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Mathematical Problems in Engineering
VolumeΒ 2011Β (2011), Article IDΒ 389803, 21 pages
http://dx.doi.org/10.1155/2011/389803
Research Article

mBm-Based Scalings of Traffic Propagated in Internet

1School of Information Science & Technology, East China Normal University, No. 500, Dong-Chuan Road, Shanghai 200241, China
2University of Macau, Av. Padre TomΓ‘s Pereira, Taipa, Macau, China
3College of Computer Science, Zhejiang University of Technology, Hangzhou 310023, China

Received 20 October 2010; Accepted 29 November 2010

Academic Editor: CristianΒ Toma

Copyright Β© 2011 Ming Li et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Scaling phenomena of the Internet traffic gain people's interests, ranging from computer scientists to statisticians. There are two types of scales. One is small-time scaling and the other large-time one. Tools to separately describe them are desired in computer communications, such as performance analysis of network systems. Conventional tools, such as the standard fractional Brownian motion (fBm), or its increment process, or the standard multifractional fBm (mBm) indexed by the local HΓΆlder function 𝐻(𝑑) may not be enough for this purpose. In this paper, we propose to describe the local scaling of traffic by using 𝐷(𝑑) on a point-by-point basis and to measure the large-time scaling of traffic by using E[𝐻(𝑑)] on an interval-by-interval basis, where E implies the expectation operator. Since E[𝐻(𝑑)] is a constant within an observation interval while 𝐷(𝑑) is random in general, they are uncorrelated with each other. Thus, our proposed method can be used to separately characterize the small-time scaling phenomenon and the large one of traffic, providing a new tool to investigate the scaling phenomena of traffic.

1. Introduction

Consider an application that sends a series of packets from the source to the destination through the Internet. Suppose a traffic series passes through 𝐼 servers from the first server with the service curve 𝑆1(𝑑) to the 𝐼th server with the service curve 𝑆𝐼(𝑑) to reach the destination. Then, the communication from the first server to the 𝐼th one can be expressed by Figure 1 (Li and Zhao [1], Li [2]), where 𝐴1𝑗(𝑑) is the arrival traffic accumulated within the time interval [0,𝑑] and 𝐷𝐼𝑗(𝑑) is the departure traffic within [0,𝑑].

389803.fig.001
Figure 1: Packets passing through a series of servers from source to destination.

Let π‘Žπ‘–π‘—(𝑑) be instantaneous arrival traffic, implying the bytes of a packet at time 𝑑 from connection 𝑗 at the input port of the server 𝑖 with the service curve 𝑆𝑖(𝑑). Then, the accumulated function regarding π‘Žπ‘–π‘—(𝑑) in the time interval [0,𝑑] is given by𝐴𝑖𝑗(ξ€œπ‘‘)=𝑑0π‘Žπ‘–π‘—(𝑑)𝑑𝑑.(1.1) We now consider the aggregated traffic π‘₯(𝑑). By aggregated traffic, we mean the following: π‘₯(𝑑)=𝑁𝑗=1π‘Žπ‘–π‘—(𝑑),(1.2) where 𝑁 is the positive number representing all connections at the input port of the server 𝑖. In this research, traffic time series π‘₯(𝑑) is in the sense of (1.2). The accumulated traffic within the interval [𝑑0,𝑑] is given byξ€œπ΄(𝑑)=𝑑𝑑0π‘₯(𝑑)𝑑𝑑.(1.3)

In the field of traffic modeling, there are two categories of traffic models. One is deterministic modeling, more precisely, bounded modeling, and the other is stochastic modeling, see Li and Borgnat [3], Michiel and Laevens [4]. Scaling plays a role in all types of traffic models, see, for example, Willinger et al. [5], Feldmann et al. [6], Jiang [7], and Papagiannaki et al. [8]. There are two types of scaling phenomena in traffic. One is the small-time scaling and the other is large-time one, see, for example, Paxson and Floyd [9]. This paper aims at investigating two types of scaling phenomena of traffic for either the bounded modeling, say 𝐴(𝑑), and the stochastic modeling of π‘₯(𝑑).

Note that a commonly used model of π‘₯(𝑑) in the wide sense stationarity is the self-similar process, that is, fractional Gaussian noise (fGn), see, for example, Stalling [10], McDysan [11], Pitts and Schormans [12], Leland et al. [13], Beran et al. [14], Tsybakov and Georganas [15], Willinger and Paxson [16], and Adas [17]. However, there is a limitation in fGn for the analysis of two scaling phenomena, namely, small scaling and large one, since it is indexed by a single parameter called the Hurst parameter 𝐻, see Paxson and Floyd [9], Tsybakov and Georganas [15], Ayache et al. [18], Li and Lim [19, 20], and Li [21–24]. Therefore, two-parameter models of traffic are needed.

In this paper, we address two types of traffic models. One is the multifractional Brownian motion (mBm), see Li et al. [25]. The other is the 2-parameter bounded model introduced by Cruz, see [26, 27], Li and Zhao [28], Raha et al. [29], Jiang and Liu [30], and Boudec and Thiran [31]. The contributions of this paper are in two aspects. (i)We claim the small-time scaling phenomenon is independent of the large-time one and vice versa based on the model of Cruz.(ii)We propose the point of view to use mBm to analyze the scaling phenomena of traffic in this way. Describing the small-time scaling phenomenon by using 𝐷(𝑑) on a point-by-point basis and to characterize the large-time scaling one by using E[𝐻(𝑑)] on an interval-by-interval basis.

The rest of the paper is organized as follows. We will give the preliminaries regarding conventional time series in Section 2, aiming at pointing out why scaling is a topic in traffic of the fractal type. We will describe the reason why the small-scaling phenomenon of traffic is independent of its large-time one in Section 3. In Section 4, we will introduce a two-parametric model of mBm towards the scaling analysis of traffic based on the local HΓΆlder function 𝐻(𝑑). Finally, Section 5 concludes the paper.

2. Preliminaries

Traffic time series on old telephony networks is in the class of the Poisson processes, such as the Poisson one and its compound ones, see Erlang [32] and Brockmeyer et al. [33]. It has been successfully used in the design of infrastructure of old telephony networks for years, see, for example, Bojkovic et al. [34], Le Gall [35], Lin et al. [36], Manfield and Downs [37], and Reiser [38]. It is such a success on old telephony networks that it has almost been taken as an axiom for modelling traffic in communication systems, see Gibson [39], Cooper [40], and Akimaru and Kawashima [41]. Due to unsatisfactory performances of the Internet, such as traffic congestions, people began doubting about the models of the Poisson type. Accordingly, they began measuring and analyzing the traffic at different sites in the Internet during different periods of times for the purpose of reevaluating general patterns of traffic, see [9, 13, 14], Paxson [42, 43], and Traffic Archive at http://www.sigcomm.org/ITA/. Experimental processing real-traffic traces exhibited that traffic is in the class of fractal time series.

The early fractal model used for traffic modelling is the self-similar process with long-range dependence (LRD), that is, fGn with LRD. For this reason, we will address the preliminaries in this section in the aspects of conventional time series, stationary self-similar process, that is, fGn, and LRD processes.

2.1. Conventional Time Series

Let {π‘₯𝑙(𝑑)}(𝑙=1,2,…) be a 2-order stationary random process, where π‘₯𝑙(𝑑)βˆˆβ„ is the 𝑙th sample function of the process, where ℝ is the set of real numbers. We use π‘₯𝑙(𝑑) to represent the process without confusion causing. Its mean in the wide sense can be expressed byπœ‡π‘ π‘₯(𝑑)=limπ‘β†’βˆž1𝑁𝑁𝑙=1π‘₯𝑙(𝑑)=const.(2.1) Its autocorrelation function (ACF) can be written by𝑅𝑠π‘₯(𝑑,𝑑+𝜏)=limπ‘β†’βˆž1𝑁𝑁𝑙=1π‘₯𝑙(𝑑)π‘₯𝑙(𝑑+𝜏)=𝑅𝑠π‘₯(𝜏).(2.2) In (2.1) and (2.2), the superscript 𝑠 implies that the mean and the ACF are computed by using spatial average. The mean and the ACF of a process expressed by time average are written by πœ‡π‘‘π‘₯(𝑑)=limπ‘‡β†’βˆž1π‘‡ξ€œπ‘‡0π‘₯𝑙𝑅(𝑑)𝑑𝑑=const,(2.3)𝑑π‘₯(𝜏)=limπ‘‡β†’βˆž1π‘‡ξ€œπ‘‡0π‘₯𝑙(𝑑)π‘₯𝑙(𝑑+𝜏)π‘‘πœ,(2.4) where the superscript 𝑑 indicates that the mean and the ACF are computed by time average.

