A. Erramilli, R. P. Singh and P. Pruthi, "An application of deterministic chaotic maps to model packet traffic," Queueing Systems, vol. 20, pp. 171-206, 1995.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....Kingdom. tra#c is also bursty, which poses major engineering challenges, at times scales spanning several orders of magnitude. In order to address these challenges it is necessary to have at one s disposal models that can mimic packet tra#c. One such model has been proposed by Erramilli et al. [3, 4]. The appeal of Erramilli s model lies in its simplicity; it is a one dimensional chaotic map f defined on the unit interval by f(x) 8 x 1 d m 1 x x d (1 d) m 2 (1 x) where d, m 1 , m 2 are parameters ; a graph of the map is shown in figure 1. To ....

A. Erramilli, R. P. Singh, and P. Pruthi. An application of deterministic chaotic maps to model packet tra#c. Queueing systems, 20:171--206, 1995.

....results in an aggregate network traffic that is long range dependent. In this note our purpose is to make a couple of remarks in connection with another intriguing idea for arriving at physical descriptions of network communication traffic, initially suggested in the networking context in [16]. This idea is related to anumber of mostly numerical investigations in the physics communitythathave found surprisingly long lived correlations in the evolution of certain chaotic dynamical systems. Recall that a stationary dynamical system is called strongly mixing if for anytwoevents the ....

....to pick up long range dependentcharacteristics from the aggregate flows that it interacts with as it traverses the network# the latter mightbe long range dependentby virtue of being the superposition of heavy tailed activity processes. 2 Deterministic dynamical systems The proposal in [16] is to model a communication traffic source (such as an Ethernet LAN, or a VBR video source) by means of a deterministic nonlinear transformation x n 1 = S(x n ) taking values in some state space X. The traffic source is modeled as having an activity level that depends on the current state. Let ....

[Article contains additional citation context not shown here]

Erramilli, A. and Singh, R. P. (1995). An application of deterministic chaotic maps to model packet traffic. Queueing Systems : Theory and Applications,Vol 20, pp. 171-206.

....shifts [9] or Dirac pulses [15] with short range dependent (SRD) stationary processes. Another approach has been to use stochastic models (such as fractional Brownian motion [26] zero rate renewal processes [35] and various other point processes [32] or deterministic models (such as chaotic maps [13]) that exhibit the LRD observed in the experimental data. However, these models are analytically difficult to handle. Furthermore, they do not provide much insight into why they are meaningful on physical grounds. This explains in part why much modeling work still relies on more traditional ....

A. Erramilli, R. P. Singh, and P. Pruthi. An Application of Deterministic Chaotic Maps to Model Packet Traffic. Queueing Systems, 20(3):171--206, 1995.

....18, 37, 47] however, the investigated systems had to be heavily simplified due the hard analytical tractability of these tra#c models. Also, the generation of artificial self similar tra#c traces for simulation purposes based on these approaches is computationally very intensive. Chaotic maps [17] are a very e#cient means for generating self similar tra#c in simulation studies, but this approach is also hardly tractable analytically. The hard tractability of analytical approaches led to several approaches to develop Markovian approaches for approximating self similar tra#c, which can ....

A. Erramilli and R. P. Singh. An application of deterministic chaotic maps to model packet tra#c. Queueing Systems, 20(I--II):171--206, 1995.

....time scales. As a result, Poisson or Markovian 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 ....

A. Erramilli, P.R. Singh, and P. Pruthi, 1995: An application of deterministic chaotic maps to model packet traffic. Queueing Systems, 20, 171--206.

....raised questions regarding the evaluation of network performance (see e.g. 7] 12] 35] Various stochastic models and techniques have been proposed for modeling the distinctive statistical nature of self similar network tra#c and its e#ects on network performance. These include Chaotic Maps [8], 31] Cox s M G 1 type models [4] 14] 28] fractional Gaussian noise [10] 28] fractional Brownian motion [25] 26] and Markovian models [32] In this work, we show that Fractal Point Processes (FPPs) provide novel tools for understanding, modeling and analyzing diverse types of ....

Erramilli, A., Singh, R. P., and Pruthi, P. An application of deterministic chaotic maps to model packet tra#c. Queueing Systems, 20:171--206, 1995.

....time scales. As a result, Poisson or Markovian models 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 ....

A. Erramilli, P.R. Singh, and P. Pruthi, 1995: An application of deterministic chaotic maps to model packet traffic. Queueing Systems, 20, 171--206.

