45:. The differences between these two phenomena are subtle. Heavy-tailed refers to a probability distribution, and long-range dependent refers to a property of a time series and so these should be used with care and a distinction should be made. The terms are distinct although superpositions of samples from heavy-tailed distributions aggregate to form long-range dependent time series.
1371:, including the selection of alternative protocols, for more effective data transport. For short files, which constitute the bulk of connection requests in heavy-tailed file size distributions of web servers, elaborate feedback control may be bypassed in favour of lightweight mechanisms in the spirit of optimistic control, which can result in improved bandwidth utilisation.
1295:
provisions in networks. To achieve a constant level of throughput or packet loss as self-similarity is increased, extremely large buffer capacity is needed. However, increased buffering leads to large queuing delays and thus self-similarity significantly steepens the trade-off curve between throughput/ packet loss and delay.
772:
the logarithm of both the range and the domain is taken, the tail of the long-tail distribution is approximately linear over many orders of magnitude. In the graph above left, the condition for the existence of a heavy-tail distribution, as previously presented, is not met by the curve labelled "Gamma-Exponential Tail".
38:. For example, if we consider the sizes of files transferred from a web server, then, to a good degree of accuracy, the distribution is heavy-tailed, that is, there are a large number of small files transferred but, crucially, the number of very large files transferred remains a major component of the volume downloaded.
771:
If the logarithm of the range of an exponential distribution is taken, the resulting plot is linear. In contrast, that of the heavy-tail distribution is still curvilinear. These characteristics can be clearly seen on the graph above to the right. A characteristic of long-tail distributions is that if
133:
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
1342:
Given the ubiquity of scale-invariant burstiness observed across diverse networking contexts, finding an effective traffic control algorithm capable of detecting and managing self-similar traffic has become an important problem. The problem of controlling self-similar network traffic is still in its
1326:
With respect to SLAs, the same level of service for heavy-tailed distributions requires a more powerful set of servers, compared with the case of independent light-tailed request traffic. To guarantee good performance, focus needs to be given to peak traffic duration because it is the huge bursts of
86:
if the series is averaged then the data begins to look smoother. However, with self-similar data, one is confronted with traces that are spiky and bursty, even at large scales. Such behaviour is caused by strong dependence in the data: large values tend to come in clusters, and clusters of clusters,
1049:
It has for long been realised that efficient and accurate modelling of various real-world phenomena needs to incorporate the fact that observations made on different scales each carry essential information. In most simple terms, representing data on large scales by its mean is often useful (such as
1021:
Second, is a transport layer cause which theorizes that the feedback between multiple TCP streams due to TCP's congestion avoidance algorithm in moderate to high packet loss situations causes self-similar traffic or at least allows it to propagate. However, this is believed only to be a significant
1001:
In general, there are three main theories for the causes of long-tail traffic (see a review of all three causes). First, is a cause based in the application layer which theorizes that user session durations vary with a long-tail distribution due to the file size distribution. If the distribution of
1374:
It was found that the simplest way to control packet traffic is to limit the length of queues. Long queues in the network invariably occur at hosts (entities that can transmit and receive packets). Congestion control can therefore be achieved by reducing the rate of packet production at hosts with
1045:
Since (unlike traditional telephony traffic) packetised traffic exhibits self-similar or fractal characteristics, conventional traffic models do not apply to networks that carry long-tail traffic. Previous analytic work done in
Internet studies adopted assumptions such as exponentially-distributed
1355:
in nature, multiple time scale traffic control exploits the long-range correlation structure present in self-similar traffic. Congestion control can be exercised concurrently at multiple time scales, and by cooperatively engaging information extracted at different time scales, achieve significant
1061:
There is not an abundance of heavy-tailed models with rich sets of accompanying data-fitting techniques. A clear model for fractal traffic has not yet emerged, nor is there any definite direction towards a clear model. Deriving mathematical models which accurately represent long-tail traffic is a
1025:
Finally, is a theorized link layer cause which is predicated based on physics simulations of packet switching networks on simulated topologies. At a critical packet creation rate, the flow in a network becomes congested and exhibits 1/f noise and long-tail traffic characteristics. There have been
1350:
The resource provisioning approach seeks to identify the relative utility of the two principal network resource types – bandwidth and buffer capacity – with respect to their curtailing effects on self-similarity, and advocates a small buffer/ large bandwidth resource dimensioning policy. Whereas
1346:
Traffic control for self-similar traffic has been explored on two fronts: Firstly, as an extension of performance analysis in the resource provisioning context, and secondly, from the multiple time scale traffic control perspective where the correlation structure at large time scales is actively
1294:
declines gradually as self-similarity increases, queuing delay increases more drastically. When traffic is self-similar, we find that queuing delay grows proportionally to the buffer capacity present in the system. Taken together, these two observations have potentially dire implications for QoS
1192:
Traffic self-similarity negatively affects primary performance measures such as queue size and packet-loss rate. The queue length distribution of long-tail traffic decays more slowly than with
Poisson sources. However, long-range dependence implies nothing about its short-term correlations which
767:
which is hyperbolic over its entire range. Complementary distribution functions for the exponential and Pareto distributions are shown below. Shown on the left is a graph of the distributions shown on linear axes, spanning a large domain. To its right is a graph of the complementary distribution
1029:
Simulation showed that long-range dependence could arise in the queue length dynamics at a given node (an entity that transfers traffic) within a communications network even when the traffic sources are free of long-range dependence. The mechanism for this is believed to relate to feedback from
1200:
The graph above right, taken from, presents a queueing performance comparison between traffic streams of varying degrees of self-similarity. Note how the queue size increases with increasing self-similarity of the data, for any given channel utilisation, thus degrading network performance.
1053:
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 long-tail traffic will be required to develop, design and dimension future multi-service networks. We seek an equivalent to the
1286:
For network queues with long-range dependent inputs, the sharp increase in queuing delays at fairly low levels of utilisation and slow decay of queue lengths implies that an incremental improvement in loss performance requires a significant increase in buffer size.
1298:
ATM can be employed in telecommunications networks to overcome second-order performance measure problems. The short fixed-length cell used in ATM reduces the delay and most significantly the jitter for delay-sensitive services such as voice and video.
1193:
affect performance in small buffers. For heavy-tailed traffic, extremely large bursts occur more frequently than with light-tailed traffic. Additionally, aggregating streams of long-tail traffic typically intensifies the self-similarity ("
1327:
requests that most degrade performance. That is why some busy sites require more headroom (spare capacity) to handle the volumes; for example, a high-volume online trading site reserves spare capacity with a ratio of three to one.
1315:. Without adaptive, optimal management and control of resources, SLAs based on response time are impossible. The capacity requirements on the site are increased while its ability to provide acceptable levels of performance and
275:
that explains how other types of random data will converge towards the form of these
Tweedie distributions, and consequently express both the variance to mean power law and a power law decay in their autocorrelation functions.
879:
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The ability to accurately forecast request patterns is an important requirement of capacity planning. A practical consequence of burstiness and heavy-tailed and correlated arrivals is difficulty in capacity planning.
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Before the heavy-tail distribution is introduced mathematically, the memoryless
Poisson distribution, used to model traditional telephony networks, is briefly reviewed below. For more details, see the article on the
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are met. The following subsection deals with the provisioning of standard network resources, and the subsection after that looks at provisioning web servers that carry a significant amount of long-tail traffic.
762:
This means that regardless of the distribution for small values of the random variable, if the asymptotic shape of the distribution is hyperbolic, it is heavy-tailed. The simplest heavy-tail distribution is the
375:
1227:) are best-effort services, so degraded performance, although undesirable, can be tolerated. However, since the connection is contracted, ATM networks need to keep delays and jitter within negotiated limits.
