B. Tsybakov and N. D. Georganas. On self-similar traffic in ATM queue: Definitions, overflow probability bound, and cell delay distribution. IEEE/ACM Trans. on Networking, 5(3):397--409, 1997.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....analysis of fractal traffic is a very important issue for network dimensioning and management. Therefore the study of queueing systems with fractal traffic input is a challenge in queueing theory. In the recent years the performance of queues with LRD or self similar input has been deeply analyzed [16, 21, 35, 37]. A collection of studies has proven that the fBm based models have a tail queue distribution that decays asymptotically like a Weibullian law [37] P[Q b] exp( b 2 2H ) 1) where is a positive constant that depends on the service rate of the queue [28, 8] This important result shows that ....

B. Tsybakov and N. D. Georganas. On self-similar traffic in ATM queue: Definitions, overflow probability bound, and cell delay distribution. IEEE/ACM Trans. on Networking, 5(3):397--409, 1997.

....input traffic. In particular, there are two important questions need to study as shown in Figure 1: One is whether the superposition of self similar processes retains the self similarity properties; the other is whether a server mechanism will change the self similarity nature of the traffic. In [12], B. Tsybakov and N. D. Georganas point out that the superposition of two uncorrelated self similar processes retain some asymptotically self similarity property. S. Vamvakos and V. Anantharam [14] consider a special case of a leaky bucket system with long range dependent input traffic, and prove ....

....Superposition of Self similar Processes In this section, we ll discuss superposition of self similar processes which is of great importance for network performance evalution. Our main concern is under what conditions the superposition of self similar streams will produce a self similar stream. In [12], statement 7 and 8 point lead to the following results: Lemma 3.1 (1) If are such uncorrelated processes that ####### # # # # ### ####### # # ##### ##### ##### ### ### # # # ### ### ### # ### # # is an asymptotically self similar process with parameter ### # # ### # ### ....

B. Tsybakov and N. D. Georganas, On self-similar traffic in ATM queues: definitions, overflow probability bound, and cell delay distribution, IEEE/ACM Trans. on Networking,Vol.5, No.3, pp.397-408, 1997.

....whether a server mechanism will change the self similarity nature of the traffic. In [40] S. Vamvakos and V. Anantharam consider a special case of a leaky bucket system with long range dependent input traffic, and prove that the output (departure) process is also long range dependent. In [38], B. Tsybakov and N. D. Georganas point out some results about the superposition of any two independent self similar process. While Norros [33] use Fractional Brown Motion(FBM) to represent the self similar traffic, A. Erramilli et al. 12] verified the effectiveness of FBM model. Based on these ....

.... Hurst parameter H = 1 ( 2) According to the Definitions above and the Theorem 2 of [39] we have the following relationships X is es s = X is 1 rd X is sas s : X is gsas s = X is as s that is, es s C 1 rd C sas s C gsas s C as s (16) Similar as in statement 7 and 8 of [38], we have Theorem 2.1 Let the uncorrelated processes X and X be strong asymptotically self similar, X with Hurst parameter Ht and X with H2, then X X is a strong asymptotically self similar process with parameter H = max Ht, H2 . Proof: X and X are uncorrelated sas s processes, then ....

B. Tsybakov and N. D. Georganas, On self-similar traffic in ATM queues: definitions, overflow probability bound, and cell delay distribution, IEEEIACM Trans. on Networking,Vol.5, No.3, pp.397-408, 1997.

....X tend to second order pure noise as m , i.e. 0, 1,2,3, rk asm k =# (3.6) The above mentioned intuition is best illustrated with the sequence of plots in Figure 3.1. All of the plots look similar and distinctively different from pure noise. Tsybakov and Georganas [25] derived an interesting statistical feature for the superposition aggregation of two self similar processes with different self similarity parameters. This derivation states that; the superposition of two exactly second order self 21 similar processes with self similarity parameters H 1 and H 2 ....

Tsybakov, B. and N. D. Georganas, On Self-Similar Traffic in ATM Queues: Definitions, Overflow Probability Bound, and Cell Delay Distribution, IEEE/ACM Transactions on Networking, Vol.5, No.3, pp.397-408, 1997.

