Abstract
In 1993, it was found out that there are modeling problems with using Markovian statistics to describe data traffic. A series of experiments on Ethernet traffic revealed that the traffic behavior was fractal-like in nature and exhibit self-similarity, i.e. the statistical behavior was similar across many different time scales (seconds, hours, etc.) [1, 3]. Also, several research studies on traffic on wireless networks revealed that the existence of self-similar or fractal properties at a range of time scale from seconds to weeks. This scale-invariant property of data or video traffic means that the traditional Markovian traffic models used in most performance studies do not capture the fratal nature of computer network traffic. This has implications in buffer and network design. For example, the buffer requirements in multiplexers and switches will be incorrectly predicted. Thus, self-similar models, which can capture burstiness (see Fig. 10.1) over several time scales, may be more appropriate.
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Problems
Problems
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10.1
(a) Explain the concept of self-similarity.
(b) What is a self-similar process?
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10.2
Show that the Brownian motion process B(t) with parameter H = 1/2 is self-similar. Hint: Prove that B(t) satisfy conditions in Eqs. (10.1) to (10.3).
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10.3
Show that the Eq. (10.14) is valid and that the variance of Pareto distribution is infinite.
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10.4
If X is a random variable with a Pareto distribution with parameters α and δ, then show that the random variable Y = ln (X/δ) has an exponential distribution with parameter α.
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10.5
Evaluate and plot σ in Eq. (10.24) for 0 < ρ < 0.2 with r = 0.01.
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Sadiku, M.N.O., Musa, S.M. (2013). Self-Similarity of Network Traffic. In: Performance Analysis of Computer Networks. Springer, Cham. https://doi.org/10.1007/978-3-319-01646-7_10
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DOI: https://doi.org/10.1007/978-3-319-01646-7_10
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