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64:
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
255:
Such power law scaling of the autocorrelation function can be shown to be biconditionally related to a power law relationship between the variance and the mean, when evaluated from sequences by the
530:, 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.:
1189:
control is able to shape source traffic into an on-average constant output stream while conserving information. Congestion control of heavy-tailed traffic is discussed in the following section.
1185:
In some cases, an increase in the Hurst parameter can lead to a reduction in network performance. The extent to which heavy-tailedness degrades network performance is determined by how well
230:
595:
1177:-stable stochastic processes for modelling traffic in broadband networks are presented. The simulations are compared to a variety of empirical data (Ethernet, WWW, VBR Video).
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1042:
based on accurate assumptions of the traffic that they carry. The dimensioning and provisioning of networks that carry long-tail traffic is discussed in the next section.
1861:
Park K., Future
Directions and Open Problems in Performance Evaluation and Control of Self-Similar Network Traffic, Department of Computer Sciences, University of Purdue.
1753:
516:
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1018:. This is currently the most popular explanation in the engineering literature and the one with the most empirical evidence through observed file size distributions.
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variation are of import to provisioning user-specified QoS. Self-similar burstiness is expected to exert a negative influence on second-order performance measures.
1840:
1154:
process, is perhaps the most successful model to date. It is demonstrated to satisfy the basic requirements of a simple, but accurate, model of long-tail traffic.
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criticisms on these sorts of models though as being unrealistic in that network traffic is long-tailed even in non-congested regions and at all levels of traffic.
401:
1266:, 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.
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Long-range dependence and its exploitation for traffic control is best suited for flows or connections whose lifetime or connection duration is long lasting.
1996:
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in the context of web client/ server interactions, the size of the file being transported (which is known at the server) is conveyed or made accessible to
1792:
Neame T., Characterisation and
Modelling of Internet Traffic Streams, Department of Electrical and Electronic Engineering, University of Melbourne, 2003.
632:
is a measure of the level of self-similarity of a time series that exhibits long-range dependence, to which the heavy-tail distribution can be applied.
1491:
Kennedy I., Lecture Notes, ELEN5007 – Teletraffic
Engineering, School of Electrical and Information Engineering, University of the Witwatersrand, 2005.
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Another approach adopted in controlling long-tail traffic makes traffic controls cognizant of workload properties. For example, when TCP is invoked in
784:
671:
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packet inter-arrivals, and conclusions reached under such assumptions may be misleading or incorrect in the presence of heavy-tailed distributions.
1954:
Tuan T., Park K., Multiple Time Scale
Congestion Control for Self-Similar Network Traffic, Department of Computer Sciences, University of Purdue.
1509:
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With long-tail traffic, the bursty behaviour may itself be bursty, which exacerbates the clustering phenomena, and degrades network performance.
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network environment will be long-range dependent. Additionally, this causal mechanism is robust with respect to changes in network resources (
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of networks, and many of the previous assumptions upon which systems have been built are no longer valid in the presence of self-similarity.
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Due to the increased demands that long-tail traffic places on networks resources, networks need to be carefully provisioned to ensure that
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an average income or an average number of clients per day) but can be inappropriate (e.g. in the context of buffering or waiting queues).
1771:
Arrowsmith D.K., Woolf M., Internet Packet Traffic Congestion in Networks, Mathematics Research Centre, Queen Mary, University of London.
302:
149:
In short-range dependent processes, the coupling between values at different times decreases rapidly as the time difference increases.
1935:
1640:
Park K.; Kim G.; Crovella M. (1996). "On the relationship between file sizes, transport protocols, and self-similar network traffic".
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No unanimity exists about which of the competing models is appropriate, but the Poisson Pareto Burst Process (PPBP), which is an M/G/
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Workload pattern complexities (for example, bursty arrival patterns) can significantly affect resource demands, throughput, and the
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892:
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Heavy-tail distributions have properties that are qualitatively different from commonly used (memoryless) distributions such as the
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takes on values from 0.5 to 1. A value of 0.5 indicates the data is uncorrelated or has only short-range correlations. The closer
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sensitive traffic streams comprising a growing fraction of network traffic, second-order performance measures in the form of “
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methods are used for producing packet traffic models which can replicate both short-range and long-range dependent streams.
