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WUSTL CSE 567M - From Poisson Processes to Self-Similarity

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http://www.cse.wustl.edu/~jain/cse567-06/ftp/traffic_models1/index.html 1 of 13From Poisson Processes to Self-Similarity: a Survey of Network Traffic ModelsMichela Becchi, [email protected] paper provides a survey of network traffic models. It starts from the description of the Poisson model, born in the context of telephony, and highlights the main reasons for its inadequacy to describe data traffic in LANs and WANs. It then details two models which have been conceived to overcome the Poisson model's limitations. In particular, the discussion focuses on the packet train model, validated in a Token Ring LAN, and on the self-similar model, used to capture traffic burstiness at several times scales in both Ethernet LANs and WANs. The discussion closes with some examples of usage of those models in LAN and WAN environments.Keywords: Traffic models, Poisson processes, stochastic processes, compound processes, renewal processes, packet trains, self-similarity, fractals.Table of Contents1. Introduction2. Traffic modeling: basic concepts3. The Poisson Model3.1 Description of the model3.2 Traffic burstiness: the limitations of the Poisson model4. The Packet Train Model5. The Self-Similar Model5.1 Spatial and time variability: from Poisson to Fractals5.2 An analytical view of self similarity6. Other Traffic Models6.1 Renewal Traffic Models6.2 Markov Traffic Models6.3 Autoregressive Traffic Models6.4 Transform-Expand-Sample7. Application of the Models7.1 Modeling LAN traffic7.2 Modeling WAN traffic8. ConclusionsReferencesList of Acronyms1. IntroductionOne important research area in the context of networking focuses on developing traffic models which can be applied to the Internet and, more generally, to any communication network. The interest towards such models is two-fold. First, traffic models are needed as input in network simulations. In turn, these simulations must be performed in order to study and validate algorithms and protocols to be applied to real traffic, and to analyze how traffic reacts to particular network conditions (e.g.: congestion, etc.). Thus, it is essential that the assumed models reflect as much as possible the relevant characteristics of the traffic it is supposed to represent. Second, a good traffic model may lead to a better understanding of the characteristics of the network traffic itself. This, in turn, can help designing routers and devices which handle network traffic. If, for instance, a model which has been well validated shows some correlation between traffic arrivals, this information can be used in order to conceive ad hoc packet handling strategies.The first traffic model, based on Poisson processes, was born in the context of telephony, where call arrivals could be considered independent and identically distributed and "holding times" followed an exponential distribution. Althoughhttp://www.cse.wustl.edu/~jain/cse567-06/ftp/traffic_models1/index.html 2 of 13initially successful and analytically simple, the Poisson model has proven not suitable to describe data traffic in modern LANs and WANs, where batch arrivals, event correlations and traffic burstiness are important factors. The use of heavy tailed distributions and of self-similarity has become more and more predominant.The goal of this paper is to point out the most important concepts at the core of the basic traffic models in use, and to show how these models are applied to LANs and WANs.The remainder of the paper is organized as follows. Section 2 introduces basic concepts about traffic modeling. Section 3 describes the Poisson model and its limitations. Section 4 presents the Packet trains model, intended to overcome some of thePoisson model's limitations and to capture correlation and locality among packet arrivals. Section 5 introduces a mathematical description of self-similarity and explains how the self-similar model differs from traditional ones and allows capturing traffic burstiness at different time scales. Section 6 lists other traffic models in used. Sections 7 show how the models presented in the paper have been applied to traffic in LANs and WANs. Finally, section 8 closes the discussion with concluding remarks.Back to Table of Contents 2. Traffic modeling: basic conceptsInternet traffic can be modeled as a sequence of arrivals of discrete entities, such as packets, cells, etc. Mathematically, this leads to the usage of two equivalent representations: counting processes and interarrival time processes. A counting process {N(t)}t=0..∞ is a continuous-time, integer-valued stochastic process, where N(t) expresses the number of arrivals in the time interval (0,t]. An interarrival time process is a non-negative random sequence {An}, where An=Tn-Tn-1 indicates the length of the interval separating arrivals n-1 and n. The two kind of processes are related through the following equation:{N(t) = n}={Tn ≤ t < Tn+1} = { nΣ k=1 Ak ≤ t <n+1Σ k=1Ak } (1)In case of compound traffic, arrivals may happen in batches, that is, several arrivals can happen at the same instant Tn. This fact can be modeled by using an additional non-negative random sequence {Bn}n=1..∞, where Bn is the cardinality of the n-th batch. The traffic model is largely defined by the nature of the stochastic processes {N(t)} and {An} chosen, which will be analyzed in the remainder of this paper.One important issue in the selection of the stochastic process is its ability to describe traffic burstiness. In particular, a sequence of arrival times will be bursty if the Tn tend to form clusters, that is, if the corresponding {An} sees a mix of relatively long and short interarrival times. Mathematically speaking, traffic burstiness is related to short-terms autocorrelations between the interarrival times. However, there is not a single widely accepted notion of burstiness [frost94traffic]; instead, several different measures are used, some of which ignore the effect of second order properties of the traffic. A first measure is the ratio of peak rate to mean rate, and has the drawback of being dependent upon the interval used to measure the rate. A second measure is the coefficient of variation cA=σ[An]/E[An] of the interarrival times. A metric considering second order properties of the traffic is the index of dispersion for counts (IDC). In particular, given an intervalof time τ, IDC(τ)=Var[N(τ)]/E[N(τ)]. Because of the relationship in Eq. (1), IDC includes in the numerator the effects of


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