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| '''Self-similar processes''' are types of [[stochastic processes]] that exhibit the phenomenon of [[self-similarity]]. A self-similar phenomenon behaves the same when viewed at different degrees of magnification, or different scales on a dimension (space or time). Self-similar processes can sometimes be described using [[heavy-tailed distribution]]s, also known as [[long-tailed distribution]]s. Example of such processes include traffic processes such as packet inter-arrival times and burst lengths. Self-similar processes can exhibit [[long-range dependency]].
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| ==Overview==
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| The design of robust and reliable networks and network services has become an increasingly challenging task in today's [[Internet]] world. To achieve this goal,
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| understanding the characteristics of Internet traffic plays a more and more critical
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| role. Empirical studies of measured traffic traces have led to the wide recognition of
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| self-similarity in network traffic.{{citation needed|date=June 2012}}
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| Self-similar [[Ethernet]] traffic exhibits dependencies over a long range of time scales. This is to be contrasted with telephone traffic which is [[Poisson distribution|Poisson]] in its arrival and departure process.<ref name="autogenerated1">{{cite web|url=http://www.cs.bu.edu/brite/user_manual/node42.html |title=Appendix: Heavy-tailed Distributions |publisher=Cs.bu.edu |date=2001-04-12 |accessdate=2012-06-25}}</ref>
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| In traditional Poisson traffic, the short-term fluctuations would average out, and a graph covering a large amount of time would approach a constant value.
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| Heavy-tailed distributions have been observed in many natural phenomena including both physical and sociological phenomena. [[Benoît Mandelbrot|Mandelbrot]] established the use of heavy-tailed distributions to model real-world [[fractal]] phenomena, e.g. Stock markets, earthquakes, climate, and the weather.{{cn|date=April 2013}}
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| Ethernet, [[world wide web|WWW]], [[Signaling System 7|SS7]], [[transmission control protocol|TCP]], [[File Transfer Protocol|FTP]], [[TELNET]] and [[VBR]]{{disambiguation needed|date=June 2012}} video (digitised video of the type that is transmitted over [[asynchronous transfer mode|ATM]] networks) traffic is self-similar.
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| Self-similarity in packetised data networks can be caused by the distribution of file sizes, human interactions and/ or Ethernet dynamics. Self-similar and long-range dependent characteristics in computer networks present a fundamentally different set of problems to people doing analysis and/or design of networks, and many of the previous assumptions upon which systems have been built are no longer valid in the presence of self-similarity.<ref>{{cite web|url=http://www.cs.bu.edu/pub/barford/ss_lrd.html |title=The Self-Similarity and Long Range Dependence in Networks Web site |publisher=Cs.bu.edu |date= |accessdate=2012-06-25}}</ref>
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| ==The Poisson distribution==
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| Before the heavy-tailed distribution is introduced mathematically, the [[Poisson process]] with a [[memorylessness|memoryless]] waiting-time distribution, used to model (among many things) traditional telephony networks, is briefly reviewed below.
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| Assuming pure-chance arrivals and pure-chance terminations leads to the following:
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| *The number of call arrivals in a given time has a Poisson distribution, i.e.:
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| ::<math>
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| P(a)= \left ( \frac{\mu^a}{a!} \right )e^{-\mu},
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| </math> | |
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| where ''a'' is the number of call arrivals in time ''T'', and <math>\mu</math> is the mean number of call arrivals in time ''T''. For this reason, pure-chance traffic is also known as Poisson traffic.
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| *The number of call departures in a given time, also has a Poisson distribution, i.e.:
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| ::<math>
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| P(d)=\left(\frac{\lambda^d}{d!}\right)e^{-\lambda},
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| </math>
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| where ''d'' is the number of call departures in time ''T'' and <math>\lambda</math> is the mean number of call departures in time ''T''.
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| *The intervals, ''T'', between call arrivals and departures are intervals between independent, identically distributed random events. It can be shown that these intervals have a negative exponential distribution, i.e.:
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| ::<math>
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| P[T \ge \ t]=e^{-t/h},\,
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| </math> | |
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| where ''h'' is the mean holding time (MHT).{{citation needed|date=June 2012}}
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| ==The heavy-tail distribution==
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| {{main|heavy-tailed distribution}}
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| A distribution is said to have a heavy tail if
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| :<math>
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| \lim_{x \to \infty} e^{\lambda x}\Pr[X>x] = \infty \quad \mbox{for all } \lambda>0.\,
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| </math>
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| One simple example of a heavy-tailed distribution is the [[Pareto distribution]].
