**2.1 Traffic modelling in multimedia networks**

The traffic modelling and its application to real traffic in operational networks, allows the implementation of research platforms that simulate future or real network critical conditions, which is particularly interesting for huge service providers. Injecting traffic series generated accordingly to mathematical models may help to evaluate several conditions in a network and certainly this may help to develop more accurate capacity planning models regarding specific QoS requirements. Such procedures also facilitate the creation of management strategies. A large number of tools on the Internet provide traffic analysis, like TG (TG), NetSpec (NetSpec), Netperf (Netperf), MGEN (MGEN) and D-ITG (D-ITG) and GTAR, Gerador de Tráfego e Analisador de QoS na Rede (Carvalho et al., 2006), FracLab (FracLab, 2011).

To model the traffic in integrated networks is necessary the use of mathematical models that allow, from its base, to infer the impact of traffic on network performance. The efficient characterization of traffic will be given by the degree of accuracy of the model in comparison with the real traffic statistical properties.

In our work, the characterization of the traffic is used as a key element in the design of complex telecommunications systems. Once characterized, the traffic on different time scales can be used in network simulations. The simulation process can reproduce the behaviour of traffic by application type, for parts of the network, by customer group or interconnections with other networks, opening the possibility to increase the knowledge of the network and making possible a better control of resources.

### **2.2 Poisson and erlang model**

The use of the Internet to transmit real-time audio and video flows increases every day. Some of these applications are transmitted at a constant rate. This kind of traffic results by sending one packet every 1/Tx seconds, where Tx is the rate of transmission in packets per second, defined by the type of the application.

In circuit switched networks, a very successfully model is based on the Poisson distribution. The Poisson traffic is characterized by exponentially distributed random variables to represent the inter-packet times. The Erlang model, broadly used in telephony systems has been successfully used for capacity planning for many years and is based in the premise that a Poisson distribution describes the traffic in this type of network.

The Poisson model was considered accurate in the early years of the packet switched networks and was heavily used for capacity planning. In the early 90's, the work of Leland(Leland et al., 1994) proved that the behavior of the Ethernet traffic was considerably different than Poisson traffics mainly regarding self-similar aspects with long-range dependence, which is not well described by short memory processes. In practice, the packet switched networks that were planned using the Poisson model, normally had an overprovision in links capacity to comply with the lack of accuracy of the model. Considering the different works about capacity planning following the work of Leland, the heavy-tail models were considered more accurate to describe the traffic in packet switched networks and consequently, they appeared as a better choice.

#### **2.3 Self-similar**

32 Telecommunications Networks – Current Status and Future Trends

carrier) service provider of IP traffic was used to collect real network traces and we

A 3G with a Metro Ethernet access was also analysed. The analysis considered a per application separation of traffic. The statistical analysis was done using a self-similarity approach, calculating the Hurst parameter using different calculation methodologies (Abry et al., 2002). Some multifractal analysis was also done as a tool to better choose the time

The results show that the proposed method is able to generate better results in terms of an on-line traffic engineering control and also to provide key information to long term capacity planning cycles. The Traffic Engineering function is detailed using some network simulations examples. Finally, some long term forecasting and short term traffic engineering

The traffic modelling and its application to real traffic in operational networks, allows the implementation of research platforms that simulate future or real network critical conditions, which is particularly interesting for huge service providers. Injecting traffic series generated accordingly to mathematical models may help to evaluate several conditions in a network and certainly this may help to develop more accurate capacity planning models regarding specific QoS requirements. Such procedures also facilitate the creation of management strategies. A large number of tools on the Internet provide traffic analysis, like TG (TG), NetSpec (NetSpec), Netperf (Netperf), MGEN (MGEN) and D-ITG (D-ITG) and GTAR, Gerador de Tráfego e Analisador de QoS na Rede (Carvalho et al.,

To model the traffic in integrated networks is necessary the use of mathematical models that allow, from its base, to infer the impact of traffic on network performance. The efficient characterization of traffic will be given by the degree of accuracy of the model in

