A New Model for Network Traffic Based on Alpha-Stable Self-similar Processes Ge Xiaohu, Zhu Guangxi, Zhu Yaoting Department of E & I, Huazhong University of Science & Technology, Wuhan, China, 430074 Abstract-This paper proposes a new network traffic and call them H-sssi processes [4] Self-similarity model based on alpha-stable processes Then the traces of manifests itself in a number of equivalent ways, the most three simulated relevant of which is the display of the Long Range Comparing the traces of the simulation data and the trace Dependence (LRD) for H-sssi processes Mandelbrot of the actual data, it is shown that the new model is better refers to this phenomenon as the Joseph effect Another than the other models in fitting with the actual data important way of self-similarity is high variability for different self-similar models are Key words: communication, network modeling, self-similar, Noah effect alpha-stable processes H-sssi processes Mandelbrot refers to this phenomenon as INTRODUCTION Recent empirical studies of high-resolution traffic measurements from a variety of different working communication networks have provided ample evidence that actual network traffic is self-similar or fractal in nature [1][2], which couldn’t be modeled by Poisson and Markov processes or variants of them For example, people have discovered that overall packet-loss decreases very slowly with the increasing of the buffer capacity, in sharp contrast to Poisson-based models where packet-loss decreases exponentially fast while the buffer size increases Moreover, packet-delay (95th percentile) always increases with the increasing of buffer capacity, again in contrast to the Poisson models where packet-delay does not exceed a fixed limit despite the increasing of the buffer size As a result, some routers and protocols designed by the Poisson model produce negative effect on network traffic So it is necessary to explore a new model which can capture the self-similarity The term “self-similar” was firstly used by Mandelbrot [3] during the 1960’s to designate those processes that are scalable over time (or space) without losing their statistical properties In other words, a continuous-time process X = {X (t ), t ≥ 0} is self-similar, with self-similarity parameter H, if it satisfies the condition: d X (t ) = c −H X (ct), ∀t ≥ 0, ∀c > 0, < H < (1) There are many different self-similar processes We typically consider those that have stationary increments, Since the discovery of the self-similarity in network traffic, many models have been put forward to describe the self-similar network, which include the ON-OFF model, the FBM model, the FARIMA model, the S4 model and so on [5][6] But they all have some drawbacks The hypotheses of the ON-OFF model are inconsistent with the fact The FBM model can’t capture features of the burstiness Because the FARIMA model is too complicated, it is impossible to use it for real-time simulation It is difficult for the S4 model to describe the LRD Therefore, it is important to find a new model of network traffic, which can exhibit the Joseph, the Noah effects and parsimonious parameters Because of the infinity of sample variance caused by the LRD, the actual network traffic modeling could not use the central limit theorem But the generalized central limit theorem could be used, since it states that if the marginal distribution of the normalized aggregation of infinitely many independent and identically distributed (i.i.d) sources converges, then it belongs to the family of alpha-stable marginal distributions, which have in general infinite variance [4] Therefore the authors try to model the network traffic based on the alpha-stable processes The material and contributions of the paper are organized as follows In Section the basic properties of alpha-stable distributions and the concept of Linear Fractional Stable Motion (LFSM) processes are introduced The new model proposed is submitted in Section Some analyses and comparisons are delivered in Section In the conclusion part the authors summarize the main its characteristic function is observations and refer to future work { E exp i θ X = exp − σ θ + i µθ ALPHA-STABLE DISTRIBUTION AND LINEAR } (3) It is the FBM process So the alpha-stable process FRACTIONAL