....process is called self similar if it behaves, up to a scaling factor, in exactly the same way at all time scales. This rather complex burst within burst structure is in sharp contrast to the classical traffic modeling assumptions. The self similar traffic models seem to be quite successful [7,8,18,19,21,23,26]. They are very appealing from the point of view that a very complex bursty and correlated traffic stream can be described by only a few parameters. In most cases three parameters are appropriate for the traffic characterization (mean rate, peakedness and Hurst parameter) The basic idea behind ....

A. Erramilli and R. P. Singh. Application of deterministic chaotic maps to model packet traffic in broadband networks. In Proc. 7th ITC Specialists Seminar, pages

....are different promising approaches to capture this complex fractal like behaviour. Norros [17,19] used a Gaussian self similar process known as the Fractional Brownian Motion. Willinger et al. 27] applied the superposition of on off sources with heavy tailed on and off periods. Erramilli et al. [8,25] studied different chaotic maps. Another hot topic of the present research is to investigate the impacts of different network control mechanisms (e.g. shaping [16,23] on self similarity properties and the dimensioning aspects of fractal traffic. The purpose of this paper is threefold. First, we ....

....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 robustness of the ....

[Article contains additional citation context not shown here]

R. P. Singh and A. Erramilli. Application of deterministic chaotic maps to model packet traffic in broadband networks. In 7th ITC Specialists Seminar, pages 8.1.1--

....temporal behavior of network traffic, and make accurate traffic modeling a challenging task. Several models have been developed to model long range dependence in network traffic, such as Fractional Gaussian Noise (FGN) processes[21] the fractal point processes[36] and methods based on chaotic maps[12][11] Since these models can not model strong short range dependence (SRD) in network traffic, they can hardly be used to model video traffic. Models, which can model both long range and short range dependence, include FARIMA models[15] a model based on Importance Sampling[18] scene based ....

A. Erramilli, R.P. Singh, and P. Pruthi. An application of deterministic chaotic maps to model packet traffic. Queueing System, 20:171--206, 1995.

.... Wallis, 1968; Norros, 1993; Veitch, 1992; Huang et al. 1995; Slimane Le Ngoc, 1995; Lau et al. 1995; Chen et al. 1996] or based on the superposition of ON OFF sources with heavy tailed ON or OFF periods [Willinger et al. 1995] and chaotic maps [Erramilli et al. 1994a; 1994b; Pruthi Erramilli, 1995]. All these models describe successfully the burstiness of the traffic but their approach is very different. The aim of this article is to develop a better understanding of chaotic maps as models of packet traffic. Packet traffic has been characterised by three parameters [Norros, 1993] As ....

.... Hurst parameter H = 3m Gamma 4) 2m Gamma 2) The autocorrelation function decays as a power law lim L 1 C(L) L Gamma(2 Gammam) m Gamma1) or equivalent the power spectrum decays as S( Gamma(2m Gamma3) m Gamma1) where is the frequency [Erramilli et al. 1994a, 1994b; Pruthi Erramilli, 1995; Schuster, 1995] The intermittent behaviour of an orbit of the map occurs if m 1 1 (or and m 2 1) The orbit changes randomly between a long lasting regular phase and an irregular phase. The intermittency map generates traffic with long OFF (or and ON) regions. The Bernoulli shift map (m 1 ....

[Article contains additional citation context not shown here]

Erramilli, A., Singh, R. P. & Pruthi, P. [1995] "An Application of Deterministic Chaotic Maps to Model Packet Traffic," Queueing Systems 20, 171-206.

....The queue length dynamics, Q n , of a deterministic server is described by Q n 1 = Q n (Q n ) M n 1 (36) where M n 1 is the number of queue arrivals at time n 1, Q n ) is the Heaviside function, Q n 0) 1 and (Q n = 0) 0, which describes the queue departures. Erramilli et al. [15] studied the case where the arrivals are described by a chaotic map of the form M n 1 = R(M n ) N n y(x n ) 37) where y(x n ) is the indicator variable given by Eqs. 1,2) and N n is the number of packet arrivals at a given time. They studied the case where N n = 2 for 13 10 5 0 5 10 ....