.... We restrict ourselves to second order self similarity, which can be tested between some finite upper and lower cut off timescales, see [2] Therefore, in this paper we also use the term self similarity to refer to scaling of second order properties over some specific timescales, see also [17] [28]. We note that certain statements of the paper are also valid in the sense of exact statistical self similarity. The mathematical analysis part of the paper proves some asymptotic results relevant to LRD processes. The ns 2 simulator [22] is used for the network simulations. Several variants of ....

B. Tsybakov and N. D. Georganas. On self-similar traffic in ATM queues: Definitions, overflow probability bound, and cell delay distribution. IEEE/ACM Transactions on Networking, 5(3):397--409, June 1997.

....traffic in terms of the underlying TCP connections multiplex. Indeed, Willinger et al. 4] show that the multiplex of on off sources with heavy tailed on off periods turns out to have selfsimilar properties, a distinguishing feature of Internet traffic [4] 5] Furthermore, Tsybakov and Georganas [6] show that as long as the on period of the individual connections is heavytailed the resulting traffic multiplex is asymptotically secondorder self similar, even though the connection arrival process is Poisson. A heavy tailed random variable R has a distribution tail with the form P (R r) ....

....where takes the values 1 2. The resulting random variable has finite mean but infinite variance. On the other hand, a stationary stochastic process in discrete time X = fX t g = fX 1 ; X 2 ; g is called asymptotically secondorder self similar with Hurst parameter H if for all k 1 [6] lim m 1 (k) 1 (k 1) 2k (k 1) 2H (2) being (k) the lag k autocorrelation of the aggregated process X , t = Xtm m 1 : Xtm ) t 1 (3) For 1=2 H 1 this means that the correlation (k) decays to zero so slowly that, k (k) 1 (4) The ....

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B. Tsybakov and N. D. Georganas. On self-similar traffic in ATM queues: definitions, overflow probability bound and cell delay distribution. IEEE/ACM Transactions on Networking, 5(3):397--409, June 1997.

....is an active c flIFIP 1996. Published by Chapman Hall research area since network dimensioning for Internet services has became a very important issue. However, performance metrics are obtained at the cell or packet level : buffer overflow probability and delay estimates under selfsimilar input [4, 7, 9, 10]. Buffer overflow probability and delay at the packet or cell level may not be an adequate QOS metric for service provisioning. Little literature exists on QOS metrics that relate Internet user satisfaction and network parameters such as end to end delay and bandwidth. David Clark addresses this ....

....interarrival times [1, 6] The transaction arrival process in the busy hours can be modeled approximately as a Poisson process. Nabe et al. 6] show that Poisson arriving heavy tailed bursts constitute an accurate traffic model for busy hours of WWW service. Tsybakov and Georganas show in [10] that Poisson arriving heavy tailed batches with constant cell rate within the batch lead to an asymptotically second order self similar process. If we consider Tsybakov and Georganas model, a transaction level analysis of a multiplex of a large number of users in a single virtual circuit can be ....

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B. Tsybakov and N. D. Georganas. On self-similar traffic in ATM queues: Definitions, overflow probability bound and cell delay distribution. IEEE/ACM Transactions on Networking, 5(3):397--409, June 1997.

.... traffic usually a memoryless (Poisson) type of traffic is assumed for the simplicity of the model [1] Measure ments of real life networks have shown however that this assumption is too optimistic, as traffic shows a fractal behaviour, which is also called self similar or long range dependent [2]. Figure 1 shows this self similarity: if the measurement resolution decreases, e.g. from 10 5 to 10 3 , the trace remains its burstiness (Poisson traces be come more smoothed when plotted with a low resolution) The proposed solution incorporates such a fractal traffic source. In the next ....

B. Tsybakov and N. D. Georganas, "On self- similar traffic in ATM queues: Definitions, overflow probability bound and cell delay

....traffic in terms of the underlying TCP connections multiplex. Indeed, Willinger et al. 4] show that the multiplex of on off sources with heavy tailed on off periods turns out to have selfsimilar properties, a distinguishing feature of Internet traffic [4] 5] Furthermore, Tsybakov and Georganas [6] show that as long as the on period of the individual connections is heavytailed the resulting traffic multiplex is asymptotically secondorder self similar, even though the connection arrival process is Poisson. A heavy tailed random variable has a distribution tail with the form ....