The process π‘₯𝑙(𝑑) is said to be ergodic if (2.5) holds,πœ‡π‘ π‘₯(𝑑)=πœ‡π‘‘π‘₯(𝑑)=πœ‡π‘₯𝑅=const,𝑠π‘₯(𝜏)=𝑅𝑑π‘₯(𝜏)=𝑅(𝜏).(2.5) In what follows, we simply use π‘₯(𝑑) to represent a random function in general.

Denote by 𝑝(π‘₯) the probability density function (PDF) of π‘₯(𝑑). Then, the probability is given by𝑃π‘₯2ξ€Έξ€·π‘₯βˆ’π‘ƒ1ξ€Έξ€Ίπ‘₯=Prob1<πœ‰<π‘₯2ξ€»=ξ€œπ‘₯2π‘₯1𝑝(πœ‰)π‘‘πœ‰.(2.6) The mean and the ACF of π‘₯(𝑑) based on PDF are given by (2.7) and (2.8), respectively, πœ‡π‘₯=ξ€œβˆžβˆ’βˆžπ‘…π‘₯𝑝(π‘₯)𝑑π‘₯,(2.7)π‘₯ξ€œ(𝜏)=βˆžβˆ’βˆžπ‘₯(𝑑)π‘₯(𝑑+𝜏)𝑝(π‘₯)𝑑π‘₯.(2.8) Let 𝑉π‘₯ be the variance of π‘₯. Then,𝑉π‘₯ξ€Ί=Eπ‘₯(𝑑)βˆ’πœ‡π‘₯ξ€»2=ξ€œβˆžβˆ’βˆžξ€·π‘₯βˆ’πœ‡π‘₯ξ€Έ2𝑝(π‘₯)𝑑π‘₯.(2.9)

If π‘₯(𝑑)βˆˆβ„, then it has the following properties.

Note 1. The PDF 𝑝(π‘₯) is light tailed. By light tailed, we mean that the integrals in (2.7) and (2.8) are convergent in the domain of ordinary functions.

Note 2. There exist πœ‡π‘₯ and 𝑉π‘₯ for π‘₯(𝑑) if the PDF of π‘₯(𝑑) is light tailed.

The Poisson distribution is an instance of light-tailed distribution, which expresses the probability of a number of events occurring in a fixed period of time if these events occur with a known average rate and independently of the time since the last event. In communication networks, one is interested in the work focused on certain random variables 𝑁 that count, among other things, a number of discrete occurrences (sometimes called β€œarrivals”) that take place during a time interval of a given length. Denote the expected number of occurrences in this interval by a positive real number πœ†. Then, the probability that there are exactly 𝑛 occurrences (𝑛=0,1,2,…) is given by the Poisson distribution below πœ†π‘(π‘₯;πœ†)=π‘˜π‘’βˆ’πœ†π‘›!.(2.10)

Note 3. The ACF of π‘₯(𝑑) with a light-tailed PDF decays fast. By β€œdecays fast,” we mean that 𝑅(𝜏) is integrable in the continuous case and summable in the discrete case in the domain of ordinary functions.

Denote by 𝑆π‘₯(πœ”) the power spectrum density (PSD) of π‘₯(𝑑). Then, 𝑆π‘₯ξ€œ(πœ”)=βˆžβˆ’βˆžπ‘…π‘₯(𝜏)π‘’βˆ’π‘—πœ”πœπ‘‘πœ.(2.11)

Thus, we have Note 4 below, which is a consequence of Note 3.

Note 4. 𝑆π‘₯ exists in the domain of ordinary functions.

The results in Notes 1–4 are usually assumptions for conventional time series as can be seen from Fuller [44], Box et al. [45], Mitra and Kaiser [46], and Bendat and Piersol [47]. We will explain below that all in Notes 1–4 may be no longer valid for LRD traffic.

2.2. Scaling Measures for Conventional Gaussian Time Series

A Gaussian process is completely determined by its second-order properties, more precisely, its mean and ACF, see Papoulis [48] and Doob [49]. Note that the mean of π‘₯(𝑑) is a measure of the global property of π‘₯(𝑑). On the other side, the variance of π‘₯(𝑑) measures its local property. These two points can be easily inferred from (2.3) and (2.9). For a Gaussian process with mean zero, one has 𝑉π‘₯=𝑅(0).(2.12) Therefore, mean and variance or ACF are two essential numeric characteristics of a Gaussian process. In fact, if π‘₯(𝑑) is Gaussian, then 1𝑝(π‘₯)=√2πœ‹π‘‰π‘₯π‘’βˆ’(π‘₯βˆ’πœ‡π‘₯)2/2𝑉π‘₯.(2.13) However, 𝑉π‘₯ or mean of traffic time series π‘₯(𝑑) may not exist in general due to LRD, see Li [22], which is a particular point of a time series with LRD (Beran [50, 51]). A simple explanation about this is 𝑉π‘₯β†’βˆž in (2.13). In the case of 𝑉π‘₯β†’βˆž, πœ‡π‘₯ in (2.13) is indeterminate. Therefore, variance and mean are no longer suitable for measuring the local property and the global one of LRD traffic.

2.3. Correlation Time of Conventional Time Series

Correlation time is defined by (Nigam [52, page 74]) 𝑑𝑐=1limπ‘‡β†’βˆžβˆ«π‘‡0π‘Ÿ(𝜏)π‘‘πœlimπ‘‡β†’βˆžξ€œπ‘‡0πœπ‘Ÿ(𝜏)𝑑𝑑.(2.14) It is a measure relating to the scaling of a random function π‘₯(𝑑). It implies that the correlation can be neglected if 𝑑𝑐≀𝑑, where 𝑑 is the time scale of interest [52]. As traffic is LRD, both the numerator and denominator on the right side of (2.14) do not exist. Therefore, correlation time that is a useful measure in conventional time series is inappropriate to be used in LRD traffic.

2.4. Brief of LRD Time Series

One says that 𝑓(𝑑) is asymptotically equivalent to 𝑔(𝑑) under the limit π‘₯→𝑐 if 𝑓(𝑑) and 𝑔(𝑑) are such that limπ‘₯→𝑐(𝑓(𝑑)/𝑔(𝑑))=1 (Murray [53]), that is, 𝑓(𝑑)βˆΌπ‘”(𝑑)(π‘‘βŸΆπ‘)iflimπ‘₯→𝑐𝑓(𝑑)𝑔(𝑑)=1,(2.15) where 𝑐 can be infinity. It has the property expressed by 𝑓(𝑑)βˆΌπ‘”(𝑑)βˆΌβ„Ž(𝑑)(π‘‘βŸΆπ‘).(2.16) In this sense, 𝑓(𝑑) is called slowly varying function if limπ‘’β†’βˆž(𝑓(𝑒𝑑)/𝑓(𝑒))=1 for all 𝑑.

A random function π‘₯(𝑑) is said to be LRD if its ACF π‘Ÿ(𝜏) is nonintegrable, while it is called short-range dependent (SRD) if π‘Ÿ(𝜏) is integrable. This implies that π‘₯(𝑑) is LRD if π‘Ÿ(𝜏)βˆΌπ‘πœβˆ’π‘(𝜏⟢∞),π‘βˆˆ(0,1),(2.17) where 𝑐>0 can be either a constant or a slowly varying function. It is SRD if π‘Ÿ(𝜏)βˆΌπ‘πœβˆ’π‘(𝜏⟢∞),𝑏>1.(2.18)

Theoretically, any series whose ACF is nonintegrable are LRD. In the field of telecommunications, however, the term of LRD traffic usually corresponds to a hyperbolically decayed ACF. Its asymptotic expression for πœβ†’βˆž is often indexed by the Hurst parameter 𝐻. That is,π‘Ÿ(𝜏)βˆΌπ‘πœ2π»βˆ’2(𝜏⟢∞),𝐻∈(0.5,1).(2.19)

Note 5. The tail of the PDF of LRD traffic is heavy according to Taqqu’s theorem, see Abry et al. [54].

According to the Fourier transform in the domain of generalized functions (Kanwal [55], Gelfand and Vilenkin [56]), one immediately obtains the Fourier transform of the right side of (2.17) given byFξ€·|𝜏|βˆ’π‘ξ€Έξ‚€=2sinπœ‹π‘2Γ(1βˆ’π‘)|πœ”|π‘βˆ’1,(2.20) where F stands for the operator of the Fourier transform. Therefore, for LRD traffic, we haveF[]π‘Ÿ(𝜏)∼|πœ”|π‘βˆ’1forπœ”βŸΆ0.(2.21)

Note 6. LRD traffic is in the class of 1/𝑓 noise (Li [57]).

