems 7:5 (0) (ndependent and dentcally dstrbuted) when the number of the nput connectons ncreases. Ther fndngs are of great mportance n desgnng and mplementng telecommuncaton software systems and hardware devces snce HTD/LRD sources wll sgnfcantly ncrease system desgn complexty and the cost of hgh-speed memory. Moreover, for some partcular cases and under certan tme scalng, Refs. [6] developed some theory to make ther smulaton fndngs justfed. However, snce the above well-known smulatons are computer based ones, the truncaton of a heavy-taled random varable used n the smulatons s nevtable due to the hardware and software lmtatons of a computer, such as, the numberng format of bnary dgts and the pseudo unform random number generators (see, e.g., Ref. [4]). Then a natural ssue arses concernng the truncaton: how well the approxmaton of a concerned objectve desgned n a smulaton wll be f a truncated random varable s used to replace the orgnal heavy-taled one? As ponted out n ths paper, the dfference of the means (or varances) between the truncated verson and the orgnal one wll stll be nfnte f the orgnal random varable s heavly taled wth nfnte mean and/or varance, whch ndcates that the truncated approxmaton may be not consstent wth the HTD/LRD based theory. In fact, some of the smulatons conducted by authors n Refs. [5~6], and etc. have revealed some gaps between the smulaton fndngs and the related HTD/LRD theory. For example, the theory developed n Ref. [6] to justfy ther smulaton fndngs s under the so-called crtcal tme scale ntroduced n Ref. [5],.e., the characterstcs of an ndvdual source becomes less mportant when the number of superposed sources ncreases. evertheless, n general, the characterstcs of the superposed lmtng process heavly depends on the ways of the tme/space scales (see, e.g., Refs. [6], [0], []). If the tme scale of nterest does not change wth superposton, thngs wll be dfferent snce ndvdual source behavor stll plays an mportant role. So, by the above observaton and our prevously establshed approxmaton theory n Ref. [0] concernng RGP wth or wthout LRD, we can provde more reasonable nterpretatons to the smulaton fndngs n Refs. [5~6], and etc., whch are based on the same queueng model and the same tme/space scalng wth both heavy-taled random varable and ts truncated counterpart as the drvng random envronmental factors. From these nterpretatons, we can see that the smulaton fndngs n Refs. [5~6], and etc. should be more close to the realty but not to the mathematcal assumptons about HTD/LRD, whch are mposed n ther models and stemmed from Ref. [9] where only bounded range of tme scales are concerned. Hence our fndngs smooth, to some extent, the gaps between HTD/LRD theory and real-world smulatons. More mportantly, our fndngs also ndcate that t s more sutable to use the truncated verson to replace a heavy-taled random varable n a smulaton and related analyss n order to be more consstent wth realty and to avod more complcated dscussons va HTD/LRD theory. The rest of ths paper s organzed as follows. In Secton, we make a comparson between a heavy-taled random varable and ts assocated truncated verson, and state the reason why the hardware and software systems n a computer can cause truncaton to a heavy-taled random varable. Moreover, n the secton, we also descrbe the possble mpact of the truncaton of a random varable on some well-known smulaton fndngs n hgh-speed networks. In Secton 3, we use our prevously establshed approxmaton theory va RGP to justfy that some well-known smulaton fndngs should be more close to the realty but not to the mposed mathematcal assumptons about HTD/LRD. Fnally, n Secton 4, we

