Abstract- In this paper we demonstrate how TCP congestion control can show chaotic behavior. We demonstrate the major features of chaotic systems in TCPlIP networks with examples. These features include un-predictability, extreme sensitivity to initial conditions and odd periodicity. Previous work has shown the fractal nature of aggregate TCPAP traffic and one explanation to this phenomenon was that traffic can be approxi-mated by a large number of ON/OFF sources where the random ON and/or OFF periods are of length described by a heavy tailed distribution. In this paper we show that this argument is not necessary to explain self-similarity, neither randomness is required. Rather, TCP itself as a deter-ministic process creates chaos, which generates self-similarity. This prop-erty is inherent in todays TCPlIP networks and it is independent of higher layer applications or protocols. The two causes: heavy tailed ONlOFF and chaotic TCP together contribute to the phenomena, called fractal nature of Internet traffic. Keywords-TCP congestion control, fractal traffic, chaotic models. I.

Observations of both Ethernet traffic and variable bit rate (VBR) video traffic have demonstrated that these traffics exhibit “self-similarity ” and/or infinit l e asymptotic index of dispersion for counts (IDC). We report here on measurements of traffic an a commercialpublic broadband network where similar characteristics have been observed. For the purpose of analysis and dimensioning of the central links of an ATM network we analyse in this paper the performance of a single server queue fed by Gaussian trafic with infjlnite IDC. The analysis leads to an approximation fo,r the performance of a queue in which the arriving trafic is “fractal ” Gaussian and consequently where there does not exist a dominant negative-exponential tail. The term ufractal ” is used here in the sense thtzt the autocovariance of the traffic exhibits self-similarity, that is to say, where the autocovariance of an aggregate of the trafic is the same, or asymptotically the same for large time lags, as the original traffic. We are not concerned with proving or exploiting this self-similarity property as such, but only with performancle analysis techniques which are effective for such processes. In order to be able to test the performance analysis formulae, we show that trafic with the same arutocovariawe as measured an a real network over a wide range of lags (sufficiently wide a range for the traffic to be equivalent from the point of view of queueing performance) can be generated as a mixture of two Gaussian AR(1) processes. In this way we demonstrate that the analytic performance formulae are accurate. 1

This is a survey paper on fluid queues, with a strong emphasis on recent attempts to represent phenomena like long-range dependence. The central model of the paper is a fluid queueing system fed by N independent sources that alternate between silence and activity periods. The distribution of the activity periods of at least one source is assumed to be long-tailed, which may give rise to long-range dependence. We consider the effect of this tail behaviour on the steady-state distributions of the buffer content at embedded points in time and at arbitrary time, and on the busy period distribution. Both exact results and bounds are discussed.

After measurements on a lan at Bellcore, it is known that data traffic is extremely variable on time scales ranging from milliseconds to days. The traffic behaves quite differently to what has been assumed until now; traffic sources were generally characterized by short term dependences but characteristics of the measured traffic have shown that it is long term dependent. Therefore, new models (such as Fractional Brownian motion, arima processes and Chaotic maps) have been applied. Although they are not easily tractable, one big advantage of these models is that they give a good description of the traffic using few parameters. In this paper, we describe a Markov chain emulating self-similarity which is quite easy to manipulate and depends only on two parameters (plus the number of states in the Markov chain). An advantage of using it is that it is possible to re-use the well-known analytical queuing theory techniques developed in the past in order to evaluate network performance. The t...

Recent measurements of packet/cell... In this paper, we first give some definitions of selfsimilarity. Then, we propose a fitting method for the self-similar traffic in terms of Markov-modulated Poisson process (MMPP). We construct an MMPP as the superposition of two-state MMPPs and fit it so as to match the variance function over several time-scales. Numerical examples show that the variance function of the self-similar process can be well represented by that of resulting MMPPs. We also examine the queueing behavior of the resulting MMPP/D/1 queueing systems. We compare the analytical results of MMPP/D/1 with the simulation ones of the queueing system with self-similar input.

This paper considers a fluid queueing system, fed by N independent sources that alternate between silence and activity periods. We assume that the distribution of the activity periods of source 1 is a regularly varying function of index i , whereas all other sources have activity period distributions with an exponential tail. In addition, we assume that the inflow rate of each of the sources, when active, exceeds the outflow rate of the buffer. Under these assumptions, we show that the tail of the buffer content distribution is regularly varying of index i +1. In the special case that i 2 (\Gamma2; \Gamma1), which implies long-range dependence of the input process, the buffer content does not even have a finite first moment. Based on the obtained results and on a conjecture for the case that the outflow rate of the buffer is not necessarily exceeded by the inflow rates of the sources, we suggest a simple, effective-bandwidth-like, connection admission rule. AMS Subject Classi...

Since the discovery in the early 1990's that packet traffic exhibits long-range-dependence there has been considerable interest in finding system that could generate reliable pseudo-traffic traces. One such system is a one-dimensional chaotic map. We examine a particular instance of such maps. It is non-smooth, but its nature is such that analytical results are easier to obtain. In particular we derive a theorem about the asymptotics of the autocorrelation. In the subsequent discussion, it is argued that to get a relevant description of packet traffic, one needs to venture beyond first- and second-order statistics.

Recent papers have pointed out that data traffic exhibits self-similarity, but self-similarity is observed only on a finite timescale. In order to account for that, we introduce the concept of pseudo long-range dependences. In this paper, we describe a Modulated Markov process producing self-similarity on a finite timescale; the process is quite easy to manipulate and depends only on three parameters (two real numbers and one integer). An advantage of using it is that it is possible to re-use the well-known analytical queuing theory techniques developed in the past in order to evaluate network performance. A quantitative method based on the decomposability theory of Courtois is given to evaluate the domain of validity where the process exhibits pseudo long-range dependences. The validation on a queuing problem is also discussed. Finally, we analyze the inputs of a statistical multiplexer in the context of a project called Scalability Enhancements for Connection-Oriented Networks (SCONE).

* This paper deals with the problem of traffic modelling for an Optical Transparent Packet Network (OTP-N), designed to carry very high bit rate traffic. The first part of the paper discusses the issue of the functional implementation of the access interface to the network, proposing two possible solutions to this problem. The second part shows how the incoming traffic is influenced by the interface. The aim is that to characterise the traffic to be carried on the OTP-N and in particular to understand whether it is equal, better shaped or worse than the input one. The results presented show that the traffic in the OTP-N is in general more "well behaving" for the network performance than the input one and therefore that the solution proposed for the access to the OTP-N is beneficial in terms of traffic shaping. 1. Introduction In existing high speed networks routing and multiplexing are generally performed electrically, optics being confined to transmission. Nevertheless the bandwidt...

This report is a brief survey of schemes used for characterization of application traffic. We discuss the different kinds of applications, and their QoS requirements. Measurement-based, Statistical model-based and Deterministic model-based schemes are used to characterize traffic, and we elucidate each scheme with an application. We look at self-similarity and, consequences of changes in network resources and flow-control algorithms on self-similar traffic. Finally, we cover a few parsimonious models for this kind of traffic.