Prüfer domain

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 distributions, also known as long-tailed distributions. Example of such processes include traffic processes such as packet inter-arrival times and burst lengths. Self-similar processes can exhibit long-range dependency.

Overview

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, understanding the characteristics of Internet traffic plays a more and more critical role. Empirical studies of measured traffic traces have led to the wide recognition of self-similarity in network traffic.Potter or Ceramic Artist Truman Bedell from Rexton, has interests which include ceramics, best property developers in singapore developers in singapore and scrabble. Was especially enthused after visiting Alejandro de Humboldt National Park.

Self-similar Ethernet traffic exhibits dependencies over a long range of time scales. This is to be contrasted with telephone traffic which is Poisson in its arrival and departure process.

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.

Heavy-tailed distributions have been observed in many natural phenomena including both physical and sociological phenomena. Mandelbrot established the use of heavy-tailed distributions to model real-world fractal phenomena, e.g. Stock markets, earthquakes, climate, and the weather.Template:Cn Ethernet, WWW, SS7, TCP, FTP, TELNET and VBRTemplate:Disambiguation needed video (digitised video of the type that is transmitted over ATM networks) traffic is self-similar.

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doing analysis and/or design of networks, and many of the previous assumptions upon which systems have been built are no longer valid in the presence of self-similarity.

The Poisson distribution

Before the heavy-tailed distribution is introduced mathematically, the Poisson process with a memoryless waiting-time distribution, used to model (among many things) traditional telephony networks, is briefly reviewed below.

Assuming pure-chance arrivals and pure-chance terminations leads to the following:

• The number of call arrivals in a given time has a Poisson distribution, i.e.:

$P(a)=\left(\frac{{\mu }^a}{a!}\right)e^{-\mu },$

where a is the number of call arrivals in time T, and $\mu$ is the mean number of call arrivals in time T. For this reason, pure-chance traffic is also known as Poisson traffic.

• The number of call departures in a given time, also has a Poisson distribution, i.e.:

$P(d)=\left(\frac{{\lambda }^d}{d!}\right)e^{-\lambda },$

where d is the number of call departures in time T and $\lambda$ is the mean number of call departures in time T.

• 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.:

$P[T\ge t]=e^{-t/h},$

where h is the mean holding time (MHT).Potter or Ceramic Artist Truman Bedell from Rexton, has interests which include ceramics, best property developers in singapore developers in singapore and scrabble. Was especially enthused after visiting Alejandro de Humboldt National Park.

The heavy-tail distribution

Mining Engineer (Excluding Oil ) Truman from Alma, loves to spend time knotting, largest property developers in singapore developers in singapore and stamp collecting. Recently had a family visit to Urnes Stave Church. A distribution is said to have a heavy tail if

$\underset{x\to \mathrm{\infty }}{\lim }{e}^{\lambda x}\mathrm{Pr}[X>x]=\mathrm{\infty }\text{for all }\lambda >0.$

One simple example of a heavy-tailed distribution is the Pareto distribution.

Modelling self-similar traffic

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.Potter or Ceramic Artist Truman Bedell from Rexton, has interests which include ceramics, best property developers in singapore developers in singapore and scrabble. Was especially enthused after visiting Alejandro de Humboldt National Park.

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.Potter or Ceramic Artist Truman Bedell from Rexton, has interests which include ceramics, best property developers in singapore developers in singapore and scrabble. Was especially enthused after visiting Alejandro de Humboldt National Park.

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.

Deriving mathematical models which accurately represent long-range dependent traffic is a fertile area of research.

Self-similar stochastic processes modeled by Tweedie distributions

Leland et al have provided a mathematical formalism to describe self-similar stochastic processes. For the sequence of numbers

$Y=(Y_i:i=0,1,2,...,N)$

with mean

$\hat{\mu }=\text{E}(Y_i)$,

deviations

$y_i=Y_i-\hat{\mu }$,

variance

${\hat{\sigma }}^2=\text{E}(y_i^2)$,

and autocorrelation function

$r(k)=\text{E}(y_i,y_{i+k})/\text{E}(y_i^2)$

with lag k, if the autocorrelation of this sequence has the long range behavior

$r(k)\sim k^{-d}L(k)$

as kBuying, selling and renting HDB and personal residential properties in Singapore are simple and transparent transactions. Although you are not required to engage a real property salesperson (generally often known as a "public listed property developers In singapore agent") to complete these property transactions, chances are you'll think about partaking one if you are not accustomed to the processes concerned.

Professional agents are readily available once you need to discover an condominium for hire in singapore In some cases, landlords will take into account you more favourably in case your agent comes to them than for those who tried to method them by yourself. You need to be careful, nevertheless, as you resolve in your agent. Ensure that the agent you are contemplating working with is registered with the IEA – Institute of Estate Brokers. Whereas it might sound a hassle to you, will probably be worth it in the end. The IEA works by an ordinary algorithm and regulations, so you'll protect yourself in opposition to probably going with a rogue agent who prices you more than they should for his or her service in finding you an residence for lease in singapore.