3 W. Da. /Journal of Computatonal Informaton Systems 7:5 (0) present the concluson of the paper and make some remarks about the mpact of a truncated heavy-taled random varable on fnancal engneerng, nsurance and tme seres analyss.. Truncated Random Varable vs. HTD/LRD n a Computer Smulaton.. Truncaton Aganst HTD/LRD Related Smulaton Consder a random varable X wth dstrbuton F(x)=P { X x } over x [p, q], where p,q (-, + ) wth p<q, and moreover, suppose [p, q]=[p, + ) f q=+ and [p, q]=(-, q] f p=-. Then we can defne the truncated random varable Y over [p, q] as follows, X f X takesvalues n[ p, q], Y = () gnored f X does not takevalues n[ p, q]. In other words, the dstrbuton of Y can be represented n a condtonal probablty form,.e., for any y [ pq, ], PX { yx, [ pq, ]} PY { y} = P{ X y X [ pq, ]} =. PX { [ pq, ]} () Thus we know that the truncated mean correspondng to X s gven as follows, E XI{ X [ p, q]} E E XI{ X [ p, q]} X [ p, q] E[ Y]. = = = E[ Y]. Smlarly, we know that the truncated varance correspondng to X s gven by ( X p q ) Var XI{ [, ]} = Var( Y ). (4) Therefore, when the random varable X s lghtly taled, we know that the followng clams are true as p and q +, PX { (, p) ( q, + )} 0, EX [ ] EY [ ] 0, Var( X ) Var( Y ) 0, whch mply that Y can be used as an approxmatng random varable for X n a smulaton. However, when the random varable X s heavly taled, the mean and/or varance of X may be nfnte. For example, f X s a Pareto dstrbuted random varable, then the dstrbuton of X has the followng expresson wth parameters β 0 and x 0, m (3) xm β for x xm, F( x) = x 0 for x < xm, (5)

4 49 W. Da. /Journal of Computatonal Informaton Systems 7:5 (0) whch mples that both the mean and varance of X wll be nfnte f β and that the mean s fnte and the varance s nfnte f < β. Moreover, the dstrbuton for the correspondng random varable Y that s truncated by a gven constant b x m can be expressed as follows, for x> q; β xm t x F ( x) = for xm x q, β xm q 0 for x< xm. Therefore t follows from (.5)-(.6) that we may have the followng clams for any gven m (6) q> x when X s Pareto dstrbuted, E[ X ] E[ Y ] = and/or Var( X ) Var( Y ) =, (7).e., the dfferences of the mean and/or varance between the heavy-taled random varable and ts truncated counterpart may be nfnte. Therefore, f E[X]= and/or Var(X)=, t s not sutable to use the truncated random varable Y as an approxmatng one for X n a smulaton no matter how large E[Y] and/or Var(Y) are. In addton, we have the observaton that the truncated random varable may take a large (maybe huge but bounded) value wth relatvely large probablty but tself s essentally not a heavy-taled random varable. So, to be concse n applcatons and to avod more complcated dscussons through applyng HTD/LRD theory when a random varable appeared n a real-world applcaton (e.g., n fnance) s obvously bounded by some large (maybe huge) constant (e.g., the total wealth n the world or n a local envronment), a good choce s to use the truncated verson of a heavy-taled random varable n a smulaton... The Reason Causng Truncaton n a Computer Smulaton In a today s computer, numberng format s based on a fxed number of bnary dgts, whch s denoted by L, e.g., L=3. So the nteger M= L - s the largest nteger accepted by the computer. Due to ths fact, some random numbers generated from pseudo random number generators employed n a smulaton wll be truncated f the assocated random varable s heavly taled. For example, consder the Pareto dstrbuted random varable X and a unformly dstrbuted random varable U over [0, ], t follows from the dscusson n page 3 of Ref. [8] or page 36 of Ref. [4] that X = F ( U) (8) where F denotes the left-contnuous nverse of F and has the followng expresson,

5 W. Da. /Journal of Computatonal Informaton Systems 7:5 (0) x m / β F ( u) =, u (0,). ( u) (9) Therefore, once we generate the unformly dstrbuted random numbers from U over (0, ), we get the Pareto dstrbuted random numbers from X over [0, ). Currently, there exst some pseudo unform random number generators such as the Kss generator whose perod s of order 95 (see, e.g., pages n Ref. [4]). evertheless, all the generated random numbers F ( u) Correspondng to u near the unty wll be truncated n a smulaton f F ( u) > M. To be clear, we also remark that there exsts a class of unformly dstrbuted random number generators wth the form of X = f( X ) such as lnear congruence generator, whose perod s bounded by M(see, n+ n e.g., pages n Ref. [4]), then the largest