M. S. Taqqu and J. Levy, "Using renewal processes to generate long-range dependence and high variability," In E. Eberlein and M. S. Taqqu, editors, Dependence in Probability and Statistics, pp. 73--89, Boston, Birkhauser, 1986.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....noise (FGN) 33] Fractional autoregressive integrated moving average (F ARIMA) processes [28] are an extension of standard ARIMA [5] processes. Compared to e.g. FBM, they have a higher flexibility in accounting for short range correlation structures as well. Aggregation approaches [45] are based on the superposition of a large number of identical renewal processes. The inter renewal time distribution has to be heavy tailed i.e. it has infinite variance, also referred to as the Noah e#ect by Mandelbrot [33] Approaches like these were applied successfully in several case ....

M. S. Taqqu and J. B. Levy. Using renewal processes to generate long-range dependence and high variability. In E. Eberlein and M. S. Taqqu, editors, Dependence in Probability and Statistics: A Survey of Recent Results, volume 11 of Progress in Probability and Statistics, pages 73--89. Birkhauser, Boston, 1986.

....assume finite variance distributions for the sojourn time in ON and OFF periods. As a result, the aggregation of large number of such sources will not have significant correlation, except possibly in the short range [35] An extension to such traditional ON OFF models was first introduced by [36, 37] (as cited in [35] by allowing the ON and OFF periods to have infinite variance (high variability or Noah Effect) The superposition of many such sources produces aggregate traffic that exhibits long range dependence (also called the Joseph Effect [35, 38] The source model used in [38] can be ....

M. Taqqu and J. Levy, "Using Renewal Processes to Generate LongRange Dependence and High Variability," Dependence in Probability and Statistics, Boston, MA, 1986, pp. 73--89.

....this gives rise to self similarity of packet counts in a mathematically rigorous manner. This formulation yields models exhibiting LRD, 1 f noise, and slowly decaying variance, and broadens the range of stochastic processes manifesting selfsimilarity beyond that achieved by On O# processes [14] [40], 43] and Gaussian processes [26] We present four fractal point processes: the fractal renewal process (FRP) the superposition of several fractal renewal processes (Sup FRP) the fractal shot noise driven Poisson process (FSNDP) and the fractal binomial noise driven Poisson process (FBNDP) ....

Taqqu, M. S. and Levy, J. B. Using renewal processes to generate long-range dependence and high variability. In E. Eberlein and M. S. Taqqu, editors, Dependence in Probability and Statistics, volume 11, pages 73--89. Birkhauser, Boston, MA, 1986.

....results but validating these findings by the analysis of actual ATM traffic in a real WAN network. Second, we investigate the mentioned three promising self similar modeling approaches to capture the observed properties, namely, the fractional Brownian traffic [17,19] superposed on off sources [13,26] and chaotic maps [8,25] Third, we give analysis results of shaped self similar traffic. We investigate the practically important question: can self similarity be removed from the traffic by shaping or not Queueing analysis of the shaped and original traffic are also presented about the ....

....and the Hurst parameter H of Z t [19] We generated the self similar sample path known as fractional Gaussian noise as presented in [21] and it is referred to as FGN in this paper. Superposed on off sources The model was first introduced by Mandelbrot [13] and later extended by Taqqu and Levy [26]. The basic idea of the model is the construction of self similar processes based on aggregating many simple on off processes with heavy tailed on and off periods [26] This sequence is referred to as the ON OFF data. Chaotic maps Erramilli and Singh proposed chaotic maps for fractal traffic ....

[Article contains additional citation context not shown here]

M. S. Taqqu and J. Levy. Using renewal processes to generate long-range dependence and high variability. Birkhauser (ed.), Dependence in Probability and Statistics, pages 73--89, Boston, 1986.

....it provides no explanation for this result. This section provides an explanation, based on measured characteristics of the Web. 5. 1 Superimposing Heavy Tailed Renewal Processes Our starting point is the method of constructing self similar processes described by Mandelbrot [16] and Taqqu and Levy [26] and summarized in [15] A self similar process may be constructed by superimposing many simple renewal reward processes, in which the rewards are restricted to the values 0 and 1, and in which the inter renewal times are heavy tailed. As described in Section 2, heavy tailed distributions have ....

