This page describes papers and talks I have given on the subject of LRD in the internet.
R. G. Clegg, A new method for queuing performance estimates Using Markov chains (submitted to journal, currently being reviewed). Available at Arxiv. This paper allows estimates for the expected queue length and probability of given queue lengths for a large class of queueing models. The models analysed works with discrete time and has the following properties: the model has two states, on and off; in an off state no traffic is generated and the model will remain in an off state with a given probability f0; in an off state the model will switch to an on state which will last for n time units with a given probability fn; in an on state, during one time unit, the model generates a non-zero integer number of work units with a given probability distribution; and exactly one unit of work is used every time unit. Given this model, the paper gives an exact solution for the expected equilibrium queue length and a system of equations for the probability of a given queue length. This paper also has important implications for the generation of LRD using Markov methods as described below.
R. G. Clegg, Markov-modulated on/off processes for long-range dependent internet traffic (submitted to journal, currently being reviewed). Available at Arxiv. This paper has not been subject to peer review and may be revised as a subject of referees' comments. The paper uses a very simple queuing model to test the queuing performance of a number of models which generate LRD. The models are tuned to replicate real-life data and their queuing performance is compared with the real-life data. The conclusion is that none of the models reviewed (including the author's own model described below) accurately replicates the queuing performance of real internet traffic.
A Practical Guide to Measuring the Hurst Parameter, International Journal of Simulation: Systems, Science & Technology 7(2) pp 3-14 2006. Available at Arxiv. This paper is intended as the title claims, as a practical guide for those who have a data set and wish to measure the Hurst parameter (the most commonly used measure of LRD). It provides links to software using R which can estimate LRD using several estimators. A basic message of this paper is that different estimators very often produce different results. Relying on just one estimator for the Hurst parameter can be extremely misleading. Similarly, investigating data at just one time resolution can lead to problems. Commonly used filtering techniques to improve estimation seem to have little effect on these problems.
C. Di Cairano-Gilfedder and R. G. Clegg, A decade of internet research: Advances in models and practices BT Technology Journal, 23 (4) p115-128 October 2005. Available as preprint. This is a summary paper describing research into modelling the internet with a bent towards LRD. It could serve as an introduction to the subject for someone interested in learning about LRD.
R. G. Clegg and M. M. Dodson, A Markov Chain based method for generating long-range dependence, Phys. Rev. E 72, 026118 2005. Available at Arxiv or at PRE (requires subscription). This paper describes a simple message for generating a binary sequence (which can be interpreted as a stream of packets and inter-packet gaps) which exhibit LRD. The method described has several advantages. The mean of the generated data and its Hurst parameter can be easily set. Data can be generated in an ongoing manner (unlike frequency domain based methods which often require specification of how many packets must be generated before the series begins). The algorithm is simple and does not have issues with finite precision arithmetic. However see the paper "Markov-modulated on/off processes for long-range dependent internet traffic" above for criticism of the model accuracy and the paper "A new method for performance estimates using Markov chains" above for criticism of a fundamental property of this model.
The first two chapters of my PhD thesis, The Statistics of Dynamic Networks (submitted 2004, accepted 2005) concern LRD. The first chapter is (I hope) a reasonably clear introduction to the subject from the perspective of an internet engineer and contains a summary of research up until 2004. I believe it would be a good introduction to the subject for someone from an engineering, mathematics or statistics background but who had not previously encountered LRD before. The second chapter describes the model from the paper A Markov Chain based method for generating long-range dependence above. The PRE paper is a better reference for that model.
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