....the values . The resulting random variable has finite mean but infinite variance. On the other hand, a stationary stochastic process in discrete time # ( 01 2 2324 is called asymptotically secondorder self similar with Hurst parameter 5 if for all 687 [6] 9; BADC FE IH 6 J . 0LK M 6 0LK J 6 N . 0LKPO (2) being A C FE 6 the lag 6 autocorrelation of the aggregated process C FE , C FE Q R FS J 2 232 J 481 7 (3) For WV X 5 ) this means that the ....

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B. Tsybakov and N. D. Georganas. On self-similar traffic in ATM queues: definitions, overflow probability bound and cell delay distribution. IEEE/ACM Transactions on Networking, 5(3):397--409, June 1997.

....models tend to yield overly optimistic performance prediction. Recent modeling works have therefore focused on obtaining parsimonious models capable of capturing the basic LRD property of traffic processes. Such approaches include chaotic maps [5] a LRD ON OFF model [18] Cox s M G 1 type models [4,8,11,17], the Fractional Brownian motion (FBm) model [9,10] fractional autoregressive integrated moving average (FARIMA) models [7,15] point processes [14] and pseudo models [1,13] An issue of much interest is whether and how multifractal can be employed to model LRD traffic. Recently Taqqu et al. ....

B. Tsybakov and N.D. Georganas,1997: On selfsimilar traffic in ATM queues: Definitions, overflow probability bound, and cell delay distribution. IEEE/ACM Trans. on Networking, 5 397--409

....[10, 19, 29] and triggers the need for new models. Several models have already been proposed in the literature, including the fractional Brownian motion [24, 25] on off sources with heavy tailed distributions for the on and or off periods [1, 2, 3, 4, 11, 18, 34] and the M=G=1 input process [17, 20, 27, 28, 32] process (these lists of references are not exhaustive as the activity in this domain is very dense) More generally, studies of queues in presence of heavy tailed distributions, initiated with the works of Cohen [7] Pakes [26] and Veraverbeke [33] can be found in a recent special issue of ....

B. Tsybakov and N. D. Georganas, On self-similar traffic in ATM queues: Definitions, overflow probability and cell delay distribution. IEEE/ACM Trans. on Networking, Vol. 5, No.3, pp. 397-409, Jun. 1997.

....tend to yield overly optimistic performance predictions. Recent modeling works have therefore focused on obtaining parsimonious models capable of capturing the basic LRD property of traffic processes. Such approaches include chaotic maps [5] a LRD ON OFF model [20] Cox s M G 1 type models [4,10,13,19], the Fractional Brownian motion (FBm) model [11,12] fractional autoregressive integrated moving average (FARIMA) models [9,17] point processes [16] and pseudo models [1,15] An issue of much interest is whether and how multifractal can be employed to model LRD traffic. Using Bellcore s LAN ....

B. Tsybakov and N.D. Georganas,1997: On selfsimilar traffic in ATM queues: Definitions, overflow probability bound, and cell delay distribution. IEEE/ACM Trans. on Networking, 5 397--409

....(LRD) if r(k) # k 2H 2 L(k) as k ##, 1 2 H 1, where L is slowly varying at infinity, i.e. lim k## [L(tk) L(k) 1, t 0 and a(x) # b(x) means a(x) b(x) # 1 as x # #. The class of LRD processes is equivalent to the class of asymptotically second order self similar processes [22] defined as follows. For all integer m # 1 let X (m) k = 1 m # km i= k 1)m 1 X i be the aggregated process with autocorrelation function r (m) k) X is called asymptotically second order self similar if lim m## r m (k) 1 2( k 1 2H 2 k 2H k 1 2H ) for all ....

TSYBAKOV, B. -- GEORGANAS, N. D.: On Self-Similar Traffic in ATM Queue: Definitions, Overflow Probability Bound, and Cell Delay Distribution. IEEE/ACM Trans. on Networking, 5 (3) (1997), pp. 397--409.