Murad S. Taqqu and Joshua B. Levy. Using renewal processes to generate long-range dependence and high variability. In Ernst Eberlein and Murad S. Taqqu, editors, Dependence in Probability and Statistics, pages 73--90. Birkhauser, 1986.

....with heavy tailed on and or off periods. Such models offer some mathematical tractability and an explanation of observed long range dependence in the packet count per unit time data. There is a large body of recent literature which uses such models for modeling network traffic. See, for example, [1, 6, 14, 28, 9, 17, 29, 34, 35, 15, 36] and the references therein. The basic fluid model, which we call the classical on off model, consists of a single idealized source feeding a server. The single channel of this model alternates between an on state, in 1991 Mathematics Subject Classification. Primary 60K25; secondary 90B15. Key ....

M.S. Taqqu and J. Levy. Using renewal processes to generate long-range dependence and high variability. In E. Eberlein and M.S. Taqqu, editors, Dependence in Probability and Statistics, pages 73--89, Boston, 1986. Birkhauser.

....level form the basis for the self similar features observed in the WAN traffic traces collected at Berkeley [35, 36] It is the (occasional) sustained inactivity or activity implied by power law distributions that manifests itself as the 1=f noise spectra or LRD. Extending the results in [30, 37], it can be shown (see [41] that aggregating a large number of such independent ON OFF sources (with ON OFF periods characterized by P (T t) t Gammaff ) will in fact generate FBM of index H = 3 Gamma ff) 2. Thus for 1 H 1=2, we have 2 ff 1, so that the distribution has a finite ....

M.S. Taqqu and J.B. Levy, "Using Renewal Processes to Generate Long-Range Dependence and High Variability ", in: Dependence in Probability and Statistics, E. Eberlein and M.S. Taqqu (Eds.), Progress in Prob. and Stat. 11, Birkhauser, Boston, pp. 73-89, 1986.

....area of traffic models which account for these second order statistical characteristics of network data. Wavelet models [1] Markovian arrival processes [2] the M=G=1 model [5] chaotic maps [6] Fractional Brownian motion [9] fractional ARIMA processes [9] and superposition of ON OFF sources [14] This work supported by DARPA under contract number F19628 98 C 0057 and by MURI contract F49620 97 1 0382 through AFOSR. are some of the models that have been suggested. Though all these models model the long range dependence and show either exact or asymptotically second order selfsimilarity, ....

....the model with a class of other sources, all of which lead to similar network performance. The limiting distribution of a self similar traffic model composed of the superposition of a large number of ON OFF sources, whose ON OFF periods are taken from a heavy tailed distribution is considered in [14] and [15] The authors show that as the number of the ON OFF sources increases and under proper scaling, the superposed process converges weakly to fractional Brownian motion in the space of continuous functions and thus it is not surprising that the tails queues fed with both these sources have ....

M. S. Taqqu and J. Levy, "Using renewal processes to generate long-range dependence and high variability," Dependence in Probability and Statistics, E. Eberlein and M. S. Taqqu, editors, Birkhauser, Boston, pp. 73-89, 1986.

....responsible for the Gaussian property of aggregated traffic by an application of the Central Limit Theorem however, it is relevant to describing multiplexed network traffic. The on off model has its roots in a certain renewal reward process introduced by Mandelbrot [46] and further studied in [63]) and provides the theoretical underpinning for much of the recent works on physical modeling of network traffic. This theoretical foundation together with the empirical evidence of heavy tailed on off durations (as, for example, given for IP flow measurements [74] represents a more low level, ....

M. S. Taqqu and J. B. Levy. Using renewal processes to generate long-range dependence and high variability. In E. Eberlein and M. S. Taqqu,editors, Progress in Prob. and Stat. Vol. 11. Birkhauser, Boston, 1996.

....Gaussian noise (FGN) 33] Fractional autoregressive integrated moving average (F ARIMA) processes [28] are an extension of standard ARIMA [5] processes. Compared to e.g. FBM, they have a higher exibility in accounting for short range correlation structures as well. Aggregation approaches [45] are based on the superposition of a large number of identical renewal processes. The inter renewal time distribution has to be heavy tailed i.e. it has in nite variance, also referred to as the Noah e ect by Mandelbrot [33] Approaches like these were applied successfully in several case ....