In our work, the characterization of the traffic is used as a key element in the design of complex telecommunications systems. Once characterized, the traffic on different time scales can be used in network simulations. The simulation process can reproduce the behaviour of traffic by application type, for parts of the network, by customer group or interconnections with other networks, opening the possibility to increase the knowledge of the network and

The use of the Internet to transmit real-time audio and video flows increases every day. Some of these applications are transmitted at a constant rate. This kind of traffic results by sending one packet every 1/Tx seconds, where Tx is the rate of transmission in packets per

In circuit switched networks, a very successfully model is based on the Poisson distribution. The Poisson traffic is characterized by exponentially distributed random variables to

simulated a similar architecture of this network using the OPNET Modeler tool.

scale.

proposal was done in a 3G networks.

2006), FracLab (FracLab, 2011).

**2.1 Traffic modelling in multimedia networks** 

comparison with the real traffic statistical properties.

making possible a better control of resources.

second, defined by the type of the application.

**2.2 Poisson and erlang model** 

One kind of traffic that appears often in wideband networks is the burst traffic. It can be generated by many applications such as compressed video services and file transfers. This traffic is characterized by periods with activity (on periods) and periods without activity (off periods). Moreover, as proved in (Perlingeiro & Ling, 2005), (Barreto, 2007), it is possible to generate self-similar traffic by the aggregation of many sources of burst traffics that presents a heavy-tailed distribution for the on period.

The self-similar model defines that a trace of traffic collected at a time scale has the same statistical characteristics that an appropriately scaled version of the traffic to a different time scale (Nichols et al., 1998). From the mathematical point of view, the self-similarity of a stochastic process in continuous time is defined as shown in Equation 1, which defines a process in continuous time X (*t*) as exactly self-similar.

$$\mathbf{x}^{\prime}\mathbf{X}(t) = \stackrel{d}{=} a^{-H}\mathbf{X}(at), a > 0 \tag{1}$$

The sample functions of a process X(*t*) and its scaled version of the *a*–HX(*at*) obtained by compressing the time axis by the factor amplitudes "a" , can not be distinguished statistically. Therefore, the moments of order n of X(*t*) are equal to the moments of order n of X (*at*), scaled by a-Hn. The Hurst parameter, H is then a key element to be identified in the traffic. For self-similar traffic, the H is greater than 0.5 and less than 1. For a Poisson traffic this value is close to 0.5. Experimental results show that this same parameter in operational networks (Perlingeiro & Ling, 2005; Carvalho et. Al., 2007) has values between 0.5 and 0.95. Then, the parameter H may be a descriptor of the degree of dependence on long traffic (Zhang et al.; 1997).

The aforementioned Hurst parameter plays a major role on the measurement of the selfsimilarity degree. The closer it is of the unity, the greatest the self-similarity degree. One of the most popular self-similar processes is the fractional Brownian motion (fBm), which is the only self-similar Gaussian process with stationary increments. The increments process of the fBm is the fractional Gaussian noise (fGn). To generate the traffic, we first create a fGn

IP and 3G Bandwidth Management Strategies Applied to Capacity Planning 35

Each point x of the support of the measure will produce a different α ( x ) , and the distribution of these exponents is what the singularity spectrum f ( α ) measures. The points for which the Hölder exponents are equal to some value α form a set, which is in turn a fractal object. The fractal dimension of this set can be calculated, and is a function of α ,

As described in (2), a function ƒ(*x*) satisfies the Hölder condition in a neighborhood of a

 *x*0 if |ƒ(*x*) - ƒ(*x*0)| ≤*c*|(*x*-*x*0)|*n* (4) And a function ƒ(*x*) satisfies a Hölder condition in an interval or in a region of the plane, for


The process of traffic characterization is a preponderant point of a feasible network project. In this section a traffic characterization framework is described. The characterization intends to describe a step by step procedure, which may be useful to understand the behavior of traffic in large networks using a mathematical model as a tool to achieve good planning. One difficult issue to characterize traffic in IP networks is the changing environment due to new applications and new services that are appearing constantly. This implies that the

all *x* and *y* in the interval or region, where *c* and *n* are constants, as in (5).

namely f ( α ).

point, where *c* and *n* are constants, as in (4).