There isn't any deal too small. Property agents who are keen to find time for any deal even if the commission is small are the ones you want on your aspect. Additionally they present humbleness and might relate with the typical Singaporean higher. Relentlessly pursuing any deal, calling prospects even without being prompted. Even if they get rejected a hundred times, they still come again for more. These are the property brokers who will find consumers what they need eventually, and who would be the most successful in what they do. 4. Honesty and Integrity

This feature is suitable for you who need to get the tax deductions out of your PIC scheme to your property agency firm. It's endorsed that you visit the correct site for filling this tax return software. This utility must be submitted at the very least yearly to report your whole tax and tax return that you're going to receive in the current accounting 12 months. There may be an official website for this tax filling procedure. Filling this tax return software shouldn't be a tough thing to do for all business homeowners in Singapore.

A wholly owned subsidiary of SLP Worldwide, SLP Realty houses 900 associates to service SLP's fast rising portfolio of residential tasks. Real estate is a human-centric trade. Apart from offering comprehensive coaching applications for our associates, SLP Realty puts equal emphasis on creating human capabilities and creating sturdy teamwork throughout all ranges of our organisational hierarchy. Worldwide Presence At SLP International, our staff of execs is pushed to make sure our shoppers meet their enterprise and investment targets. Under is an inventory of some notable shoppers from completely different industries and markets, who've entrusted their real estate must the expertise of SLP Worldwide.

If you're looking for a real estate or Singapore property agent online, you merely need to belief your instinct. It is because you don't know which agent is sweet and which agent will not be. Carry out research on a number of brokers by looking out the internet. As soon as if you find yourself certain that a selected agent is dependable and trustworthy, you'll be able to choose to utilize his partnerise find you a house in Singapore. More often than not, a property agent is considered to be good if she or he places the contact data on his web site. This is able to imply that the agent does not thoughts you calling them and asking them any questions regarding properties in Singapore. After chatting with them you too can see them of their office after taking an appointment.

Another method by way of which you could find out whether the agent is sweet is by checking the feedback, of the shoppers, on the website. There are various individuals would publish their comments on the web site of the Singapore property agent. You can take a look at these feedback and the see whether it will be clever to hire that specific Singapore property agent. You may even get in contact with the developer immediately. Many Singapore property brokers know the developers and you may confirm the goodwill of the agent by asking the developer. and where L(k) is a slowly varying function at large values of k, this sequence is called a self-similar process.

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:

$Y_i^{(m)}=(Y_{im-m+1}+...+Y_{im})/m$.

The variance determined from this sequence will scale as the bin size changes such that

$\text{var}[Y^{(m)}]={\hat{\sigma }}^2{m}^{-d}$

if and only if the autocorrelation has the limiting form

$\underset{k\to \mathrm{\infty }}{\lim }r(k)/k^{-d}=(2-d)(1-d)/2$.

One can also construct a set of corresponding additive sequences

$Z_i^{(m)}=m{Y}_i^{(m)}$,

based on the expanding bins,

$Z_i^{(m)}=(Y_{im-m+1}+...+Y_{im})$.

Provided the autocorrelation function exhibits the same behavior, the additive sequences will obey the relationship

$\text{var}[Z_i^{(m)}]=m^2\text{var}[Y^{(m)}]=({\hat{\sigma }}^2/{\hat{\mu }}^{2-d})\text{E}[Z_i^{(m)}{]}^{2-d}$

Since $\hat{\mu }$ and ${\hat{\sigma }}^2$ are constants this relationship constitutes a variance-to-mean power law (Taylor's law), with p=2-d.

Tweedie distributions are a special case of exponential dispersion models, a class of models used to describe error distributions for the generalized linear model.

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,

$\text{var}(Y)=a[\text{E}(Y){]}^p,$

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 compound Poisson–gamma distribution.

The additive form of the Tweedie compound Poisson-gamma model has the cumulant generating function (CGF),

$K_p^{*}(s;\theta ,\lambda )=\lambda {\kappa }_p(\theta )[(1+s/\theta {)}^{\alpha }-1]$,

where

${\kappa }_p(\theta )=\frac{\alpha -1}{\alpha }{\left(\frac{\theta }{\alpha -1}\right)}^{\alpha }$,

is the cumulant function, α is the Tweedie exponent

$\alpha =\frac{p-2}{p-1}$,

s is the generating function variable, θ is the canonical parameter and λ is the index parameter.

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,

$\mathrm{v}\mathrm{a}\mathrm{r}(Z)\propto \mathrm{E}(Z{)}^p$.

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.

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-like convergence effect known as the 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 domain of attraction of a Tweedie model.

The Tweedie convergence theorem can be used to explain the origin of the variance to mean power law, 1/f noise and multifractality, features associated with self-similar processes.

Network performance

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. 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.Potter or Ceramic Artist Truman Bedell from Rexton, has interests which include ceramics, best property developers in singapore developers in singapore and scrabble. Was especially enthused after visiting Alejandro de Humboldt National Park.

Self-similar traffic exhibits the persistence of clustering which has a negative impact on network performance.

• With Poisson traffic (found in conventional telephony networks), clustering occurs in the short term but smooths out over the long term.
• With self-similar traffic, the bursty behaviour may itself be bursty, which exacerbates the clustering phenomena, and degrades network performance.

Many aspects of network quality of service depend on coping with traffic peaks that might cause network failures, such as

• Cell/packet loss and queue overflow
• Violation of delay bounds e.g. in video
• Worst cases in statistical multiplexing

Poisson processes are well-behaved because they are 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.

See also

• Long-tail traffic

References

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.

External links

• A site offering numerous links to articles written on the effect of self-similar traffic on network performance.