M. S. Taqqu and J. B. Levy. Using renewal processes to generate long-range dependence and high variability. In E. Eberlein and M. S. Taqqu, editors, Dependence in Probability and Statistics: A Survey of Recent Results, volume 11 of Progress in Probability and Statistics, pages 73-89. Birkhauser, Boston, 1986.

....Two such models, the exactly (second order) self similar fractional Gaussian noise process and the asymptotically (secondorder) fractional ARIMA process, will be presented next. We also discuss a construction of self similar models, originally due to Mandelbrot [71] and later extended in [92, 60], which appears to be promising in terms of providing a physical explanation for the self similarity property in high speed packet traffic (see [97] 2.6.4 Fractional Gaussian Noise A fractional Gaussian noise [70] is a stationary Gaussian process, X = fX k g 1 k=1 , with mean X , variance ....

....are invariant under time shifts. By aggregating M iid copies, W (1) W (2) Delta Delta Delta ; W (M) of W , one obtains the process W = fW k (M)g k0 , given by W k (M) 8 : 0; k = 0 k X n=1 M X m=1 W (m) n ; k 0 It can be shown [71, 92] that for k and M both large and k M , the process W behaves like a fractional Brownian motion. More precisely, the process W , properly normalized, converges to the integrated version of fractional Gaussian noise, the notion of convergence being that of finite dimensional distributions. Thus, ....

Taqqu, M.S. and Levy, J.B., "Using Renewal Processes to Generate Long-Range Dependence and High Variability", Dependence in Probability and Statistics, Eberlein, E and Taqqu, M.S. (Eds.), Progress in Prob. and Stat., Vol. 11, Birkhauser, Boston, 1986, 73--89.

....file system traces. However, we have not yet attempted to explain the underlying cause of this observed phenomenon. Willinger et al. 28] proposed a physical explanation of observed self similarity in Ethernet LAN traffic, based on theory developed initially by Mandelbrot[14] and then Taqqu and Levy[25]. The theory states that the aggregation of many ON OFF sources, each exhibiting a characteristic known as the Noah effect, results in self similar total traffic. An individual source is classified as being an ON OFF source if the traffic from that source appears to contain highly variable ....

....The Noah effect refers to the high variability of the ON and OFF periods. If the distribution of ON and OFF period lengths from individual sources is heavy tailed, 2 then the aggregate traffic exhibits the Noah effect, and can be shown to exhibit self similarity. The theory presented in [25] makes the simplifying assumption that events within an ON period are evenly distributed. The ON OFF source model is thus similar to packet train models often used to model network 2 A heavy tailed distribution is typically one which exhibits infinite variance. An example of a heavy tailed ....

Taqqu, M., and Levy, J. Using renewal processes to generate long-range dependence and high variability. In Dependence in Probability and Statistics (Boston, MA, 1986), E. Eberlein and M. Taqqu, Eds., pp. 73--89.

....in the above sense cannot be modelled in this traditional way. Various models have been suggested to capture these effects. They range from traditional queueing models to sophisticated on off models [23,22,24] Markov modulated queues [25,26] shot noise models [31] and fractional Brownian motion [32,49,50]. Greiner et al. Telecommunication traffic and subexponential distributions 3 The aim of this article is to clarify the various notions of heavy tailed distributions as used in the queueing and network area, to describe the consequences of subexponential input distributions to the ....

....of on off models, i.e. during on periods a source is active, traffic is transmitted, during off periods no transmission happens. First letting the number of sources tend to 1 and then time, the limit process is fractional Brownian motion (see Taqqu, Willinger and Sherman [48] Taqqu and Levy [49]) This is an extension of a model first introduced by Mandelbrot [35] 30 Greiner et al. Telecommunication traffic and subexponential distributions Another class of models, also with a selfsimilar limit process, has been considered by Kurtz [31] as an alternative to the classical workload ....