In summary, from the point of view of the assumption of Gaussian distribution, we say that the tail of the PDF of LRD traffic may be so heavy that its πœ‡π‘₯ and 𝑉π‘₯ do not exist. Owing to this meaning of the heavy tails, the ACF of traffic decays so slow in a hyperbolical manner such that it is nonintegrable. Consequently, a random variable that represents a traffic time series can be no longer considered to be independent, hence, LRD or long memory. On the other hand, the PSD of LRD traffic obeys a power law, see (2.21), hence, 1/𝑓 noise.

2.5. Brief of Self-Similar Time Series

A random function π‘₯(𝑑) is said to be self-similar if it satisfies the definition of self-similarity given byπ‘₯(π‘Žπ‘‘)β‰‘π‘Žπ»π‘₯(𝑑),π‘Ž>0,(2.22) where ≑ denotes equality in the sense of probability distribution.

Note 7. The concept of LRD differs from that of self-similarity (Li [23]).

Note 8. The self-similarity described by (2.22) is in the global sense.

The commonly used self-similar model of traffic is fGn in the stationary case and fBm in the nonstationary case. We will brief them in the next subsection.

2.6. fGn and fBm for Traffic with LRD

fGn is an only stationary increment process with self-similarity (Samorodnitsky and Taqqu [58]). We discuss it in this subsection towards exhibiting the limitation of fGn in describing two types of scaling phenomena of traffic.

Let 𝐡(𝑑) be Brownian motion (Bm). Let 𝐡𝐻(𝑑) be the fBm of the Weyl integral type with the Hurst parameter 𝐻∈(0,1). Let Ξ“(β‹…) be the Gamma function. Then,𝐡𝐻(𝑑)βˆ’π΅π»1(0)=ξ‚»ξ€œΞ“(𝐻+1/2)0βˆ’βˆžξ€Ί(π‘‘βˆ’π‘’)π»βˆ’0.5βˆ’(βˆ’π‘’)π»βˆ’0.5ξ€»ξ€œπ‘‘π΅(𝑒)+𝑑0(π‘‘βˆ’π‘’)π»βˆ’0.5ξ‚Ό.𝑑𝐡(𝑒)(2.23)

The function 𝐡𝐻(𝑑) has the following properties.(i)𝐡𝐻(0)=0.(ii)The increments 𝐡𝐻(𝑑+𝑑0)βˆ’π΅π»(𝑑0) are Gaussian.(iii)Var[𝐡𝐻(𝑑+𝑑0)βˆ’π΅π»(𝑑0)]=𝑉𝐻𝑑2𝐻, where 𝑉𝐻=E{[𝐡𝐻(1)]2}.(iv)E{[𝐡𝐻(𝑑2)βˆ’π΅π»(𝑑1)]2}=E{[𝐡𝐻(𝑑2βˆ’π‘‘1)βˆ’π΅π»(0)]2}=E{[𝐡𝐻(𝑑2βˆ’π‘‘1)]2}=𝑉𝐻(𝑑2βˆ’π‘‘1)2𝐻. (v)E{[𝐡𝐻(𝑑2)βˆ’π΅π»(𝑑1)]2}=𝑉𝐻(𝑑2)2𝐻+𝑉𝐻(𝑑1)2π»βˆ’2π‘Ÿ[𝐡𝐻(𝑑2),𝐡𝐻(𝑑1)].

Thus, the ACF of 𝐡𝐻(𝑑), denoted by π‘Ÿπ΅π»,π‘Š(𝑑,𝑠), is given byπ‘Ÿπ΅π»,π‘Šπ‘‰(𝑑,𝑠)=𝐻(𝐻+1/2)Ξ“(𝐻+1/2)|𝑑|2𝐻+|𝑠|2π»βˆ’|π‘‘βˆ’π‘ |2𝐻,(2.24) where𝑉𝐻𝐡=Var𝐻(1)=Ξ“(1βˆ’2𝐻)cosπœ‹π»πœ‹π».(2.25)

Denote by 𝑆𝐡𝐻,π‘Š(𝑑,πœ”) the PSD of 𝐡𝐻(𝑑). Then (Flandrin [59])𝑆𝐡𝐻,π‘Š1(𝑑,πœ”)=|πœ”|2𝐻+1ξ€·1βˆ’21βˆ’2𝐻cos2πœ”π‘‘.(2.26) From the above, we see that either the ACF or the PDF of 𝐡𝐻(𝑑) is time varying. Therefore, 𝐡𝐻(𝑑) is nonstationary.

Note that 𝐡𝐻(𝑑) is self-similar because it satisfies the definition of self-similarity. In fact,𝐡𝐻(π‘Žπ‘‘)β‰‘π‘Žπ»π΅π»(𝑑),βˆ€π‘Ž>0,(2.27) where ≑ denotes equality in the sense of probability distribution.

From (2.26), one sees that the PSD of 𝐡𝐻(𝑑) is divergent at πœ”=0, exhibiting a case of 1/𝑓 noise, see Csabai [60] for the early work of 1/𝑓 noise in traffic theory. The relationship between the fractal dimension of fBm, denoted by 𝐷fBm, and its Hurst parameter, denoted by 𝐻fBm, is given by𝐷fBm=2βˆ’π»fBm.(2.28)

Note that the increment series, 𝐡𝐻(𝑑+𝑠)βˆ’π΅π»(𝑑), is fGn. Thus, the ACF of the discrete fGn (dfGn) is given byπ‘‰π‘Ÿ(π‘˜)=𝐻2ξ€Ί(π‘˜+1)2π»βˆ’2π‘˜2𝐻+(π‘˜βˆ’1)2𝐻.(2.29) Since the ACF is an even function, we haveπ‘‰π‘Ÿ(π‘˜)=𝐻2||π‘˜||ξ€Έ+12𝐻+||||π‘˜||||βˆ’12𝐻||π‘˜||βˆ’22𝐻,(2.30) where π‘˜βˆˆβ„€. Denote by 𝐢𝐻(𝜏;πœ€) the ACF of fGn in the continuous case. Then,𝐢𝐻𝑉(𝜏;πœ€)=π»πœ€2π»βˆ’22|𝜏|πœ€ξ‚Ά+12𝐻+||||𝜏|πœ€|||βˆ’12𝐻|||πœβˆ’2πœ€|||2𝐻,(2.31) where πœ€>0 is used by smoothing fBm so that the smoothed fBm is differentiable.

The PSD of dfGn was derived out quite early by SinaΔ­ [61]. It is given by𝑆dfGn(πœ”)=2𝐢𝑓(1βˆ’cosπœ”)βˆžξ“π‘›=βˆ’βˆž||||2πœ‹π‘›+πœ”βˆ’2π»βˆ’1,(2.32) where 𝐢𝑓=𝑉𝐻(2πœ‹)βˆ’1sin(πœ‹π»)Ξ“(2𝐻+1) and πœ”βˆˆ[βˆ’πœ‹,πœ‹]. The PSD of fGn is (see Li and Lim [62])𝑆fGn(πœ”)=𝑉𝐻sin(π»πœ‹)Ξ“(2𝐻+1)|πœ”|1βˆ’2𝐻,(2.33) which exhibits that fGn belongs to the class of 1/𝑓 noises.

Note that 0.5[(𝜏+1)2π»βˆ’2𝜏2𝐻+(πœβˆ’1)2𝐻] can be approximated by 𝐻(2π»βˆ’1)(𝜏)2π»βˆ’2, in fact, that is, the finite second-order difference of 0.5(𝜏)2𝐻. Approximating it with the second-order differential of 0.5(𝜏)2𝐻 yieldsξ€Ί0.5(𝜏+1)2π»βˆ’2𝜏2𝐻+(πœβˆ’1)2π»ξ€»β‰ˆπ»(2π»βˆ’1)(𝜏)2π»βˆ’2.(2.34)

From the above, one immediately sees that fGn contains three subclasses of time series. In the case of 𝐻∈(0.5,1), the ACF is nonsummable and the corresponding series is of LRD. For 𝐻∈(0,0.5), the ACF is summable and fGn in this case is of SRD. FGn reduces to white noise when 𝐻=0.5.

Among LRD processes, fGn has its advantage in traffic modeling. For example, it can be used to easily represent two types of traffic series, namely, self-similar process and processes with LRD. Note that LRD is a global property of traffic. However, in principle, self-similarity is a local property of traffic, which is measured by fractal dimension 𝐷.

Denote 𝐷fGn and 𝐻fGn the fractal dimension and the Hurst parameter of fGn, respectively. Then, one has (Li [23])π‘ŸfGn(0)βˆ’π‘ŸfGn(𝜏)βˆΌπ‘|𝜏|2𝐻fGnfor|𝜏|⟢0.(2.35) Therefore, one gets (Li et al. [63])𝐷fGn=2βˆ’π»fGn.(2.36) Hence, for fGn type traffic, the local properties of traffic happen to be reflected in the global ones as noticed in mathematics by Mandelbrot [64].