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| ==Modelling self-similar traffic==
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| Since (unlike traditional telephony traffic) packetised traffic exhibits self-similar or fractal characteristics, conventional traffic models do not apply to networks which carry self-similar traffic.{{citation needed|date=June 2012}}
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| With the convergence of voice and data, the future multi-service network will be based on packetised traffic, and models which accurately reflect the nature of self-similar traffic will be required to develop, design and dimension future multi-service networks.{{citation needed|date=June 2012}}
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| Previous analytic work done in Internet studies adopted assumptions such as exponentially-distributed packet inter-arrivals, and conclusions reached under such assumptions may be misleading or incorrect in the presence of heavy-tailed distributions.<ref name="autogenerated1"/>
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| Deriving mathematical models which accurately represent long-range dependent traffic is a fertile area of research.
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| ==Self-similar stochastic processes modeled by [[Tweedie distributions]]==
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| Leland ''et al'' have provided a mathematical formalism to describe self-similar stochastic processes.<ref name=Leland1994>{{cite journal|last=Leland|first=W E|coauthors=Leland, W. E., M. S. Taqqu, W. Willinger, and D. V. Wilson|title=On the self-similar nature of ethernet traffic|journal=IEE/ACM Trans. Networking|year=1994|volume=2|pages=1-15 }}</ref> For the sequence of numbers
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| : <math>Y=(Y_i :i=0,1,2,...,N)</math>
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| with mean
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| : <math>\hat{\mu}=\text{E}(Y_i)</math>,
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| deviations
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| : <math>y_i = Y_i - \hat{\mu} </math>,
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| variance
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| : <math>\hat{\sigma}^2=\text{E}(y_i^2)</math>,
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|
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| and autocorrelation function
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| : <math>r(k)=\text{E}(y_i,y_{i+k})/\text{E}(y_i^2)</math>
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| with lag ''k'', if the [[autocorrelation]] of this sequence has the long range behavior
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| : <math>r(k)\sim k^{-d} L(k) </math>
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| as ''k''{{math|<VAR>→∞</VAR>}} and where ''L(k)'' is a slowly varying function at large values of ''k'', this sequence is called a '''self-similar process'''.
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| The '''method of expanding bins''' can be used to analyze self-similar processes. Consider a set of equal-sized non-overlapping bins that divides the original sequence of ''N'' elements into groups of ''m'' equal-sized segments (''N/m'' is integer) so that new reproductive sequences, based on the mean values, can be defined:
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| : <math>Y_i^{(m)}=(Y_{im-m+1}+...+Y_{im})/m</math>.
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| The variance determined from this sequence will scale as the bin size changes such that
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| : <math>\text{var}[Y^{(m)}]=\hat{\sigma}^2 m^{-d}</math>
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|
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| if and only if the autocorrelation has the limiting form<ref name=Tsybakov1997>Tsybakov B & Georganas ND (1997) On self-similar traffic in ATM queues: definitions, overflow probability bound, and cell delay distribution. ''IEEE/ACM Trans Networking'' 5, 397–409</ref>
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| : <math>\lim_{k \to \infty}r(k)/k^{-d} = (2-d)(1-d)/2</math>.
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| One can also construct a set of corresponding additive sequences
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| : <math>Z_i^{(m)} = mY_i^{(m)}</math>,
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| based on the expanding bins,
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| : <math>Z_i^{(m)}=(Y_{im-m+1}+...+Y_{im})</math>.
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| Provided the autocorrelation function exhibits the same behavior, the additive sequences will obey the relationship
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| : <math>\text{var}[Z_i^{(m)}]=m^2 \text{var}[Y^{(m)}]=(\hat{\sigma}^2 /\hat{\mu}^{2-d})\text{E}[Z_i^{(m)}]^{2-d}</math>
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| Since <math>\hat{\mu}</math> and <math>\hat{\sigma}^2</math> are constants this relationship constitutes a '''variance-to-mean [[power law]]''' ([[Taylor's law]]), with ''p''=2-''d''.<ref name=Kendal2011b>Kendal WS & Jørgensen BR (2011) Tweedie convergence: a mathematical basis for Taylor's power law, ''1/f'' noise and multifractality. ''Phys. Rev E'' 84, 066120</ref>
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| '''Tweedie distributions''' are a special case of [[exponential dispersion model]]s, a class of models used to describe error distributions for the [[generalized linear model]].<ref
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| name="Jørgensen-1997">{{cite book
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| | author = Jørgensen, Bent
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| | year = 1997
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| | title = The theory of dispersion models
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| | publisher = Chapman & Hall
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| | isbn = 978-0412997112
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| }}</ref>
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| These Tweedie distributions are characterized by an inherent [[scale invariance]] and thus for any [[random variable]] ''Y'' that obeys a Tweedie distribution, the [[variance]] var(''Y'') relates to the [[mean]] E(''Y'') by the power law,
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| : <math>\text{var}\,(Y) = a[\text{E}\,(Y)]^p ,</math>
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| where ''a'' and ''p'' are positive constants. The exponent ''p'' for the variance to mean power law associated with certain self-similar stochastic processes ranges between 1 and 2 and thus may be modeled in part by a [[Tweedie distributions|Tweedie compound Poisson–gamma distribution]].<ref name=Kendal2011b></ref>
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| The additive form of the Tweedie compound Poisson-gamma model has the [[cumulant|cumulant generating function]] (CGF),
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| : <math>K^*_p(s;\theta,\lambda) = \lambda\kappa_p(\theta)[(1+s/\theta)^\alpha-1]</math>,
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| where
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| : <math>\kappa_p(\theta) = \dfrac{\alpha-1}{\alpha} \left(\dfrac{\theta}{\alpha-1}\right)^\alpha</math>,
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| is the [[cumulant]] function, ''α'' is the Tweedie exponent
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| : <math>\alpha=\dfrac{p-2}{p-1}</math>,
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| ''s'' is the generating function variable, ''θ'' is the [[Exponential family|canonical parameter]] and ''λ'' is the index parameter.