....length is given by a deterministic transformation in the variables (x n 2 (0; 1) Q n 2 Z) see Fig. 5a) The queue length distribution P (Q = L) is obtained by the marginal distribution of the iterates of Q n and can be computed numerically. Supported by numerical experiments Erramilli et al. [15] found that in the case that the arrivals are heavy tailed OFF distributed and geometrical ON distributed, the queue length decays as P (Q q) q 1 d q . In the case that the arrivals are heavy tailed in their ON and OFF periods then the queue length decays as a power law P (Q q) q 1 ....

A. Erramilli, R. P. Singh, and P. Pruthi. An Application of Deterministic Chaotic Maps to Model Packet Trac. Queueing Systems, 20:171-206, 1995.

....termed procession and is the effect of the control of the dynamical system which permits the transfer of the data between source and sink. 3. DYNAMICAL SYSTEMS APPROACH TO TELETRAFFIC An ON OFF traffic source can be modelled using the family of one dimensional chaotic maps (Erramilli, 1994a, Erramilli, 1994b, Pruthi 1995a) x F x F x x x d d x d F x x x d d d x n n n n n m m n n n n m m n = 1 1 1 1 2 2 2 1 1 2 2 1 0 1 1 1 ( e e e e (1) with parameters m m 1 2 3 2 2 , e e 1 2 1 , and 0 1 d . The ON OFF ....

Erramilli, A. Singh, R.P. and Pruthi, P. (1995) An application of deterministic chaotic maps to model packet traffic. Queueing Systems. Vol 20, 171-206.

..... In [19] Paxson and Floyd show that superposition of on off sources that have a fixed rate in the on period and have a heavy tailed distribution for the on and off period lengths can be used to model LRD X . Erramilli, Singh and Pruthi use deterministic nonlinear chaotic maps to define a LRD X [5]. Andersen and Nielsen propose a Markovian approach in which an LRD X is obtained by superposing a number of two state Markov Modulated Poisson Processes (MMPPs) 1] with the resultant arrival process being an MMPP. The advantage of this last method is that in addition to allowing the modeling of ....

A. Erramilli, R. P. Singh and P. Pruthi, "An application of deterministic chaotic maps to model packet traffic," Queuing Systems, vol. 20, pp. 171206, 1995.

....notion of selfsimilarity and long range dependence in network traffic. This has spurred research in the 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 ....

A. Erramilli, R. P. Singh and P. Pruthi, "An application of deterministic chaotic maps to model packet traffic," Queueing Systems, vol. 20, pp. 171-206, 1995.

....the same way at all time scales. This rather complex burst within burst structure of the investigated packet traffic has also been reported by these studies which is in sharp contrast to the classical traffic modeling assumptions. The self similar traffic models seem to be quite successful [7, 8, 19, 18, 22, 24, 27]. They are very appealing from the point of view that a very complex bursty and correlated traffic stream can be described by only a few parameters. In most cases three parameters are appropriate for the traffic characterization (mean rate, peakedness and Hurst parameter) The basic idea behind ....

A. Erramilli and R. P. Singh. Application of deterministic chaotic maps to model packet traffic in broadband networks. In Proc. 7th ITC Specialists Seminar, pages 8.1.1--8.1.3., Morristown, NJ, 1990.

....15, 18, 37, 47] however, the investigated systems had to be heavily simpli ed due the hard analytical tractability of these trac models. Also, the generation of arti cial self similar trac traces for simulation purposes based on these approaches is computationally very intensive. Chaotic maps [17] are a very ecient means for generating self similar trac in simulation studies, but this approach is also hardly tractable analytically. The hard tractability of analytical approaches led to several approaches to develop Markovian approaches for approximating self similar trac, which can easily ....

A. Erramilli and R. P. Singh. An application of deterministic chaotic maps to model packet trac. Queueing Systems, 20(I{II):171-206, 1995.

.... Bestavros proved how WWW as the major contributor to current Internet traffic can cause long range dependence and self similarity [4] Chaotic maps appeared as efficient and parsimonious methods to generate packet traffic on the link source level, see for example the work by Erramilli and Singh [7] and the same authors with Pruthi [6] The drawback of link source level models is that they disregard one of the major properties of todays Internet, namely that the majority (80 90 ) of traffic is generated and controlled by the TCP protocol, which is adaptive in nature. The consequence of ....