The above discussions exhibit that the standard fGn as well as fBm has its limitation in traffic modeling because it uses a single parameter 𝐻 to characterize two different phenomena, that is, small-time scaling and large-time one. The former is a local property and the latter is a global one.

3. Large-Time Scaling of Traffic Is Independent of Its Small-Time One

Traffic π‘₯(𝑑) is greater than zero, that is,π‘₯(𝑑)β‰₯0,π‘‘βˆˆ(0,∞).(3.1) The above holds because π‘₯(𝑑) is arrival traffic. In addition,π‘₯min≀π‘₯(𝑑)≀π‘₯max,(3.2) where π‘₯min(𝑑) and π‘₯max(𝑑) are constants restricted by the IEEE standard without technical reasons except the need to limit delays. For instance, the Ethernet protocol forces all packets of π‘₯(𝑑) to have π‘₯min=64 bytes and π‘₯max=1518 bytes without considering the Ethernet preamble and header (Stalling [10]).

Due to the functionality of TCP, traffic appears β€œburstiness” (see Tobagi et al. [65]) or intermittency and non-Poisson (Jain and Routhier [66], Jiang and Dovrolis [67], and Papagiannaki [8]). The burstiness has considerable effects on system performances, see, for example, Nain [68], Draief and Mairesse [69], NΓ©meth et al. [70], Li and Zhao [71], Jiang et al. [72], Wang et al. [73], and Starobinski and Sidi [74].

The following measure introduced by Cruz [26, 27] characterizes the bound of the burstiness of traffic0≀lim𝑑→𝑑0ξ€œπ‘‘π‘‘0π‘₯(𝑑)π‘‘π‘‘β‰€πœŽ.(3.3) The integral expressed in (3.3) does not make sense if lim𝑑→𝑑0βˆ«π‘‘π‘‘0π‘₯(𝑑)𝑑𝑑≠0 for the continuous π‘₯(𝑑) even in the field of Lebesgue’s integrals, see Bartle and Sherbert [75] and Trench [76]. However, it makes sense when it is considered in the domain of generalized functions. A simple way to explain (3.3) is lim𝑑→𝑑0ξ€œπ‘‘π‘‘0ξ€œπ‘₯(𝑑)𝑑𝑑=𝑑𝑑0𝜎1π›Ώξ€·π‘‘βˆ’π‘‘0𝑑𝑑,(3.4) where 𝜎1β‰€πœŽ and 𝛿(𝑑) is the Dirac-𝛿 function. Equation (3.3) represents the burstiness bound of π‘₯(𝑑), which is a local behavior of traffic.

Note that 𝜎 is 𝑑0 dependent. Therefore, we may rewrite (3.3) by the following expression:0≀lim𝑑→𝑑0ξ€œπ‘‘π‘‘0𝑑π‘₯(𝑑)π‘‘π‘‘β‰€πœŽ0ξ€Έ.(3.5) The above exhibits that traffic has highly local irregularity or high burstiness as observed by Feldmann et al. [6], Papagiannaki et al. [8], Paxson and Floyd [9], Jiang and Dovrolis [67], Willinger et al. [77], and Estan and Varghese [78]. Such a local irregularity considerably affects the polices or performances of telecommunication systems, such as queuing (see, e.g., Nain [68] and Draief and Mairesse [69]), end-to-end delay, see, for example, NΓ©meth et al. [70], Li and Zhao [71], Jiang et al. [72], Wang et al. [73], and Starobinski and Sidi [74], resource allocation (see, e.g., Gravey et al. [79]), anomaly detection (Tian and Li [80]), and admission control (Knightly and Shroff [81], Raha et al. [82], and Jia et al. [83]), just naming a few.

Another measure introduced by Cruz [26, 27] describes the bound of the average rate of traffic. It is given by0≀limπ‘‘β†’βˆžβˆ«π‘‘π‘‘0π‘₯(𝑑)π‘‘π‘‘π‘‘βˆ’π‘‘0β‰€πœŒ=constant.(3.6) Note that the bound of the average rate expressed above describes a global property of traffic. It implies that the bound of the average rate of traffic is robust as 𝜌 is a constant. This is in agreement with the experimental observations stated by Feldmann et al. [6], Willinger et al. [22], and Paxson and Floyd [9].

The above exhibits, taking into account (3.5) and (3.6) together, that the accumulated traffic within [𝑑0,𝑑] is bounded byξ€œπ‘‘π‘‘0𝑑π‘₯(𝑒)π‘‘π‘’β‰€πœŽ0ξ€Έξ€·+πœŒπ‘‘βˆ’π‘‘0ξ€Έ.(3.7) Equation (3.7) implies that traffic has scaling phenomena in two folds. One is small-time scaling and the other large one.

Note 1. Parameter 𝜎 is independent of 𝜌.

Note 2. From Note 1, we see that the small-time scaling of traffic is independent of the large-time one.

We now further explain the point in Note 2 from the point of view of fractal time series. Denote the autocorrelation function (ACF) of traffic byπ‘Ÿπ‘₯[](𝜏)=Eπ‘₯(𝑑)π‘₯(𝑑+𝜏),(3.8) where 𝜏 is the lag. Then, π‘Ÿπ‘₯(𝜏) for small lags, more precisely, for πœβ†’0, if π‘Ÿπ‘₯(𝜏) is sufficiently smooth on (0,∞), is given byπ‘Ÿπ‘₯(0)βˆ’π‘Ÿπ‘₯(𝜏)βˆΌπ‘|𝜏|𝛼,(3.9) where 𝑐 is a constant and 𝛼 is the fractal index of π‘₯(𝑑). The fractal dimension of π‘₯(𝑑), denoted by 𝐷, is given by𝛼𝐷=2βˆ’2,(3.10) see Adler [84], Hall and Roy [85], Chan et al. [86], Kent and Wood [87], Gneiting and Schlather [88], Lim and Li [89], and Li et al. [90]. The parameter 𝐷 is used to describe the local irregularity of traffic. It is in terms of small-time scaling of traffic, see Li [21–24] and Li and Lim [19, 20]. From (2.19), we have𝑏𝐻=1βˆ’2.(3.11) The parameter 𝐻 is utilized to characterize the global property, more precisely, LRD, of traffic from a view of fractals.

Note 3. Generally, 𝐷 is independent of 𝐻.

Note 4. Owing to Note 3, we infer that the small-time scaling of traffic is independent of the large-time one in general.

The above discussions exhibit that it may be more flexible to characterize two types of scaling phenomena of traffic by using two independent parameters. One is for large-time scaling and the other for small-time scaling.

4. Applying mBm to the Scaling Analysis of Traffic

From the previous discussions, we suggest that it is natural to use two independent measures to describe two types of scaling phenomena that are independent of each other. Conventionally, fBm as well as its increment process, that is, fGn, is indexed by a single parameter 𝐻, alternatively by 𝐷=2βˆ’π». Thus, there is a limitation for them to independently characterize the scaling phenomena of two. This limitation was empirically noticed by Paxson and Floyd [9]. Lately, it was noticed by Ayache et al. [18] from the point of view of the multifractional Brownian motion (mBm).

In this research, we are interested in the work in mBm by Peltier and Levy-Vehel [91, 92] as well as Benassi et al. [93] to generalize the standard fBm by replacing the constant 𝐻 with the HΓΆlder function 𝐻(𝑑). Li et al. [94] applied 𝐻(𝑑) to describe the multifractality of traffic. Although [91–93, 95] explained the local self-similarity characterized by using 𝐻(𝑑) and Ayache et al. [18] discussed their method to measure the LRD of a random function, those works may not be enough for traffic because the small-time scaling is independent of the large-time one as we explained previously. As a matter of fact, it is quite awkward to use 𝐻(𝑑) to describe two scaling phenomena of traffic because 𝐻(𝑑) is linearly correlated with the fractal dimension 𝐷(𝑑) with the expression 𝐷(𝑑)=2βˆ’π»(𝑑) [91, 92]. To overcome the difficulty to capture the large scaling phenomena of traffic in the global sense, we introduce the measure expressed by E[𝐻(𝑑)]. Based on this, we propose our opinion like this; using 𝐷(𝑑) to represent the small scaling of traffic on a point-by-point basis and E[𝐻(𝑑)] to characterize the large scaling of traffic in the global sense, respectively. The key point of our opinion is that 𝐷(𝑑) and E[𝐻(𝑑)] are independent of each other.

In the rest of this section, we will brief the mBm in Section 4.1. Then, in Section 4.2, we will demonstrate the applications of 𝐷(𝑑) and E[𝐻(𝑑)] to real-traffic traces.