1991:
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Everything you always wanted to know about self-similar network traffic and long-range dependency, but were ashamed to ask
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Many aspects of network quality of service depend on coping with traffic peaks that might cause network failures, such as
1068:, even long-range dependent Gaussian models, are unable to accurately model current Internet traffic. Classical models of
111:
94:
Heavy-tail distributions have been observed in many natural phenomena including both physical and sociological phenomena.
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traffic exhibits dependencies over a long range of time scales. This is to be contrasted with telephone traffic which is
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60:
The design of robust and reliable networks and network services has become an increasingly challenging task in today's
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Reference to additional information on the effect of long-range dependency on network performance can be found in the
127:
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23:
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Smith R. (2011). "The Dynamics of Internet Traffic: Self-Similarity, Self-Organization, and Complex Phenomena".
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Self-similar traffic exhibits the persistence of clustering which has a negative impact on network performance.
1087:
A number of models have been proposed for the task of modelling long-tail traffic. These include the following:
1080:, or at least weak dependence. Poisson and Markov related processes have, however, been used with some success.
30:
to regions far from the mean or median. A more formal mathematical definition is given below. In the context of
2001:
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259:. This variance to mean power law is an inherent feature of a family of statistical distributions called the
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Zukerman M., ARC Centre for Ultra Broadband Information Networks, EEE Dept., The University of Melbourne,
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Neame T., ARC Centre for Ultra Broadband Information Networks, EEE Dept., The University of Melbourne,
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diminishes. Techniques to control and manage long-tail traffic are discussed in the following section.
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Development of Procedures to Analyze Queuing Models with Heavy-Tailed Interarrival and Service Times
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encountered by user requests, in terms of higher average response times and higher response time
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Readers interested in a more rigorous mathematical treatment of the subject are referred to the
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Park K., Kim G., Crovella M., On the Effect of Traffic Self-similarity on Network Performance.
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factor at relatively short timescales and not the long-term cause of self-similar traffic.
243:, α is a parameter in the interval (0,1) and the ~ means asymptotically proportional to as
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Information on the fundamentals of statistics and probability theory can be found in the
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Packet-switching-based services, such as the Internet (and other networks that employ
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In long-range processes, the correlations at longer time scales are more significant.
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The number of call departures in a given time also has a Poisson distribution, i.e.:
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Heavy-tailed distributions, generalised source coding and optimal web layout design
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Assuming pure-chance arrivals and pure-chance terminations leads to the following:
42:
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networks), clustering occurs in the short term but smooths out over the long term.
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1069:
874:{\displaystyle p(x)=\alpha k^{\alpha }x^{-\alpha -1},\ \alpha ,k>0,\ x\geq k}
753:{\displaystyle P\sim x^{-\alpha },\ {\text{as}}\ x\to \infty ,0<\alpha <2}
83:
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file sizes is heavy-tailed then the superposition of many file transfers in a
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Willinger, W., Govindan, R., Jamin, S., Paxson, V. & Shenker, S. (2002).
1654:
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The number of call arrivals in a given time has a Poisson distribution, i.e.:
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1780:
Resnick S.I., Heavy Tail Modeling and Teletraffic Data, Cornell University.
1739:
1712:
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Proceedings of 1996 International Conference on Network Protocols (ICNP-96)
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School of Information Technology and Engineering, George Mason University,
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Internet Control and Inference Tools at the Edge: Network Traffic Modelling
485:{\displaystyle P(d)=\left({\frac {\lambda ^{d}}{d!}}\right)e^{-\lambda },}
142:
Long-range and short-range dependent processes are characterised by their
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is to 1, the greater the degree of persistence or long-range dependence.
407:. For this reason, pure-chance traffic is also known as Poisson traffic.