A. Erramilli and R. P. Singh, "Application of Deterministic Chaotic Maps to Model Packet Traffic in Broadband Networks", Proc. 7th ITC Specialist Seminar, Morristown, NJ, 8.1.1-8.1.3, 1990

....shifts [9] or Dirac pulses [15] with short range dependent (SRD) stationary processes. Another approach has been to use stochastic models (such as fractional Brownian motion [27] zero rate renewal processes [36] and various other point processes [33] or deterministic models (such as chaotic maps [13]) that exhibit the LRD observed in the experimental data. However, these models are analytically difficult to handle. Furthermore, they do not provide much insight into why they are meaningful on physical grounds. This explains in part why much modeling work still relies on more traditional ....

A. Erramilli, R. P. Singh, and P. Pruthi. An Application of Deterministic Chaotic Maps to Model Packet Traffic. Queueing Systems, 20(3):171--206, 1995. 25

....in seeing whether such a result is also true of the On Off storage model. It is certainly reasonable to expect queues with tails that are at least Weibullian, since now bursts of very long duration have in addition a large rate. Power law tails have been found numerically by Erramilli et al. [8] for a queueing system modelled by certain chaotic mappings of the plane, where the deterministically generated traffic is essentially On Off like with weak correlations between neighbouring silence and active periods. General expressions for Q(x) or for bounds on Q(x) are given in [2] for both ....

....was found for the distribution of queue content for the limiting system where each source contributes a single burst. This is in accordance with speculation based on the M=G=1 system with service times of infinite variance, which is the analogous system for instantaneous arrivals. See also [8] for suggestive numerical results in a related context. In sharp contrast, normalising by reducing the peak inflow rate h with increasing N for each source whilst leaving the distributions for the On and Off periods unchanged leads to Weibullian tail behaviour [2] Although the queue content ....

A.Erramilli, R.P.Singh, P.Pruthi, An Application of Deterministic Chaotic Maps to Model Packet Traffic, Queueing Systems 20 (1995) 171-206.

.... significantly on queueing delays [18] and that certain simplified analytical models of single server queues incorporating LRD corroborate this by exhibiting virtual work distributions with tails which decay slower than the exponential decay familiar from Markovian models [27] 11] 32] 13] [28]. In the absence however of a complete programme of data analysis, informed model selection, parameter estimation and finally model verification against relevant performance criteria, the connection between LRD and performance metrics cannot be fully understood. Among the outstanding issues in ....

.... Brownian Storage Model of Norros [27] and for the superposition of a large number of small peak rate ON Off fluid sources [11] Even more extreme behaviour is found for superpositions of large peak rate ON Off fluid sources, that is power law tails with infinite expectation [32] See also [13] and [28] for related results) Our extended analysis of the Bellcore data reveals some striking features with important implications for the choice of a model capable of capturing the work arrival process. We also discuss the issue of mono vs multi fractal modelling, and conclude that the data traces are ....

A.Erramilli, R.P.Singh, P.Pruthi, An Application of Deterministic Chaotic Maps to Model Packet Traffic, Queueing Systems 20, (1995) pp.171-206.

....questions regarding the evaluation of network performance (see e.g. 7] 12] 35] Various stochastic models and techniques have been proposed for modeling the distinctive statistical nature of self similar network traffic and its effects on network performance. These include Chaotic Maps [8], 31] Cox s M=G=1 type models [4] 14] 28] fractional Gaussian noise [10] 28] fractional Brownian motion [25] 26] and Markovian models [32] In this work, we show that Fractal Point Processes (FPPs) provide novel tools for understanding, modeling and analyzing diverse types of ....

Erramilli, A., Singh, R. P., and Pruthi, P. An application of deterministic chaotic maps to model packet traffic. Queueing Systems, 20:171--206, 1995.

....[8] or Dirac pulses [13] with short range dependent (SRD) stationary processes. Another approach has been to use stochastic models (such as fractional Brown ian motion [24] zero rate renewal processes [30] and various other point processes [28] or deterministic models (such as chaotic maps [12]) that exhibit the LRD observed in the experimental data. However, these models are analytically diOEcult to handle. Furthermore, they do not provide much insight into why they are meaningful on physical grounds. This explains in part that much modeling work still relies on more traditional ....

A. Erramilli, R. P. Singh, and P. Pruthi. An Appli cation of Deterministic Chaotic Maps to Model Packet TraOEc. Queueing Systems: Theory and Applications, to appear, 1995.