4.1. mBm of 𝐻(𝑑) Type

Note that the above (2.27) implies that the local irregularity of a random function 𝑋(𝑑) is globally the same. That, nevertheless, may not meet the real case of traffic. As a matter of fact, if 𝐷 of a traffic function π‘₯(𝑑) is a constant, 𝜎 of π‘₯(𝑑) in (3.3) is a constant too. This is a unifractal case, which is obviously in contradiction with real traffic as 𝜎 is time dependent, see (3.5).

One simple way to investigate the multifractality of traffic is to use mBm. Replacing the constant 𝐻 with a time-dependent function 𝐻(𝑑), where 𝑑>0 and 𝐻∢[0,∞]β†’(π‘Ž,𝑏)βŠ‚(0,1) is also called the local HΓΆlder exponent, see Peltier and Levy-Vehel [91, 92] and Benassi et al. [93], yields1𝑋(𝑑)=ξ€œΞ“(𝐻(𝑑)+1/2)0βˆ’βˆžξ€Ί(π‘‘βˆ’π‘ )𝐻(𝑑)βˆ’1/2βˆ’(βˆ’π‘ )𝐻(𝑑)βˆ’1/2ξ€»ξ€œπ‘‘π΅(𝑠)+𝑑0(π‘‘βˆ’π‘ )𝐻(𝑑)βˆ’1/2𝑑𝐡(𝑠),(4.1) where 𝐡(𝑑) is the standard Bm. The variance of 𝐡𝐻(𝑑) is given byEξ€Ί(𝑋(𝑑))2ξ€»=𝑉𝐻(𝑑)|𝑑|2𝐻(𝑑),(4.2) where 𝑉𝐻(𝑑)=Ξ“(2βˆ’π»(𝑑))cos(πœ‹π»(𝑑))πœ‹π»(𝑑)(2𝐻(𝑑)βˆ’1).(4.3) Without lose of generality, one may normalize 𝐡𝐻(𝑑) such that E[(𝑋(𝑑))2]=|𝑑|2𝐻(𝑑) by replacing 𝑋(𝑑) with 𝑋(𝑑)/𝑉𝐻(𝑑).

Unless otherwise stated, 𝑋(𝑑) denotes the normalized process in what follows. The explicit expression of the covariance of 𝑋(𝑑) can be calculated byE𝑋𝑑1𝑋𝑑2𝐻𝑑=𝑁1𝑑,𝐻2||𝑑1||𝐻(𝑑1)+𝐻(𝑑2)+||𝑑2||𝐻(𝑑1)+𝐻(𝑑2)βˆ’||𝑑1βˆ’π‘‘2||𝐻(𝑑1)+𝐻(𝑑2)ξ‚„,(4.4) where𝑁𝐻𝑑1𝑑,𝐻2=Γ𝑑2βˆ’π»1ξ€Έξ€·π‘‘βˆ’π»2ξ€·πœ‹π»ξ€·π‘‘ξ€Έξ€Έcosξ€·ξ€·1𝑑+𝐻2ξ€Έξ€Έ/2ξ€Έξ€Έπœ‹π»ξ€·π‘‘ξ€·ξ€·1𝑑+𝐻2𝐻𝑑/2ξ€Έξ€·1𝑑+𝐻2ξ€Έξ€Έβˆ’1.(4.5) With the assumption that 𝐻(𝑑) is 𝛽-HΓΆlder function such that0<inf(𝐻(𝑑))≀sup(𝐻(𝑑))<min(1,𝛽),(4.6) one may have 𝐻(𝑑+πœ†π‘’)β‰ˆπ»(𝑑) for πœ†β†’0. Thus, the local covariance function of the normalized mBm has the following limiting form for πœβ†’0:E[]∼1𝑋(𝑑+𝜏)𝑋(𝑑)2ξ€·|𝑑+𝜏|2𝐻(𝑑)+|𝑑|2𝐻(𝑑)βˆ’|𝜏|2𝐻(𝑑)ξ€Έ.(4.7) The variance of the increment process for πœβ†’0 becomesEξ€½[]𝑋(𝑑+𝜏)βˆ’π‘‹(𝑑)2ξ€ΎβˆΌ|𝜏|2𝐻(𝑑),(4.8) which implies that the increment processes of mBm are locally stationary. It follows that the local Hausdorff dimension of the graphs of mBm is given by[][]dim{𝑋(𝑑),π‘‘βˆˆπ‘Ž,𝑏}=2βˆ’min{𝐻(𝑑),π‘‘βˆˆπ‘Ž,𝑏},(4.9) for each interval [π‘Ž,𝑏]βŠ‚π‘…+.

Regarding the computation of 𝐻(𝑑), we need a sequence π‘†π‘˜(𝑗) expressed by the local growth of the increment process, π‘†π‘˜π‘š(𝑗)=π‘βˆ’1𝑗+π‘˜ξ“π‘—=0||||𝑋(𝑖+1)βˆ’π‘‹(𝑖),1<π‘˜<𝑁,(4.10) where π‘š is the largest integer not exceeding 𝑁/π‘˜. The local HΓΆlder function 𝐻(𝑑) at point 𝑗𝑑=(π‘βˆ’1)(4.11) is given by (see Peltier and Levy-Vehel [91], Muniandy et al. [95], and Li et al. [94])ξ‚€βˆšπ»(𝑑)=βˆ’logπœ‹/2π‘†π‘˜ξ‚(𝑗)log(π‘βˆ’1).(4.12) The local box or Hausdorff dimension denoted by 𝐷(𝑑) is equal to𝐷(𝑑)=2βˆ’π»(𝑑).(4.13) That is,ξ‚€βˆšπ·(𝑑)=2+logπœ‹/2π‘†π‘˜ξ‚(𝑗)log(π‘βˆ’1).(4.14)

4.2. Scaling Analysis of Traffic Using mBm

The function 𝐷(𝑑) in (4.14) characterizes the local irregularity of traffic on a point-by-point basis or the small-time scaling of traffic.

Note that 𝐻(𝑑) may be used to describe the LRD of traffic on a point-by-point basis, see Peltier and Levy-Vehel [91]. From a view of applications, it is desired to represent the LRD, which is a global property of traffic at large time scales, on an interval-by-interval basis. As a matter of fact, from a practical view of the Internet traffic, one is interested in the LRD measure, say 𝐻, to investigate how traffic at time 𝑑 is correlated with that at 𝜏 apart from 𝑑. Thus, the LRD at time 𝑑 on a point-by-point basis, that is, 𝐻(𝑑), may be difficult to be used in practice. In addition to this, since the local irregularity of traffic is independent of its LRD while 𝐻(𝑑) linearly correlates to 𝐷(𝑑) (see (4.13)), 𝐻(𝑑) may be unsatisfactory to characterize the LRD property of traffic. Therefore, we propose the following expression to describe the LRD of traffic:π»π‘š[]=E𝐻(𝑑),(4.15) where the subscript π‘š implies the mean.

Note 1. E[𝐻(𝑑)] should be understood on an interval-by-interval basis.

Note 2. E[𝐻(𝑑)] is uncorrelated with 𝐷(𝑑). Denote by corr as a correlation operator. Then, considering that π»π‘š is a constant, we have []corr{E𝐻(𝑑),𝐷(𝑑)}=0.(4.16)

Note 3. According to (4.13), we have ||||corr{𝐻(𝑑),𝐷(𝑑)}=1.(4.17) Equation (4.17) exhibits that 𝐻(𝑑) is completely correlated with 𝐷(𝑑).

We show two demonstrations of real-traffic traces named DEC-PKT-1.TCP and DEC-PKT-2.TCP that were recorded at Digital Equipment Corporation (DEC) in March 1995. Figure 2 plots its first 1025 data of traffic DEC-PKT-1.TCP, which is denoted by π‘₯(𝑖) to imply the size of the 𝑖th packet (𝑖=0,1,…). Figure 3 shows its 𝐷(𝑖) of the first 8193 data points. The value of π»π‘š=E[𝐻(𝑖)] for DEC-PKT-1.TCP equals to 0.756 in the range of 𝑖=0,…,8192. Figures 4 and 5 are plots for DEC-PKT-2.TCP, where π»π‘š=0.754. The plots in Figures 3 and 5 exhibit that traffic has highly local irregularity as discussed by Li and Lim [19] on an interval-by-interval basis.

389803.fig.002
Figure 2: Traffic trace DEC-PKT-1.TCP in packet size.
389803.fig.003
Figure 3: 𝐷(𝑖) of DEC-PKT-1.TCP for 𝑖=0,…,8192. π»π‘š=0.756.
389803.fig.004
Figure 4: Traffic trace DEC-PKT-2.TCP in packet size.
389803.fig.005
Figure 5: 𝐷(𝑖) of DEC-PKT-2.TCP for 𝑖=0,…,8192. π»π‘š=0.754.