M.S. Taqqu and J. Levy. Using renewal processes to generate long-range dependence and high variability. In E. Eberlein and M.S. Taqqu, editors, Dependence in Probability and Statistics, pages 73--89. Birkhauser, Boston, 1986.

....to (second order) self similarity of packet counts in a mathematically rigorous manner. This formulation yields models exhibiting LRD, 1=f noise, and slowly decaying variance, and broadens the range of stochastic processes manifesting selfsimilarity far beyond that achieved by On Off processes [11, 25, 26] and Gaussian processes [15] In this work we first establish that our FPP models are indeed practical. To show practicality, we successfully apply our FPP models to two qualitatively different sets of fractal empirical data (Bellcore traces) This is possible because the broad range of FPP models ....

M. S. Taqqu and J. B. Levy. Using renewal processes to generate long-range dependence and high variability. In E. Eberlein and M. S. Taqqu, editors, Dependence in Probability and Statistics, volume 11, pages 73--89. Birkhauser, Boston, MA, 1986.

....provides a possible explanation, based on measured characteristics of the Web. 5. 1 Superimposing Heavy Tailed Renewal Processes Our starting point is the method of constructing self similar processes described in [30] which is a refinement of work done by Mandelbrot [15] and Taqqu and Levy [28]. A self similar process may be constructed by superimposing many simple renewal reward processes, in which the rewards are restricted to the values 0 and 1, and in which the inter renewal times are heavy tailed. As described in Section 2, a heavy tailed distribution has infinite variance and the ....

Murad S. Taqqu and Joshua B. Levy. Using renewal processes to generate long-range dependence and high variability. In Ernst Eberlein and Murad S. Taqqu, editors, Dependence in Probability and Statistics, pages 73--90. Birkhauser, 1986.

....Meier Hellstern et al. 13] in studies of individual ISDN data traffic sources. These statements are consistent with the observation that in self similar traffic, bursts occur over all durations, and there is no characteristic length or time scale for traffic bursts. Based on theoretical results [16] which suggest that aggregating a large number of ON OFF sources with the same heavy tailed distribution in the two states results in a self similar process, Willinger makes the conjecture that heavy tailed ON OFF behavior provides the physical basis for the self similarity observed in packet data ....

....1 ( e e (EQ 9) EQ 10) As discussed in Section 2, the sojourn time distributions of the active and inactive periods behave as [8] To this extent, this map is a representation of the ON OFF behavior noted by Willinger [17] in preliminary studies of single Ethernet sources. Taqqu and Levy [16] derive results which indicate that aggregating a large number of heavy tailed ON OFF sources will in the limit lead to Fractional Brownian Motion (FBM) and the Hurst index H is given by H = 3m 4) 2m 2) 3 2 m 2] Considerable numerical evidence also shows that the output is long range ....

[Article contains additional citation context not shown here]

M.S. Taqqu and J.B. Levy, "Using Renewal Processes to Generate Long Range Dependence and High Variability", Dependence in Probability and Statistics, Progress in Prob. and Vol 11, Birkhauser; Boston (1986), pp. 73-89.

....L evy stable motion S (t) See Taqqu [81] for an overview. The use of on o models (or more generally, of renewal reward processes) to generate fractional Brownian motion and or symmetric L evy stable motion was originally proposed by Mandelbrot [49] in an economic context (see Taqqu and Levy [82] for a description) Resnick and Samorodnitsky [71] provide an overview of di erent approaches to the subject. 3.3 Mathematical framework II: The in nite source Poisson model An alternative to the superposition of on o sources is the in nite source Poisson model. In this model, sources arrive ....

M. S. Taqqu and J. Levy. Using renewal processes to generate longrange dependence and high variability. In E. Eberlein and M. S. Taqqu, editors, Dependence in Probability and Statistics, pages 73-89, Boston, 1986. Birkhauser.

....variance) interarrival times and finite variance rewards. If the number of summands grows to infinity, then, after rescaling, the limit turns out to be a fractional Brownian motion (fBm) This has been already observed in an economics context by Mandelbrot (1969) The result is also described in Taqqu and Levy (1986). An on o# version of the model has been adapted to the telecommunications context by Taqqu et al. 1997) Recall that fBm BH with H # (0, 1) is a Gaussian self similar process with stationary increments. The self similarity means that BH (at) and a H BH (t) have the same finite dimensional ....