....queues generated with the true and the simulated sample, respectively, as could be expected on the basis of their visual dissimilarity. A more theoretical explanation is that the outlook of this sample is much closer to the single source models with heavy tailed on and off periods studied in [6] where the corresponding queue length was found to have a power tail instead of the Weibull tail of our model. As a side product, the somewhat interesting observation was made that the removal of all negative arrivals (in the finest resolution available in that case, 2 Gamma5 seconds) from the ....

A. Erramilli, R.P. Singh, and P. Pruthi. An application of deterministic chaotic maps to model packet traffic. Submitted for publication, 1994.

No context found.

A. Erramilli, R. P. Singh, and P. Pruthi, "An application of deterministic chaotic maps to model packet traffic," Queueing Syst., vol. 20, pp. 171--206, 1995.

....favors a revision of the traditional ON OFF source model in which the sojourn times are not modeled by the familiar exponential distribution, but instead by Pareto distributions with finite means and infinite variances. An equivalent description in terms of chaotic maps is also feasible (see [10], 25] 24] In the chaotic map formulation, the source state is represented by a continuous variable whose evolution in discrete time is described by a low order, nonlinear dynamical system. The packet generation process is now modeled by stipulating that a source generates a batch of packets ....

....of methods that can permit the routine analysis of fractal queueing systems, simulation methods are suggested for routine analysis. The generation and statistical analysis of traffic with long range dependence heavy tails is an active area of current research (see for example [14] 27] 28] [10], 24] There are nevertheless several analytical results which provide considerable insights into the engineering impacts of fractal traffic. These are discussed next. IV. Engineering Impacts In this section, we will briefly review current insights into the performance and engineering impacts ....

A. Erramilli, R.P. Singh and P. Pruthi. An Application of Deterministic Chaotic Maps to Model Packet Traffic. Queueing Systems (to appear).

....d; q) models (see [15, 19] are examples of asymptotically self similar processes with selfsimilarity parameter H = d 1=2; 0 d 1=2. For other approaches to characterizing the fractal properties of measured packet traffic that make explicit use of certain fractal dimension descriptors, see [9, 7, 10]. 3 Experimenting with Measured Ethernet Traffic Traces While the studies mentioned above convincingly establish the presence of LRD over a wide range of time scales in packet traffic processes (see also Figure 1) its significance to queueing performance and traffic engineering may not be clear ....

....discrepancies become increasingly apparent (in addition, the range and decay constant depend on the length of the data trace) Due to the substantial weight at large x in the distribution, these discrepancies will have practical implications. Heavy tailed queueing behavior is also reported in [9], based on numerical and analytical studies of models that formulate ON OFF source and queueing behavior in terms of chaotic maps. Thus there is considerable support to the simulation results indicating that the tails of the queue length distributions decay much more slowly than the exponential ....

A. Erramilli, R.P. Singh and P. Pruthi, "An Application of Deterministic Chaotic Maps to Model Packet Traffic", Queueing Systems, Vol. 20, pp. 171-206, 1995.

....tails, by variances that decay as a fractional power of the sample size, by a power spectrum that is divergent near the origin, and by correlations that are long range dependent. A number of analytical and experimental studies have established the performance significance of these features [4][8][14] More recent measurement work has focused on the physical basis of the self similarity observed in the full range of packet based networks. Based on a preliminary analysis of individual sources on an Ethernet, Willinger [17] observes that individual sources can be represented by the familiar ....

....such ON OFF behavior. We are motivated in part by experiences in a number of other disciplines in which chaotic maps have been used as efficient generators of fractal processes. Our approach is based on earlier work on the application of deterministic chaotic maps to model traffic flows [5] 6] 7][8]. Specifically, we consider the simplest class of chaotic systems, known as one dimensional (1 D) chaotic maps, in which the evolution of a state variable x over discrete time n is described by a deterministic nonlinear transformation . We can model packet traffic sources using such maps by ....

[Article contains additional citation context not shown here]

A. Erramilli, R.P. Singh, and P. Pruthi, "An Application of Deterministic Chaotic Maps to Model Packet Traffic," to appear Queueing Systems and their Applications, 1995.

.... [165] aggregation of simple short range dependent models) and [135, 294, 414, 416] wavelet analysis) Further models are considered in [16,19,176,313,379,386,403,417] A radically different approach to modeling self similar phenomena relies on ideas from the theories of chaos and fractals [73,97,118 120,124,125,171,250,287,337,344,345,377]; for a general discussion on chaos, probability and statistics, see [29, 46, 47] An overview of statistical inference methods for self similar models and random processes with long range dependence can be found in [22, 24] the papers [392 394] listing additional techniques. More specifically, ....