5. Conclusions

The key idea in this paper is to describe small-time scaling and large-time one of traffic, separately. Following this idea, we have explained the limitation of the standard mBm in this regard because the local irregularity of traffic because 𝐷(𝑑) of mBm linearly relates to its 𝐻(𝑑). To relax this restriction, we suggest to use 𝐷(𝑑) to describe the local irregularity of traffic on a point-by-point basis for the small scaling phenomenon and propose to use E[𝐻(𝑑)], instead of 𝐻(𝑑), to represent the LRD of traffic for the large scaling phenomenon on an interval-by-interval basis, providing a promising candidate to study the scaling phenomena of traffic. The present results, in methodology, may be applied to random data in related issues, for example, those in [96–111], for the scaling analysis.

Acknowledgments

This work was partly supported by the National Natural Science Foundation of China (NSFC) under the Project Grant nos. 60573125, 60873264, 61070214, and 60870002, the 973 plan under the Project no. 2011CB302800/2011CB302802, NCET, and the Science and Technology Department of Zhejiang Province (nos. 2009C21008, 2010R10006, 2010C33095, Y1090592).

References

  1. M. Li and W. Zhao, β€œSufficient condition for min-plus deconvolution to be closed in the service-curve set in computer networks,” International Journal of Computers, vol. 1, no. 3, pp. 163–166, 2007.
  2. M. Li, β€œRecent results on the inverse of min-plus convolution in computer networks,” International Journal of Engineering and Interdisciplinary Mathematics, vol. 1, no. 1, pp. 1–9, 2009. View at Publisher Β· View at Google Scholar
  3. M. Li and P. Borgnat, β€œForeword to the special issue on traffic modeling, its computations and applications,” Telecommunication Systems, vol. 43, no. 3-4, pp. 145–146, 2010. View at Publisher Β· View at Google Scholar Β· View at Scopus
  4. H. Michiel and K. Laevens, β€œTeletraffic engineering in a broad-band era,” Proceedings of the IEEE, vol. 85, no. 12, pp. 2007–2033, 1997. View at Publisher Β· View at Google Scholar Β· View at Scopus
  5. W. Willinger, R. Govindan, S. Jamin, V. Paxson, and S. Shenker, β€œScaling phenomena in the internet: critically examining criticality,” Proceedings of the National Academy of Sciences of the United States of America, vol. 99, no. 1, pp. 2573–2580, 2002. View at Publisher Β· View at Google Scholar Β· View at Scopus
  6. A. Feldmann, A. C. Gilbert, W. Willinger, and T. G. Kurtz, β€œThe changing nature of network traffic: scaling phenomena,” ACM SIGCOMM Computer Communication Review, vol. 28, no. 2, pp. 5–29, 1998. View at Scopus
  7. Y. Jiang, β€œPer-domain packet scale rate guarantee for expedited forwarding,” IEEE/ACM Transactions on Networking, vol. 14, no. 3, pp. 630–643, 2006. View at Publisher Β· View at Google Scholar Β· View at Scopus
  8. K. Papagiannaki, R. Cruz, and C. Diot, β€œNetwork performance monitoring at small time scales,” in Proceedings of the 2003 ACM SIGCOMM Internet Measurement Conference (IMC '03), pp. 295–300, Miami, Fla, USA, October 2003. View at Scopus
  9. V. Paxson and S. Floyd, β€œWide area traffic: the failure of Poisson modeling,” IEEE/ACM Transactions on Networking, vol. 3, no. 3, pp. 226–244, 1995. View at Publisher Β· View at Google Scholar Β· View at Scopus
  10. W. Stalling, High-Speed Networks: TCP/IP and ATM Design Principles, Prentice Hall, Upper Saddle River, NJ, USA, 2nd edition, 2002.
  11. D. McDysan, QoS & Traffic Management in IP & ATM Networks, McGraw-Hill, Chicago, Ill, USA, 2000.
  12. J. M. Pitts and J. A. Schormans, Introduction to ATM Design and Performance: with Applications Analysis Software, John Wiley & Sons, New York, NY, USA, 2nd edition, 2000.
  13. W. E. Leland, M. S. Taqqu, W. Willinger, and D. V. Wilson, β€œOn the self-similar nature of Ethernet traffic (extended version),” IEEE/ACM Transactions on Networking, vol. 2, no. 1, pp. 1–15, 1994. View at Publisher Β· View at Google Scholar Β· View at Scopus
  14. J. Beran, R. Sherman, M. S. Taqqu, and W. Willinger, β€œLong-range dependence in variable-bit-rate video traffic,” IEEE Transactions on Communications, vol. 43, no. 234, pp. 1566–1579, 1995. View at Publisher Β· View at Google Scholar Β· View at Scopus
  15. B. Tsybakov and N. D. Georganas, β€œSelf-similar processes in communications networks,” IEEE Transactions on Information Theory, vol. 44, no. 5, pp. 1713–1725, 1998. View at Zentralblatt MATH Β· View at Scopus
  16. W. Willinger and V. Paxson, β€œWhere mathematics meets the internet,” Notices of the American Mathematical Society, vol. 45, no. 8, pp. 961–970, 1998. View at Zentralblatt MATH
  17. A. Adas, β€œTraffic models in broadband networks,” IEEE Communications Magazine, vol. 35, no. 7, pp. 82–89, 1997. View at Scopus
  18. A. Ayache, S. Cohen, and J. Levy Vehel, β€œCovariance structure of Multifractional Brownian motion, with application to long range dependence,” in Proceedings of the IEEE Interntional Conference on Acoustics, Speech, and Signal Processing (ICASSP '00), vol. 6, pp. 3810–3813, June 2000. View at Scopus
  19. M. Li and S. C. Lim, β€œModeling network traffic using generalized Cauchy process,” Physica A, vol. 387, no. 11, pp. 2584–2594, 2008. View at Publisher Β· View at Google Scholar Β· View at Scopus
  20. M. Li and S. C. Lim, β€œPower spectrum of generalized Cauchy process,” Telecommunication Systems, vol. 43, no. 3-4, pp. 219–222, 2010. View at Publisher Β· View at Google Scholar Β· View at Scopus
  21. M. Li, Teletraffic Modeling Relating to Generalized Cauchy Process: Empirical Study, VDM, SaarbrΓΌcken, Germany, 2009.
  22. M. Li, β€œGeneration of teletraffic of generalized Cauchy type,” Physica Scripta, vol. 81, no. 2, Article ID 025007, 10 pages, 2010. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at Scopus
  23. M. Li, β€œSelf-Similarity and long-range dependence in teletraffic,” in Proceedings of the 9th WSEAS International Conference on Multimedia Systems and Signal Processing, pp. 19–24, Hangzhou, China, May 2009.
  24. M. Li, β€œEssay on teletraffic models (I),” in Proceedings of the 9th WSEAS International Conference on Applied Computer and Computational Science (ACACOS '10), pp. 130–135, Hangzhou, China, April 2010.
  25. M. Li, S. C. Lim, and H. Feng, β€œA novel description of multifractal phenomenon of network traffic based on generalized cauchy process,” in Proceedings of the 7th International Conference on Computational Science (ICCS '07), Y. Shi, van Albada, Sloot, and Dongarra, Eds., vol. 4489 of Lecture Notes in Computer Science, pp. 1–9, Springer, May 2007.
  26. R. L. Cruz, β€œA calculus for network delayβ€”I: network elements in isolation,” IEEE Transactions on Information Theory, vol. 37, no. 1, pp. 114–131, 1991. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at Scopus
  27. R. L. Cruz, β€œA calculus for network delay. II. Network analysis,” IEEE Transactions on Information Theory, vol. 37, no. 1, pp. 132–141, 1991.
  28. M. Li and W. Zhao, β€œRepresentation of a stochastic traffic bound,” IEEE Transactions on Parallel and Distributed Systems, vol. 21, no. 9, pp. 1368–1372, 2010. View at Publisher Β· View at Google Scholar Β· View at Scopus
  29. A. Raha, S. Kamat, X. Jia, and W. Zhao, β€œUsing traffic regulation to meet end-to-end deadlines in ATM networks,” IEEE Transactions on Computers, vol. 48, no. 9, pp. 917–935, 1999. View at Publisher Β· View at Google Scholar Β· View at Scopus
  30. Y.-M. Jiang and Y. Liu, Stochastic Network Calculus, Springer, Berlin, Germany, 2008.
  31. J.-Y. Le Boudec and P. Thiran, Network Calculus, vol. 2050 of Lecture Notes in Computer Science, Springer, Berlin, German, 2001. View at Publisher Β· View at Google Scholar
  32. A. K. Erlang, β€œTelefon-ventetider. et stykke sandsynlighedsregning,” Matematisk Tidsskrift B, pp. 25–42, 1920.
  33. E. Brockmeyer, H. L. HalstrΓΈm, and A. Jensen, β€œThe life and works of A. K. Erlang,” Transactions of the Danish Academy of Technical Sciences, vol. 1948, no. 2, pp. 23–100, 1948. View at Zentralblatt MATH