pseudo unform random number except the unty, whch L L generated from these generators, are bounded by a constant c = ( )/. Therefore, f < β n (9), we can see that the F ( u) correspondng to the generated random number u s always less than M for relatvely small x m, whch ndcates that these generators themselves also have the truncaton ablty to a heavy-taled random varable snce all the values of ( F ( c), ) to the Pareto dstrbuted random varable are truncated..3. Impacts on Some Well-known Smulatons Related to HTD/LRD The above observatons ndcate that many exstng smulatons and related analyss based on HTD/LRD appeared n communcaton systems, fnancal engneerng, and etc. (see, e.g., Refs. [5~6], [9]) may need some new nterpretatons. In fact, some of these smulatons are well-known and have shown some gaps between the smulaton fndngs and the related HTD/LRD theory. Concretely, n the early 990s, one of the most nterestng fndngs n hgh-speed networks s the one that traffc data were shown to possess LRD (see, e.g., Refs. [], [3], [9]). In partcular, The authors n Refs. [9] showed that O/OFF processes wth heavy-taled O-perod dstrbutons exhbt LRD, whch s based on detaled statstcal analyss of real-tme traffc measurements on large tme scales but bounded by M= L - n seconds from Ethernet LA's at the level of ndvdual sources. Snce then, stochastc modelng assocated wth LRD data has become an actve area of research. For examples, n the early 000s, a number of scentsts n Bell Labs focused on traffc data and related queueng modelng based on statstcal multplexng. Ther achevements (see, Refs. [5~6], and etc.) through large-scale computer and statstcal experments ndcate that the packet nterarrval tmes and packet szes tend to..d. when the number of the nput connectons ncreases. Ther fndngs are of great mportance n desgnng and mplementng telecommuncaton software systems and hardware devces snce HTD/LRD sources wll sgnfcantly

6 493 W. Da. /Journal of Computatonal Informaton Systems 7:5 (0) ncrease system desgn complexty and the cost of hgh-speed memory. Moreover, for some partcular cases and under certan tme scalng, the authors of Ref. [6] developed some theory to make ther smulaton fndngs justfed. For example, a partcular queueng model used n ther theory s based on the followng assumptons: the dstrbutons of O- and OFF-perods are Pareto and exponental respectvely and packet szes are constant. They showed that the sequence of probabltes that steady state unfnshed works exceed a threshold tend to the correspondng probablty assumng Posson nput process when the number of nput sources tends to nfnty, whch concdes wth ther smulaton studes. evertheless, as ponted out n Ref. [6], ther theory heavly depends on the so-called crtcal tme scale ntroduced n Ref. [5], under whch the characterstcs of an ndvdual source become less mportant. If the tme scale of nterest does not change wth superposton, then thngs wll be dfferent snce ndvdual source behavor stll plays an mportant role. Interested readers are also drected to Ref. [] for more detals about the nfluence on LRD caused by dfferent tme scales. ow, snce the smulatons conducted n Refs. [5~6] and etc. are computer based ones, the truncaton of a heavy-taled random varable as dscussed n Secton s nevtable, we can provde some dfferent nterpretatons about the fndngs n ther smulatons. Further, our nterpretatons seem to be more consstent between ther smulaton fndngs and our approxmaton theory developed n Ref. [0]. For example, under the same queueng model and the same tme/space scalng, f the O-perod random varables wth Pareto dstrbutons are truncated, then the queueng length process correspondng to large superposton number of nput sources s roughly equal to the one wth nput traffc whose nterarrval tmes are..d., whch concdes wth ther smulaton fndngs n Refs. [5~6], and etc. evertheless, f the O-perod random varables are not truncated, then the queueng length process correspondng to large superposton number of nput sources s roughly equal to the one whose nput traffc s of LRD. From ths example, we can see that whether or not a heavy-taled random varable s truncated wll have a sgnfcant mpact on a smulaton. In the followng secton, we wll use a generalzed queueng model to concretely justfy our fndngs based on the theory developed n Refs. [0] and the observaton found n the current paper concernng the truncaton of a heavy-taled random varable n a smulaton. 