267:
explains how certain types of random data converge towards the form of a
72:
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Performance Evaluation of a Queue Fed by a Poisson Pareto Burst Process
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phenomena, e.g. Stock markets, earthquakes, and the weather. Ethernet,
99:
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Barford P., Floyd S., Computer Science Department, Boston University,
1688:"Scaling phenomena in the Internet: Critically examining Criticality"
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119:
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Department of Electrical and Computer Engineering, Rice University,
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Modelling of long-tail traffic is necessary so that networks can be
98:
established the use of heavy-tail distributions to model real-world
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The Self-similarity and Long Range Dependence in Networks Web site
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370:{\displaystyle P(a)=\left({\frac {\mu ^{a}}{a!}}\right)e^{-\mu },}
251:
Long-range dependence as a consequence of mathematical convergence
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1838:
Planning for growth: A proven methodology for capacity planning
768:
functions over a smaller domain, and with a logarithmic range.
183:
function is often assumed to have the specific functional form,
1892:"On the relevance of long-range dependence in network traffic"
969:{\displaystyle F(x)=P=1-\left({\frac {k}{x}}\right)^{\alpha }}
1419:
Zhu X., Yu J., Doyle J., California Institute of Technology,
34:
a number of quantities of interest have been shown to have a
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126:
video (digitised video of the type that is transmitted over
41:
Many processes are technically long-range dependent but not
1437:
Medina A., Computer Science Department, Boston University,
164:
function of short-range dependent processes decays quickly.
1970:, Department of Computer Sciences, University of Purdue.
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Air Force Research Laboratory, Information Directorate,
1933:
Introduction to ATM switching, RAD Data Communications
1197:") rather than smoothing it, compounding the problem.
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Poisson processes are well-behaved because they are
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which is self-similar but not long-range dependent.
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87:etc. This can have far-reaching consequences for
138:Short-range dependence vs. long-range dependence
1806:Traffic Modelling and Related Queueing Problems
1692:Proceedings of the National Academy of Sciences
666:A distribution is said to be heavy-tailed if:
518:is the mean number of call departures in time
8:
1968:Self-Similar Network Traffic and its Control
1752:: CS1 maint: multiple names: authors list (
1234:With Poisson traffic (found in conventional
403:is the mean number of call arrivals in time
1871:Jitter analysis of ATM self-similar traffic
1576:Heavy-tailed distributions and implications
1303:Web site provisioning for long-tail traffic
239:) is the autocorrelation function at a lag
1347:exploited to improve network performance.
1282:Network provisioning for long-tail traffic
1112:Markov Modulated Poisson Processes (MMPP)
779:of a heavy-tail distribution is given by:
176:function summed over all lags is infinite.
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1157:Finally, results from simulations using
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225:{\displaystyle \rho (k)\sim k^{-\alpha }}
213:
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1252:Violation of delay bounds e.g. In video
1204:In the modern network environment with
643:Typical values of the Hurst parameter,
79:in its arrival and departure process.
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1109:Poisson Pareto Burst Processes (PPBP)
1058:model for circuit switched networks.
498:is the number of call departures and
261:Tweedie exponential dispersion models
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1890:Grossglauser M.; Bolot J.C. (1999).
1541:Linington P.F., University of Kent,
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590:{\displaystyle P=e^{\frac {-t}{h}},}
280:The Poisson distribution and traffic
271:there exists a related theorem, the
26:is one that assigns relatively high
1899:IEEE/ACM Transactions on Networking
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1249:Cell/packet loss and queue overflow
1106:Infinite Markov Modulated Processes
1030:routing effects in the simulation.
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383:is the number of call arrivals and
130:networks) traffic is self-similar.
1997:Tails of probability distributions
1076:rely heavily on the assumption of
982:represents the smallest value the
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157:function over all lags is finite.
885:cumulative distribution function
603:is the Mean Holding Time (MHT).
1216:” such as delay variation and
1147:{\displaystyle {\mathcal {1}}}
997:What causes long-tail traffic?
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1338:Controlling long-tail traffic
651:Any pure random process has
1597:Advances in Complex Systems
1255:Worst cases in statistical
1072:such as Poisson and finite
1034:Modelling long-tail traffic
614:The heavy-tail distribution
273:Tweedie convergence theorem
2018:
1062:fertile area of research.