**3. Traffic characterization** 

Fig. 2. Characterization process.

sequence based on the method presented in (Norros, 1995). Each sample of the sequence represents the number of packets to be sent on a time interval of size T. The size of the time interval and the mean of the sequence generated will depend on the traffic rate.

#### **2.4 Multifractal traffic**

As self-similar models, multifractals are multiscale process with rescaling properties, but with the main difference of being built on **multiplicative** schemes(Incite, 2011). In this way, they are highly non-Gaussian and are ruled by different limiting laws than the additive CLT (Central Limit Theorem). Therefore, multifractals can provide mathematical models to many world situations such as Internet traffic loads, web file requests, geo-physical data, images and many others. The Hölder function is defined by the h(t) function.

In the self similar model, also called as monofractal, the Hurst parameter is a global property that quantifies the process changes according to changes in the scale. For multifractal traffic, however, the Hurst parameter becomes less efficient in this characterization and another metric is needed to perform the scaling analysis of the sample regularity.

There are several ways to infer the scaling behavior of traffic, one way is widely used by local singularities of the function. A singular point is defined as a point in an equation, curve, surface, etc., which have transitions or becomes degenerate (Ried et al., 2000). It is quite common that the singular points of the signal containing essential information on network traffic packets.

In order to identify the singularities of a signal, it is necessary to measure the regularity of the same point, which will reflect in burst periods occurring at all traffic scales. In (Gilbert & Seuret, 2000) some examples can be found about the point and the exponents of the local Hölder values making possible to check the degree of uniqueness of network traffic.

According to Veira, (Veira et al., 2000) the Hölder exponent is capable to describe the degree of a singularity. Considering a function *f : R*→ *R*, with x0 as real number, and **α** a stricted real positive number. It can be assumed that *f* belongs to *C***α(***x***0)** if a polynomial *Pm* with degree **n < α**, as shown in (2).

$$\left| f(\mathbf{x}) - Pm(\mathbf{x} - \mathbf{x}\_0) \right| \le C \left| \mathbf{x} - \mathbf{x}\_0 \right|^a \tag{2}$$

As described in (Ludlam, 2004) a multifractal measure P can be characterized by calculating the distribution f ( α ) , known as the multifractal, or singularity, spectrum where α is the local Hölder exponente (Clegg, 2005 ; Castro e Silva, 2004 ; Vieira, 2006 ). This measure can be also shown as a probability density function P ( x ), in this case, the local Hölder exponente (; Gilbert & Seuret, 2000) is dened ad in (7).

$$\mathfrak{a}\left(\mathbf{x}\right) \equiv \lim \mathbf{l} \to 0 \log \mathbf{P}\left(\mathcal{B}\left(\mathbf{l}, \mathbf{x}\right)\right) \log \mathbf{l} \tag{3}$$

where ℬ ( l , x ) is a box centred at x with radius l , and P ( ℬ ) is the probability density integrated over the box ℬ . It describes the scaling of the probability within a box, centred on a point x , with the linear size of the box.

Each point x of the support of the measure will produce a different α ( x ) , and the distribution of these exponents is what the singularity spectrum f ( α ) measures. The points for which the Hölder exponents are equal to some value α form a set, which is in turn a fractal object. The fractal dimension of this set can be calculated, and is a function of α , namely f ( α ).

As described in (2), a function ƒ(*x*) satisfies the Hölder condition in a neighborhood of a point, where *c* and *n* are constants, as in (4).

$$\mathbf{x}\_0 \text{ if } |f(\mathbf{x}) - f(\mathbf{x}\_0)| \le \mathbf{c} \mid (\mathbf{x} - \mathbf{x}\_0) \mid \mathbf{n} \tag{4}$$

And a function ƒ(*x*) satisfies a Hölder condition in an interval or in a region of the plane, for all *x* and *y* in the interval or region, where *c* and *n* are constants, as in (5).

$$|f(\mathbf{x}) \cdot f(\mathbf{y})| \le c \|\mathbf{x} \cdot \mathbf{y}\|^n \tag{5}$$