  34. Z. Bojkovic, M. Bakmaz, and B. Bakmaz, β€œOriginator of teletraffic theory,” Proceedings of the IEEE, vol. 98, no. 1, pp. 123–127, 2009. View at Publisher Β· View at Google Scholar Β· View at Scopus
  35. F. Le Gall, β€œOne moment model for telephone traffic,” Applied Mathematical Modelling, vol. 6, no. 6, pp. 415–423, 1982. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at Scopus
  36. P. M. Lin, B. J. Leon, and C. R. Stewart, β€œAnalysis of circuit-switched networks employing originating-office control with spill-forward,” IEEE Transactions on Communications, vol. 26, no. 6, pp. 754–765, 1978. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at Scopus
  37. D. R. Manfield and T. Downs, β€œOn the one-moment analysis of telephone traffic networks,” IEEE Transactions on Communications, vol. 27, no. 8, pp. 1169–1174, 1979. View at Publisher Β· View at Google Scholar Β· View at Scopus
  38. M. Reiser, β€œPerformance evaluation of data communication systems,” Proceedings of the IEEE, vol. 70, no. 2, pp. 171–196, 1982. View at Publisher Β· View at Google Scholar Β· View at Scopus
  39. J. D. Gibson, Ed., The Communications Handbook, IEEE Press, 1997.
  40. R. B. Cooper, Introduction to Queueing Theory, North-Holland, New York, NY, USA, 2nd edition, 1981.
  41. H. Akimaru and K. Kawashima, Teletraffic: Theory and Applications, Springer, Berlin, Germany, 1993.
  42. V. Paxson, Measurements and analysis of end-to-end internet dynamics, Ph.D. thesis, University of California, Berkeley, Calif, USA, 1997.
  43. V. Paxson, β€œGrowth trends in wide-area TCP connections,” IEEE Network, vol. 8, no. 4, pp. 8–17, 1994.
  44. W. A. Fuller, Introduction to Statistical Time Series, Wiley Series in Probability and Statistics: Probability and Statistics, John Wiley & Sons, New York, NY, USA, 2nd edition, 1996.
  45. G. E. P. Box, G. M. Jenkins, and G. C. Reinsel, Time Series Analysis, Prentice Hall, Englewood Cliffs, NJ, USA, 3rd edition, 1994.
  46. S. K. Mitra and J. F. Kaiser, Handbook for Digital Signal Processing, John Wiley & Sons, New York, NY, USA, 1993.
  47. J. S. Bendat and A. G. Piersol, Random Data: Analysis and Measurement Procedure, John Wiley & Sons, New York, NY, USA, 3rd edition, 2000.
  48. A. Papoulis, Probability, Random Variables, and Stochastic Processes, McGraw-Hill Series in Electrical Engineering. Communications and Information Theory, McGraw-Hill, New York, NY, USA, 1997.
  49. J. L. Doob, β€œThe elementary Gaussian processes,” Annals of Mathematical Statistics, vol. 15, pp. 229–282, 1944. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  50. J. Beran, Statistics for Long-Memory Processes, vol. 61 of Monographs on Statistics and Applied Probability, Chapman and Hall, New York, NY, USA, 1994.
  51. J. Beran, β€œStatistical methods for data with long-range dependence,” Statistical Science, vol. 7, no. 4, pp. 404–416, 1992. View at Publisher Β· View at Google Scholar
  52. N. C. Nigam, Introduction to Random Vibrations, The MIT Press, Cambridge, Mass, USA, 1983.
  53. J. D. Murray, Asymptotic Analysis, vol. 48 of Applied Mathematical Sciences, Springer, New York, NY, USA, 2nd edition, 1984.
  54. P. Abry, P. Borgnat, F. Ricciato, A. Scherrer, and D. Veitch, β€œRevisiting an old friend: on the observability of the relation between long range dependence and heavy tail,” Telecommunication Systems, vol. 43, no. 3-4, pp. 147–165, 2010. View at Publisher Β· View at Google Scholar Β· View at Scopus
  55. R. P. Kanwal, Generalized Functions: Theory and Applications, BirkhΓ€user, Boston, Mass, USA, 3rd edition, 2004.
  56. I. M. Gelfand and K. Vilenkin, Generalized Functions, vol. 1, Academic Press, New York, NY, USA, 1964.
  57. M. Li, β€œFractal time seriesβ€”a tutorial review,” Mathematical Problems in Engineering, vol. 2010, Article ID 157264, 26 pages, 2010. View at Zentralblatt MATH
  58. G. Samorodnitsky and M. S. Taqqu, Stable Non-Gaussian Random Processes, Stochastic Modeling, Chapman & Hall, New York, NY, USA, 1994.
  59. P. Flandrin, β€œOn the spectrum of fractional Brownian motions,” IEEE Transactions on Information Theory, vol. 35, no. 1, pp. 197–199, 1989. View at Publisher Β· View at Google Scholar Β· View at Scopus
  60. I. Csabai, β€œ1/f noise in computer network traffic,” Journal of Physics A, vol. 27, no. 12, article 004, pp. L417–L421, 1994. View at Publisher Β· View at Google Scholar Β· View at Scopus
  61. Y. G. SinaΔ­, β€œSelf-similar probability distributions,” Theory of Probability and Its Applications, vol. 21, no. 1, pp. 63–80, 1976. View at Zentralblatt MATH
  62. M. Li and S. C. Lim, β€œA rigorous derivation of power spectrum of fractional Gaussian noise,” Fluctuation and Noise Letters, vol. 6, no. 4, pp. C33–C36, 2006. View at Publisher Β· View at Google Scholar Β· View at Scopus
  63. M. Li, W. Zhao, and S. Chen, β€œFGN based telecommunication traffic models,” WSEAS Transactions on Computers, vol. 9, no. 7, pp. 706–715, 2010.
  64. B. B. Mandelbrot, The Fractal Geometry of Nature, W. H. Freeman, San Francisco, Calif, USA, 1982.
  65. F. A. Tobagi, R. W. Peebles, and E. G. Manning, β€œModeling and measurement techniques in packet communication networks,” Proceedings of the IEEE, vol. 66, no. 11, pp. 1423–1447, 1978. View at Publisher Β· View at Google Scholar Β· View at Scopus
  66. R. Jain and S. A. Routhier, β€œPacket trains-measurements and a new model for computer network traffic,” IEEE Journal on Selected Areas in Communications, vol. 4, no. 6, pp. 986–995, 1986. View at Publisher Β· View at Google Scholar Β· View at Scopus
  67. H. Jiang and C. Dovrolis, β€œWhy is the internet traffic bursty in short time scales?” in Proceedings of the International Conference on Measurement and Modeling of Computer Systems (SIGMETRICS '05), vol. 33, pp. 241–252, June 2005. View at Publisher Β· View at Google Scholar Β· View at Scopus
  68. P. Nain, β€œImpact of bursty traffic on queues,” Statistical Inference for Stochastic Processes, vol. 5, no. 3, pp. 307–320, 2002. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  69. M. Draief and J. Mairesse, β€œServices within a busy period of an M/M/1 queue and dyck paths,” Queueing Systems, vol. 49, no. 1, pp. 73–84, 2005. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at Scopus
  70. F. NΓ©meth, P. Barta, R. SzabΓ³, and J. BΓ­rΓ³, β€œNetwork internal traffic characterization and end-to-end delay bound calculus for generalized processor sharing scheduling discipline,” Computer Networks, vol. 48, no. 6, pp. 910–940, 2005. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at Scopus
  71. C. Li and W. Zhao, β€œStochastic performance analysis of non-feedforward networks,” Telecommunication Systems, vol. 43, no. 3-4, pp. 237–252, 2010. View at Publisher Β· View at Google Scholar Β· View at Scopus
  72. Y. Jiang, Q. Yin, Y. Liu, and S. Jiang, β€œFundamental calculus on generalized stochastically bounded bursty traffic for communication networks,” Computer Networks, vol. 53, no. 12, pp. 2011–2021, 2009. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at Scopus
  73. S. Wang, D. Xuan, R. Bettati, and W. Zhao, β€œToward statistical QoS guarantees in a differentiated services network,” Telecommunication Systems, vol. 43, no. 3-4, pp. 253–263, 2010. View at Publisher Β· View at Google Scholar Β· View at Scopus
  74. D. Starobinski and M. Sidi, β€œStochastically bounded burstiness for communication networks,” IEEE Transactions on Information Theory, vol. 46, no. 1, pp. 206–212, 2000. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at Scopus
  75. R. G. Bartle and D. R. Sherbert, Introduction to Real Analysis, John Wiley & Sons, New York, NY, USA, 3rd edition, 2000.
  76. W. F. Trench, Introduction to Real Analysis, Pearson Education, 2003.