....practice. There exist numerous methods to date for generating self similar traffic traces. Exact methods, which are based on the Durbin Levinson algorithm [37, 394] are discussed in [24, 146, 196, 198, 394] They are generally impractical for long time series. Approximate methods are described in [30, 74, 120, 215, 245, 252, 258, 285, 290, 331, 334, 337, 356, 374, 376, 404,412]; some of these methods rely on earlier results reported in [77, 165,391] derived for a different purpose and re interpreted here in the context of synthetic traffic generation. These methods are generally very fast and feasible for even A Bibliographical Guide 7 very long time series. However, ....

A. Erramilli, R. P. Singh, and P. Pruthi. An application of deterministic chaotic maps to model packet traffic. Queueing Systems, 20:171--206, 1995.

....the actual data. In the following, we will use the Ethernet example to illustrate how answering this question for self similar traffic models leads to new insight into the dynamics of network traffic. In mathematical terms, developing an approach originally suggested by Mandelbrot [12] see also [16, 4]) it is shown in [19] that the superposition of many ON OFF sources, each of which exhibits a phenomenon called the Noah Effect , results in self similar aggregate traffic. Mapping these results into the well known framework of ON OFF source models (also known as packet train models , ....

A. Erramilli, R.P. Singh and P. Pruthi, "An Application of Deterministic Chaotic Maps to Model Packet Traffic", Queueing Systems, 1995 (to appear).

....aggregate packet traffic in terms of the nature of the traffic generated by the individual sources source destination pairs that contribute to the aggregate packet stream. Developing an approach originally suggested by Mandelbrot [31] see also Taqqu and Levy [42] and Erramilli, Singh and Pruthi [13]) they show that the superposition of many ON OFF sources, each of which exhibits a phenomenon called the Noah Effect , results in self similar aggregate traffic. Being able to phrase the results in the wellknown framework of ON OFF source models (also known as packet train models , introduced ....

....with timescale is a key characteristic of self similar traffic processes, and the parameter a in the FBM model is important, and should not be ignored in performance evaluation. Another early approach to modeling self similar traffic is based on chaotic maps (Erramilli, Singh and Pruthi [13] [14] The idea underlying this approach is to extend the state space to a continuum, and to describe the evolution of a continuous state variable x n over discrete time by means of a nonlinear (chaotic) map f( The packet generation process is now modeled by stipulating that a source ....

[Article contains additional citation context not shown here]

A. Erramilli, R.P. Singh, and P. Pruthi, "An Application of Deterministic Chaotic Maps to Model Packet Traffic," to appear Queueing Systems and their Applications, 1995.

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A. Erramilli, R. P. Singh and P. Pruthi, "An application of deterministic chaotic maps to model packet traffic," Queueing Systems, vol. 20, pp. 171-206, 1995.

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A. Erramilli, R. P. Singh and P. Pruthi, An application of deterministic chaotic maps to model packet traffic, Queuing Systems 20 (1995) 171-206.

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A. Erramilli, R. P. Singh, and P. Pruthi. An application of deterministic chaotic maps to model packet tra#c. Queueing Systems, 20:171--206, 1995.

No context found.

A. Eramilli, R.P. Singh, "An Application of Deterministic Chaotic Maps to Model Packet Traffic," Queueing Systems, Vol.20, pp. 171--206, 1995.

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A. Erramilli, R. P. Singh, and P. Pruthi, \An Application of Deterministic Chaotic Maps to Model Packet Trac," Queueing Systems 20, pp. 171-206, 1995.

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A. Erramilli, R. P. Singh, P. Pruthi, "An application of deterministic chaotic maps to model packet traffic", to appear in QUESTA, 1995.

No context found.

A. Erramilli, R. P. Singh, "Application of Deterministic Chaotic Maps to Model Packet Traffic in Broadband Networks", Proc. 7th ITC Specialists Seminar, Morristown, NJ, 8.1.1-8.1.3, 1990.

No context found.

A. Erramilli, R. P. Singh, "Application of Deterministic Chaotic Maps to Model Packet Traffic in Broadband Networks", Proc. 7th ITC Specialists Seminar, Morristown, NJ, 8.1.1-8.1.3, 1990.

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