160:As the lag increases, the
1619:10.1142/S0219525911003451
1351:resource provisioning is
777:probability mass function
24:heavy-tailed distribution
1655:10.1109/ICNP.1996.564935
1393:Traffic generation model
1275:service level agreements
620:exponential distribution
511:{\displaystyle \lambda }
257:method of expanding bins
36:long-tailed distribution
1170:{\displaystyle \alpha }
48:Additionally, there is
32:teletraffic engineering
1713:10.1073/pnas.012583099
1332:external links section
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1398:Tweedie distributions
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1103:Iterated Chaotic Maps
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265:central limit theorem
247:approaches infinity.
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1992:Stochastic processes
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396:{\displaystyle \mu }
387:
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287:Poisson distribution
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68:in network traffic.
1704:2002PNAS...99.2573W
1356:performance gains.
1181:Network performance
765:Pareto distribution
269:normal distribution
172:The area under the
89:network performance
1938:2004-12-04 at the
1876:2005-02-16 at the
1843:2012-10-23 at the
1581:2005-12-15 at the
1561:2005-03-15 at the
1512:2011-05-26 at the
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1142:
1074:Markov processes
1016:network topology
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583:
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569:
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231:
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220:
2017:
2016:
2012:
2011:
2010:
2008:
2007:
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2002:Autocorrelation
1977:
1976:
1975:
1974:
1965:
1958:
1953:
1946:
1940:Wayback Machine
1930:
1926:
1894:
1889:
1888:
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1880:. utdallas.edu.
1878:Wayback Machine
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1384:
1369:transport layer
1340:
1305:
1284:
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1159:
1158:
1132:
1131:
1099:Brownian motion
1066:Gaussian models
1036:
999:
984:random variable
942:
941:
891:
890:
816:
806:
783:
782:
696:
670:
669:
658:Phenomena with
627:Hurst parameter
616:
570:
564:
535:
534:
526:The intervals,
500:
499:
466:
452:
442:
436:
416:
415:
385:
384:
351:
337:
327:
321:
301:
300:
282:
263:. Much as the
253:
209:
189:
188:
181:autocorrelation
174:autocorrelation
162:autocorrelation
155:autocorrelation
153:The sum of the
140:
66:self-similarity
58:
50:Brownian motion
12:
11:
5:
2015:
2013:
2005:
2004:
1999:
1994:
1989:
1979:
1978:
1973:
1972:
1956:
1944:
1924:
1905:(5): 629–640.
1882:
1863:
1849:
1822:
1810:
1794:
1782:
1773:
1759:
1698:(3): 2573–80.
1678:
1663:
1632:
1603:(6): 905–949.
1587:
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144:autocovariance
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57:
54:
13:
10:
9:
6:
4:
3:
2:
2014:
2003:
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1998:
1995:
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1399:
1396:
1394:
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1388:Elephant Flow
1386:
1385:
1381:
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1376:
1375:long queues.
1372:
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1123:
1120:
1118:Matrix models
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1111:
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1004:client/server
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887:is given by:
886:
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71:Self-similar
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29:
28:probabilities
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1748:cite journal
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1257:multiplexing
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1987:Teletraffic
1218:packet loss
1097:Fractional
1091:Fractional
1070:time series
1040:provisioned
146:functions.
84:time-series
20:long-tailed
1981:Categories
1931:Biran G.,
1404:References
1292:throughput
1208:and other
1206:multimedia
1195:burstiness
1187:congestion
986:can take.
96:Mandelbrot
82:With many
1966:Park K.,
1610:0807.3374
1365:protocols
1353:open-loop
1343:infancy.
1264:stateless
1236:telephony
1165:α
1082:Nonlinear
1008:bandwidth
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866:≥
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730:∞
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550:≥
506:λ
475:λ
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207:∼
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1718:JSTOR
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1623:S2CID
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