Taqqu, M. S. & Levy, J. (1986), Using renewal processes to generate long-range dependence and high variability, in E. Eberlein & M. S. Taqqu, eds, `Dependence in Probability and Statistics', Birkhauser, Boston, pp. 73-- 89.

....the ON and OFF periods did not strictly alternate: they were i.i.d. and hence an ON period could be followed by other ON periods, and an OFF period by other OFF periods. The model was a relatively straightforward extension of the one first introduced by Mandelbrot [31] and Taqqu and Levy [46]. The processes we describe here have strictly alternating ON and OFF periods and agree therefore with the ON OFF source models commonly considered in the communications literature. The ON and OFF periods, moreover, may have different distributions, either with infinite or finite variance (a ....

....distributions. Heuristically, Theorem 1 states that the mean level TM( 1 = 1 2 ) t provides the main contribution for large M and T . Fluctuations from that level are given by the fractional Brownian motion oe lim BH (t) scaled by a lower order factor T H L(T ) 1=2 M 1=2 . As in [46], it is essential that the limits be performed in the order indicated. Also note that 1 ff min 2 implies 1=2 H 1, i.e. long range dependence. Thus, the main ingredient that is needed to obtain an H 1=2 is the heavy tailed property F jc (x) j x Gammaff j L j (x) as x 1; 1 ff j ....

M. S. Taqqu and J. Levy. Using Renewal Processes to Generate Long-Range Dependence and High Variability. In E. Eberlein and M. S. Taqqu, editors, Dependence in Probability and Statistics, pp. 73--89, Boston, 1986. Birkhauser.

....Ethernet LAN traffic (e.g. see [12, 13] involving the traffic generated by the individual sources or source destination pairs that make up the aggregate packet stream. Developing an approach originally suggested by Mandelbrot [15] and brought to the attention of probabilists by Taqqu and Levy [19], we presented (without proof) a result that states that the superposition of many strictly alternating independent and identically distributed ON OFF sources (also known as packet trains ; e.g. see [8] each of which exhibits a phenomenon called the Noah Effect , results in self similar ....

....truncated stable distributions) The ON and OFF periods are not required to have the same distribution. In this paper, we provide the proof of this fundamental result in self similar traffic modeling as stated in [22] The proof does not follow from the work of Mandelbrot [15] or Taqqu and Levy [19]; it is more delicate and requires a different approach and new methodologies. By presenting the mathematical results in the well known framework of the popular ON OFF sources or packet train models, we are able to identify the Noah Effect as the essential point of departure from traditional to ....

[Article contains additional citation context not shown here]

M. S. Taqqu and J. Levy. Using Renewal Processes to Generate Long-Range Dependence and High Variability. In E. Eberlein and M. S. Taqqu, editors, Dependence in Probability and Statistics, pp. 73--89, Boston, 1986. Birkhauser.

No context found.

M. S. Taqqu and J. Levy, "Using renewal processes to generate long-range dependence and high variability," In E. Eberlein and M. S. Taqqu, editors, Dependence in Probability and Statistics, pp. 73--89, Boston, Birkhauser, 1986.

No context found.

M. S. Taqqu and J. Levy, "Using renewal processes to generate long-range dependence and high variability," Dependence in Probability and Statistics, E. Eberlein and M. S. Taqqu, editors, Birkhauser, Boston, pp. 73-89, 1986.

No context found.

Taqqu, M., Levy, J., Using renewal processes to generate long-range dependence and high variability. Dependence in probability and statistics (Oberwolfach, 1985.

No context found.

M. S. Taqqu and J. B. Levy, \Using Renewal Processes to Generate Long-Range Dependence and High Variability," Dependence in Probability and Statistics, E. Eberlein and M.S. Taqqu, eds., pp. 73-89, Birkhauser, Boston, 1985.

No context found.

Taqqu, M., and J. Levy (1986): "Using Renewal Processes to Generate Long Range Dependence ", in Dependence in Probability and Statistics, E. Eberlein and M.S. Taqqu, eds., 73-89, Birkhauser.

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