....long range dependence of traffic in the view of optimal approximation. KeywordsModeling, long range dependence 1. Introduction Recent researches have shown that the behaviors of the traffic on LAN and WAN are well modeled by second order self similar processes with long range dependence (LRD) [1 2]. Second order self similar processes are classified into two classes [1 2] One is exactly second order self similar model and the other asymptotically second order self similar model. 1] pointed out that exactly second order self similar model is not enough to model real traffic. Hence, ....

....long range dependence 1. Introduction Recent researches have shown that the behaviors of the traffic on LAN and WAN are well modeled by second order self similar processes with long range dependence (LRD) 1 2] Second order self similar processes are classified into two classes [1 2]. One is exactly second order self similar model and the other asymptotically second order self similar model. 1] pointed out that exactly second order self similar model is not enough to model real traffic. Hence, asymptotically secondorder self similar processes are considered in the paper. ....

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Boris Tsybakov, Nicolas D. Goergannas, On SelfSimilar Traffic in ATM Queues: Definitions, Overflow Probability Bound, and Cell Delay Distributions, IEEE Trans. on Networking, vol. 5, no. 3, June 1997, pp. 397-409

....connections arrive at time t. X(t) X i2Z X i (t) 1 then counts how many connections are active at time t. Alternatively, X(t) can be viewed as the aggregate (over flows) traffic rate emitted at time instance t. The behavior of X(t) and its generalized brethren can be analyzed directly [43, 65, 66], but a more succinct and elegant approach that reveals the influence of heavy tailedness on long range dependence can be found in a result due to Cox involving the M G 1 queueing system [12] An M G 1 queue is defined to be the busy server process where connection arrivals are Poisson and each ....

B. Tsybakov and N. D. Georganas. On self-similar traffic in ATM queues: Definitions, overflow probability bound and cell delay distribution. IEEE/ACM Trans. Networking, 5(3):379--409, 1997.

.... been carried out to understand the implications of long memory in network traffic [20, 24, 25, 38] fGN models were used to analyze buffer requirements in switches and multiplexors [26] predictability of congestion in networks [19] cell transfer delay and cell loss probabilities in ATM networks [29] and VBR video characteristics [18] Based on the study, the general conclusion seems to be that the long range dependence plays a significant role under low network utilization and large buffer sizes. On the other hand, short range dependence is more significant under high network utilization ....

B.Tsybakov and N.D.Georganas. On Self-Similar Traffic in ATM Queues:Definitions, Overflow Probability Bound, and Cell Delay Distributions. IEEE/ACM Transactions on Networking, 5(3)397-409, 1997.

....batch over several time slots, creating long range dependence. The service time is always an order of magnitude below the delay considering a 10 Mbps. link, so that the contribution of network load is not as important as the delay contribution. See also [5] This relates to the model presented in [7], where a second order self similar process is constructed with the counting process of Poisson arriving bursts with heavy tailed distribution. Small values of the window size will make the file transmission time last longer, producing heavy tail behavior and thus resembling the model presented ....

....where a second order self similar process is constructed with the counting process of Poisson arriving bursts with heavy tailed distribution. Small values of the window size will make the file transmission time last longer, producing heavy tail behavior and thus resembling the model presented in [7]. For large window sizes, the traffic at the output of the server resembles the Poisson process at the input of the queue, whereas in the small window case the duration of the multiplexed traffic bursts becomes heavy tailed and, therefore, we are faced with a self similar process. 8 Conclusions ....

B. Tsybakov and N. D. Georganas. On self-similar traffic in ATM queues: Definitions, overflow probability bound and cell delay distribution. IEEE/ACM Transactions on Networking, 5(3):397--409, June 1997.

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B. Tsybakov and N. D. Georganas. On self-similar traffic in ATM queue: Definitions, overflow probability bound, and cell delay distribution. IEEE/ACM Trans. on Networking, 5(3):397--409, 1997.

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B. Tsybakov and N. D. Georganas. On self-similar traffic in ATM queues: Definitions, overflow probability bound, and cell delay distribution. IEEE/ACM Transactions on Networking, 5(3):397--409, June 1997.

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Tsybakov, B. and Georganas, N. D. (1997a). On Self-Similar Traffic in ATM Queues: Definitions, Overflow Probability Bound and Cell Delay Distribution. IEEE/ACM Trans. on Networking.

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