  77. W. Willinger, M. S. Taqqu, R. Sherman, and D. V. Wilson, β€œSelf-similarity through high-variability: statistical analysis of Ethernet LAN traffic at the source level,” IEEE/ACM Transactions on Networking, vol. 5, no. 1, pp. 71–86, 1997. View at Publisher Β· View at Google Scholar Β· View at Scopus
  78. C. Estan and G. Varghese, β€œNew directions in traffic measurement and accounting: focusing on the elephants, ignoring the mice,” ACM Transactions on Computer Systems, vol. 21, no. 3, pp. 270–313, 2003. View at Publisher Β· View at Google Scholar Β· View at Scopus
  79. A. Gravey, J. Boyer, K. Sevilla, and J. Mignault, β€œResource allocation for worst case traffic in ATM networks,” Performance Evaluation, vol. 30, no. 1-2, pp. 19–43, 1997. View at Publisher Β· View at Google Scholar Β· View at Scopus
  80. K. Tian and M. Li, β€œA reliable anomaly detector against low-rate DDOS attack,” International Journal of Electronics and Computers, vol. 1, no. 1, pp. 1–6, 2009.
  81. E. W. Knightly and N. B. Shroff, β€œAdmission control for statistical QoS: theory and practice,” IEEE Network, vol. 13, no. 2, pp. 20–29, 1999. View at Publisher Β· View at Google Scholar Β· View at Scopus
  82. A. Raha, W. Zhao, S. Kamat, and W. Jia, β€œAdmission control for hard real-time connections in ATM LANs,” IEE Proceedings, vol. 148, no. 4, pp. 1–12, 2001. View at Publisher Β· View at Google Scholar Β· View at Scopus
  83. X. Jia, W. Zhao, and J. Li, β€œAn integrated routing and admission control mechanism for real-time multicast connections in ATM networks,” IEEE Transactions on Communications, vol. 49, no. 9, pp. 1515–1519, 2001. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at Scopus
  84. R. J. Adler, The Geometry of Random Fields, John Wiley & Sons, Chichester, UK, 1981, Wiley Series in Probability and Mathematical Statistic.
  85. P. Hall and R. Roy, β€œOn the relationship between fractal dimension and fractal index for stationary stochastic processes,” The Annals of Applied Probability, vol. 4, no. 1, pp. 241–253, 1994. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  86. G. Chan, P. Hall, and D. S. Poskitt, β€œPeriodogram-based estimators of fractal properties,” The Annals of Statistics, vol. 23, no. 5, pp. 1684–1711, 1995. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  87. J. T. Kent and A. T. A. Wood, β€œEstimating the fractal dimension of a locally self-similar gaussian process by using increments,” Journal of the Royal Statistical Society Series B, vol. 59, no. 3, pp. 679–699, 1997. View at Zentralblatt MATH Β· View at Scopus
  88. T. Gneiting and M. Schlather, β€œStochastic models that separate fractal dimension and the hurst effect,” SIAM Review, vol. 46, no. 2, pp. 269–282, 2004. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at Scopus
  89. S. C. Lim and M. Li, β€œA generalized Cauchy process and its application to relaxation phenomena,” Journal of Physics A, vol. 39, no. 12, pp. 2935–2951, 2006. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at Scopus
  90. M. Li, W. Jia, and W. Zhao, β€œA whole correlation structure of asymptotically self-similar traffic in communication networks,” in Proceedings of the 1st International Conference on Web Information Systems Engineering (WISE '00), pp. 461–466, Hong Kong, June 2000. View at Publisher Β· View at Google Scholar
  91. R. F. Peltier and J. Levy-Vehel, β€œA new method for estimating the parameter of fractional Brownian motion,” Tech. Rep. RR 2696, INRIA, 1994.
  92. R. F. Peltier and J. Levy-Vehel, β€œMultifractional Brownian motion: definition and preliminaries results,” Tech. Rep. RR 2645, INRIA, 1995.
  93. A. Benassi, S. Jaffard, and D. Roux, β€œElliptic gaussian random processes,” Revista Matematica Iberoamericana, vol. 13, no. 1, pp. 19–90, 1997. View at Zentralblatt MATH Β· View at Scopus
  94. M. Li, S. C. Lim, and W. Zhao, β€œInvestigating multi-fractality of network traffic using local Hurst function,” Advanced Studies in Theoretical Physics, vol. 2, no. 10, pp. 479–490, 2008. View at Scopus
  95. S. V. Muniandy, S. C. Lim, and R. Murugan, β€œInhomogeneous scaling behaviors in Malaysian foreign currency exchange rates,” Physica A, vol. 301, no. 1–4, pp. 407–428, 2001. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at Scopus
  96. M. Scalia, G. Mattioli, and C. Cattani, β€œAnalysis of large-amplitude pulses in short time intervals: application to neuron interactions,” Mathematical Problems in Engineering, vol. 2010, Article ID 895785, 2010. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  97. C. Cattani, β€œFractals and hidden symmetries in DNA,” Mathematical Problems in Engineering, vol. 2010, Article ID 507056, 2010. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  98. C. Cattani, β€œHarmonic wavelet approximation of random, fractal and high frequency signals,” Telecommunication Systems, vol. 43, no. 3-4, pp. 207–217, 2010. View at Publisher Β· View at Google Scholar Β· View at Scopus
  99. E. G. Bakhoum and C. Toma, β€œDynamical aspects of macroscopic and quantum transitions due to coherence function and time series events,” Mathematical Problems in Engineering, vol. 2010, Article ID 428903, 2010. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at Scopus
  100. E. G. Bakhoum and C. Toma, β€œMathematical transform of traveling-wave equations and phase aspects of quantum interaction,” Mathematical Problems in Engineering, vol. 2010, Article ID 695208, 15 pages, 2010. View at Zentralblatt MATH
  101. D. She and X. Yang, β€œA new adaptive local linear prediction method and its application in hydrological time series,” Mathematical Problems in Engineering, vol. 2010, Article ID 205438, 15 pages, 2010. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  102. M. Humi, β€œAssessing local turbulence strength from a time series,” Mathematical Problems in Engineering, vol. 2010, Article ID 316841, 2010. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at Scopus
  103. M. Dong, β€œA tutorial on nonlinear time-series data mining in engineering asset health and reliability prediction: concepts, models, and algorithms,” Mathematical Problems in Engineering, vol. 2010, Article ID 175936, 2010. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  104. M. Li and J.-Y. Li, β€œOn the predictability of long-range dependent series,” Mathematical Problems in Engineering, vol. 2010, Article ID 397454, 9 pages, 2010. View at Zentralblatt MATH
  105. Z. Liu, β€œChaotic time series analysis,” Mathematical Problems in Engineering, vol. 2010, Article ID 720190, 2010. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at Scopus
  106. G. Toma, β€œSpecific differential equations for generating pulse sequences,” Mathematical Problems in Engineering, vol. 2010, Article ID 324818, 11 pages, 2010. View at Zentralblatt MATH
  107. O. M. Abuzeid, A. N. Al-Rabadi, and H. S. Alkhaldi, β€œFractal geometry-based hypergeometric time series solution to the hereditary thermal creep model for the contact of rough surfaces using the Kelvin-Voigt medium,” Mathematical Problems in Engineering, vol. 2010, Article ID 652306, 22 pages, 2010. View at Publisher Β· View at Google Scholar
  108. M. Li, β€œChange trend of averaged Hurst parameter of traffic under DDOS flood attacks,” Computers and Security, vol. 25, no. 3, pp. 213–220, 2006. View at Publisher Β· View at Google Scholar Β· View at Scopus
  109. C. Zhang, X. Bai, J. Teng, D. Xuan, and W. Jia, β€œConstructing low-connectivity and full-coverage three dimensional sensor networks,” IEEE Journal on Selected Areas in Communications, vol. 28, no. 7, pp. 984–993, 2010. View at Publisher Β· View at Google Scholar
  110. M. Li, C. Cattani, and S.-Y. Chen, β€œViewing sea level by a one-dimensional random function with long memory, accepted for publication,” Mathematical Problems in Engineering, vol. 2011, Article ID 654284, 13 pages, 2011. View at Publisher Β· View at Google Scholar
  111. M. Li, β€œA class of negatively fractal-dimensional Gaussian random functions,” Mathematical Problems in Engineering, vol. 2011, Article ID 291028, 18 pages, 2011.