3. Demonstraton Through a Generalzed Queueng System In the study of Ref. [0], the author establshed some approxmaton theory to justfy several heavy traffc lmt theorems under dfferent tme/space scalng for a queue wth superposed LRD O/OFF nput sources and general servce tme dstrbuton. In ths secton, we employ one of the lmt theorems, whch s under a tme/space scalng smlar to the dffusve scalng as used n the conventonal central lmt theorem, to demonstrate our fndngs as stated at the end of Secton.3. The rest of ths secton s organzed as follows. In Subsectons 3.-3., we summarze the queueng model and refne the lmt theorem requred for the current purpose, and moreover, and n Subsecton 3.3, we present our nterpretatons and ther mplcaton. 3.. The Queueng Model In the queueng system under consderaton, the servce tmes are assumed to be generally dstrbuted and there are..d. O/OFF nput sources. Each O/OFF source n {,..,} conssts of ndependent strctly

7 W. Da. /Journal of Computatonal Informaton Systems 7:5 (0) alternatng O- and OFF-perods, moreover, t transmts packets to a server accordng to a Posson process wth nterarrval tme sequence { un( ), } and rate λ f t s O and remans slent f t s OFF. The lengths of the O-perods are dentcally dstrbuted and so are the lengths of OFF-perods, and furthermore, both of ther dstrbutons can be heavy-taled wth nfnte varance. Specfcally, for any dstrbuton F, we denote by F = F the complementary (or rght tal) dstrbuton, and by F and F the dstrbutons for O-and OFF-perods wth probablty densty functons f and f respectvely. Ther means and varances are denoted by μ and σ for =,. In what follows, we assume that as x, α ether F ( x)~ x L( x) wth < α < or σ <, (0) where ~ denotes nearly equals and L > 0 s a slowly varyng functon at nfnty, that s, L ( tx) lm = for any t > 0. x L( x) ote that the mean μ s always fnte but the varance σ s nfnte when α <, and furthermore, one dstrbuton may have fnte varance and the other has an nfnte varance snce F and F are allowed to be dfferent. The szes of transmtted packets (servce tmes) form an..d. random sequence { v ( ) = v( )/ μ, }, where μ s the rate of transmsson correspondng to each and {(): v } s an..d. random sequence wth mean and varance σ v, moreover, { v (): } s ndependent of the arrval processes. To derve our queueng dynamcal equaton, we ntroduce more notatons. For a sngle source n {,, }, t follows from the explanaton n Ref. [] that the alternatng O/OFF perods can be descrbed by a statonary bnary process W = { W ( t), t 0}, where n n f nput traffc s n an O - perod at tme t, Wn () t = 0 f nput traffc s n an OFF - perod at tme t. () Moreover, the mean of W n s gven by { } γ = EW [ ( t)] = P W( t) = = n n μ μ + μ ()

8 495 W. Da. /Journal of Computatonal Informaton Systems 7:5 (0) In addton, we let A () t be the total number of packets transmtted to the server by tme t summed over all sources, S () t be the total number of packets that fnshed servce at the server f t keeps busy by tme t. Then the queue length process Q () t (the number of packets ncludng the one beng served at the server at tme t) can be represented by Q () t = A () t = S ( B ()) t (3) where we assume that the ntal queue length s zero for convenence, B () t s the cumulatve amount of tme that the server s busy by tme t. In the followng analyss, we wll employ the frst-n-frst-out (FIFO) and non-dlng servce dscplne under whch the server s never dle when there are packets watng to be served. 3.. The Requred Lmt Theorem For the above system, we are nterested n the behavor of the scaled queueng process Q () under the condton that the load of the server closely approaches the servce capacty when the source number gets large enough, where () Q (). (4) Q Moreover, for convenence, we adapt some notatons from Ref. [7]. When < α <, set a = ( Γ( α ))/( α ). Whenσ <, set α =, L and a =. In addton, defne σ / b L ( x) α α = lm t. (5) x L ( x) If 0 < b < (mplyng α = α and b= lm x L( x) / L( x) ), set αmn = α, f, on the other hand, b = 0 or b =, ( μ ab+ μ a ) π = and L = L ( ) (4 ) 3 μ+ μ Γ αmn ; (6) μ a π = and L = L ( ) (4 ) max mn 3 μ+ μ Γ αmn mn (7)