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| The first and second derivatives of the CGF, with ''s=0'', yields the mean and variance, respectively. One can thus confirm that for the additive models the variance relates to the mean by the power law,
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| : <math>\mathrm{var} (Z)\propto \mathrm{E}(Z)^p</math>.
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| Whereas this Tweedie compound Poisson-gamma CGF will represent the [[probability density function]] for certain self-similar stochastic processes, it does not return information regarding the long range correlations inherent to the sequence ''Y''.
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| Nonetheless the Tweedie distributions provide a means understand the possible origins of self-similar stochastic processes for reason of their role as foci for a [[central limit theorem|central limit-like]] [[convergence (mathematics)|convergence]] effect known as the [[Tweedie distributions|Tweedie convergence theorem]]. In nontechnical terms this theorem tells us that any exponential dispersion model that asymptotically manifests a variance-to-mean power law is required to have a variance function that comes within the [[Attractor|domain of attraction]] of a Tweedie model.
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| The Tweedie convergence theorem can be used to explain the origin of the [[Taylor's law|variance to mean power law]], [[pink noise|''1/f'' noise]] and [[Multifractal system|multifractality]], features associated with self-similar processes.<ref name=Kendal2011b>Kendal WS & Jørgensen BR (2011) Tweedie convergence: a mathematical basis for Taylor's power law, ''1/f'' noise and multifractality. ''Phys. Rev E'' 84, 066120</ref>
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| ==Network performance==
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| Network performance degrades gradually with increasing self-similarity. The more self-similar the traffic, the longer the queue size. The queue length distribution of self-similar traffic decays more slowly than with Poisson sources.
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| However, long-range dependence implies nothing about its short-term correlations which affect performance in small buffers. Additionally, aggregating streams of self-similar traffic typically intensifies the self-similarity ("burstiness") rather than smoothing it, compounding the problem.{{citation needed|date=June 2012}}
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| Self-similar traffic exhibits the persistence of [[Burst transmission|cluster]]ing which has a negative impact on network performance.
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| *With Poisson traffic (found in conventional [[telephony]] networks), clustering occurs in the short term but smooths out over the long term.
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| *With self-similar traffic, the bursty behaviour may itself be bursty, which exacerbates the clustering phenomena, and degrades network performance.
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| Many aspects of network quality of service depend on coping with traffic peaks that might cause network failures, such as
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| *Cell/packet loss and queue overflow
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| *Violation of delay bounds e.g. in video
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| *Worst cases in statistical [[multiplexing]]
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| Poisson processes are well-behaved because they are [[stateless server|stateless]], and peak loading is not sustained, so queues do not fill. With long-range order, peaks last longer and have greater impact: the equilibrium shifts for a while.<ref>{{cite web|url=http://www.cs.kent.ac.uk/people/staff/pfl/presentations/longrange/ |title=Everything you always wanted to know about Self-Similar Network Traffic and Long-Range Dependency, but were ashamed to ask* |publisher=Cs.kent.ac.uk |date= |accessdate=2012-06-25}}</ref>
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| ==See also==
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| *[[Long-tail traffic]]
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| ==References==
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| {{Reflist}}
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| ==External links==
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| *[http://www.columbia.edu/~ww2040/A12.html A site offering numerous links to articles] written on the effect of self-similar traffic on network performance.
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| [[Category:Teletraffic]]
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| [[Category:Stochastic processes]]
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| [[Category:Time series analysis]]
| |
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