9 W. Da. /Journal of Computatonal Informaton Systems 7:5 (0) where mn s the ndex f b = (e.g. f α < α ) and s the ndex f b = 0, max denotng the other ndex. Moreover, for each, let the servce rate μ correspondng to some postve constant θ be gven by = +. (8) μ λγ θ In addton, suppose that the dstrbutons of F and F satsfy F ( x) ( =,) s absolutely contnuous n terms of x; (9) The densty f ( x) ( =,) of F satsfes lm f ( x) <. (0) + x 0 Theorem 3.. Under condtons (8)-(0) and as, Q () converges n dstrbuton under Skorohod topology to a reflectng Gaussan process Q () gven by Q () = A ( γ + ) λt () S ( λγ ) θ + I () 0 () where the three processes A ( γ ), S ( λγ ) and T () are ndependent each other, and furthermore, A ( γ ) s a Brownan moton wth mean zero and varance functon λγ, S( λγ ) s also a Brownan moton wth mean zero and varance functon λγσ, T () v s a Gaussan process wth a.s. contnuous sample paths, mean zero and statonary ncrements, whose covarance functon Var( T ( t)) can be expressed as t L t as t for H H ~ π ( ), = π t for all t 0 and H = () where H s the Hurst parameter defned by H= process wth I (0) = 0 and satsfes 0 (3 α ) /. Moreover, I () mn s a non-decreasng QsdI () () s = 0. (3)

10 497 W. Da. /Journal of Computatonal Informaton Systems 7:5 (0) The Interpretatons and Ther Implcaton In ths subsecton, we suppose that the O-perods n the queueng system are of Pareto dstrbuton wth < β and thus we know that the O-perod s heavly taled wth mean μ <, varance σ =, α = β, L ( ) x m dstrbuton wth mean = x β. Moreover, we suppose that the OFF perods are of exponental μ <, varance σ = μ <, α = / μ, L( x) = and suppose that α > α. Then t follows from (5) that b = and then we know that αmn = α = β, whch mples that H defned n Theorem 3. takes value over (/, ). Hence t follows from Theorem 3. that the correspondng lmtng queue length process Q s a reflectng Gaussan process wth LRD. Therefore, f we consder Q as an approxmatng physcal queue length process, then the correspondng nput process superposed from large number of O/OFF sources should be of LRD. However, f we consder the O-perods are of the correspondng truncated Pareto dstrbuton, the assocated mean and varance are both fnte. Thus, smlar to the above dscusson, we know that H=/, whch means by Theorem 3. that the derved lmtng queue length process Q s a reflectng Brownan moton (RBM), whch does not have the LRD (nterested readers are also drected to Refs. [8~9] for more detals about Brownan approxmatons). evertheless, the constant π n () may be very large. Therefore, f we consder Q as an approxmatng queue length process, then the correspondng nput process superposed from large number of O/OFF sources should not be of LRD and the packet nterarrval tmes more look lke beng generated from an..d sequence of random varables, whch concdes wth what were observed n the smulatons conducted by Refs. [4~6], and etc. Fnally, from the practcal vewpont, t should be reasonable that the O-perods and OFF-perods from lower speed end users are bounded by some large (maybe huge) constant (e.g., even the range of tme scales used n the mplementatons of Ref. [9] are bounded by M L = ). Therefore, the fndngs based on ther smulatons n Refs. [5~6] and etc. should be more close to the realty but not to ther mathematcal assumptons mposed about HTD/LRD due to the truncaton effect n ther smulatons. More mportantly, ther assumptons on HTD/LRD are stemmed from Ref. [9] where only bounded tme scales are concerned. So t s more sutable to use the truncated verson of a heavy-taled random varable n a smulaton to be consstent wth practce and to avod more complcated HTD/LRD related analyss.

11 W. Da. /Journal of Computatonal Informaton Systems 7:5 (0) Conclusons and Future Work In ths paper, we have answered the queston arsng n a smulaton or a real-word applcaton, whch ndcates that the approxmaton of a concerned objectve through a truncated random varable to replace the orgnal heavy-taled one may be not consstent wth what should be f the HTD/LRD assumptons are mposed. So, based on ths observaton and our prevously establshed approxmatng theory about RGP wth or wthout LRD, we provde more reasonable nterpretatons to some well-known large-scale computer and statstcal experments n characterzng the superposton of nternet traffc sources. More mportantly, our fndngs also ndcate that, n a smulaton and related analyss, t s more sutable to use a truncated random varable to replace a heavy-taled one n order to be more consstent wth realty and to avod more complcated dscussons va HTD/LRD theory. In addton to telecommuncaton, the studes concernng HTD/LRD have appeared n many felds such as fnancal engneerng, nsurance, and tme seres analyss (see, e.g., Refs. [~3]). For the purpose of computer orented smulatons and real-world applcatons, t s also nevtable to handle the truncaton problem when a heavy-taled random varable s concerned. Acknowledgement Ths project s supported by atural Scence Foundaton of Chna wth Grant o References [] F. Avram, Z. Palmowsk, and M. Pstorus. On the optmal dvdend problem for a spectrally negatve Levy process. Annals of Appled Probablty, 007, 7(): 56~80. [] O. E. Barndorff-elsen and. Shephard. on-gaussan Ornsten-Uhlenbeck-based models and some of ther uses n fnancal mathematcs. J. R. Stat. Soc. Ser. B, 00, 63: 67~4. [3] P. J. Brockwell and R. A. Davs. Tme Seres: Theory and Methods, Sprnger-Verlag, ew York, 99. [4] J. Cao, W. S. Cleverland, D. Ln, and D. X. Sun. Internet traffc tends to Posson and ndependent as the load ncreases. onlnear Estmaton and Classfcaton, eds. C. Holmes, D. Denson, M. Hansen, B. Yu, and B. Mallck, Sprnger, ew York, 00. [5] J. Cao, W. S. Cleverland, D. Ln, and D. X. Sun. The effect of statstcal multplexng on the long-range dependence of Internet packet traffc. Tech Report, Bell Labs, Murray Hll, J, U.S.A., 00. [6] J. Cao and K. Ramanan. A Posson lmt for the unfnshed work superposed pont processes. Tech Report. Bell Labs, Murray Hll, J, U.S.A., 00.. [7] J. D. Cavanaugh and T.J. Salo. Internetwotkng wth ATM WAS. In Wllam Stallngs, edtor, Advances n Local and Metropoltan Area etworks, IEEE Computer Socety Press, 994. [8] J. G. Da and W. Da. A heavy traffc lmt theorem for a class of open queueng networks wth fnte buffers. Queueng Systems, 999, 3: 5~40. [9] W. Da. Brownan Approxmatons for Queueng etworks wth Fnte Buffers: Modelng, Heavy Traffc Analyss and umercal Implementatons. Ph.D Thess, The Georga Insttute of Technology, 996. Also publshed n UMI Dssertaton Servces, A Bell & Howell Company, Mchgan, U.S.A., 997. [0] W. Da. Heavy traffc lmt theorems for a queue wth Posson O/OFF long-range depedent sources and general servce tme dstrbuton. Preprnt, 004, nvted talk at 004 Annual Conference of Jangsu Probablty and Statstcal Socety, also presented at 005 Internatonal Conference of Management and Applcatons, Chengdu, Chna and 006 IFORMS Hong Kong Internatonal Conference, Hong Kong, Chna. [] W. E. Leland, M. S. Taqqu, W. Wllnger, and D. V. Wlson. On the self-smlar nature of Ethernet traffc. IEEE/ACM Trans. etw., 994, : ~5.

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