STABLE MOTION includes all properties of the FBM process, at the same 2.1 The Definition and Properties of the Alpha-stable Distribution Since densities and distribution functions are not time it has properties of non-Gaussian case Tail Approximation: Let X ~ Sα (σ , β , µ ) with < α < , then, as known in closed form for most stable distributions, they α + β C x −α α P {X > x } ~ σ P {X < − x } ~ σ α − β C x −α α are generally specified by their characteristic functions Definition [4] 1: A random variable X is said to have a stable distribution if there are < α ≤ 2, σ > 0, − ≤ β ≤ , and µ ∈ R parameters so that its characteristic function has the following form: α α πα exp− σ θ 1 − iβ (signθ ) tan + iµθ 2 E expiθX = if α ≠1 exp{− σ θ (1 + iβ (signθ ) lnθ ) + iµθ} if α =1 Parameter α 1−α Γ ( − α ) cos( πα / ) Cα = 2 π (4) if α ≠1 if α =1 So alpha-stable distributions have the property of heavy tailed 2.2 The Linear Fractional Stable Motion (2) There are different extensions of fractional Brownian motion to the alpha-stable case The one that is most if θ > if θ = if θ < 1 signθ = 0 − x→∞ commonly used is the LFSM process [4] The well-balanced LFSM processes are continuous-time stochastic processes is called characteristic exponent and specifies the level of burstiness in the distribution The distribution can be skewed if the skewness parameter β is different with zero α and β together determine the shape of the distribution Variables σ and µ are called follows: {Lα ,H ,−∞ < t < ∞} ( defined as ) H−1/α H−1/α ∞ Lα,H (t) = ∫−∞ (t − x)+ − ((− x)+ ) ⋅ Ms (dx) (5) Where < α < , < H < , H ≠ α , and M s is an the dispersion and the mean or median of the distribution alpha-stable random measure on R with Lebesque A random variable X that follows an alpha-stable control measure The new network traffic model advanced distribution with the above parameters is denoted by X ~ Sα (σ , β , µ ) in this paper is based on the Linear Fractional Stable Noise scale and location parameters, respectively, and express If α = , the alpha-stable distribution reduces to the Gaussian distribution (the parameter is β nonexistent), ( ) N α' , H ( i ) = h d ∗ S σ( α, β) , ( i ) ∑ Km = k =1 h d (k / m ) S σ( α, β) , (LFSN) processes, and the LFSN processes are the increment processes of the LFSM processes The discrete-time LFSN processes are given as follows: x d − ( x − )d hd ( x ) = x d (i − k / m ) (6) d = H −1/α if 1< x if 0< x ≤1 It is said that the LFSN processes are the LRD if H > α , positive real constants, −1 < β < ， and Nα'' , H (i ) expresses the discrete-time trace of 1-stable LFSN S1(,α1,)0 THE NEW MODEL OF NETWORK TRAFFIC ~ and S1(,α1,0) are two i.i.d random variables with common or the Short Range Dependence (SRD) if H < 1 α In order to capture the changes of network traffic, based on the LFSN processes and its property of stability, distribution Sα (1,1,0) this paper brings forward a new network traffic model The form is as follows: M (i ) = c1 N α , β , H (i ) + c ( c h = 1 d ∗ ANALYSIS AND COMPARISON In order to analyze the performance of the new model, the '' = c1 h d ∗ S (α ) 1, β ,0 actual network data (file Oct89Ext.TL) is used for comparison, )( i ) + which was collected by Leland at Bellcore Morristown Research c2 + β / α (α ) S ,1 , (i) 1/α ~ (α ) − − β S ,1 , and Engineering facility [2] The data file contains 1,000,000 packets of network traffic All packets are divided into ten data sets each having 100,000 packets, and then the number of packets passing in a range of ten (7) seconds in every data set is counted Thus ten new data sets are obtained, in each of which every element expresses the number of passing packets in 10s time scale The quantile method is used + c2 where M (i ) denotes the volume of traffic carried by the for estimating parameters of the new model in every new data set [7] Consequently the Table.1 is got as the results of estimation network element in the time unit i , c1 and c are Table.1 The parameters of alpha-stable processes measured by actual network data Data sets Sequence of packets α β σ µ 1~100000 packets 1.3635 1.0000 13.5331 4.8294 100001~200000 packets 1.4921 1.0000 36.5022 39.6485 200001~300000 packets 1.8455 1.0000 55.8680 112.9420 300001~400000 packets 1.3322 0.8869 49.3299 156.1060 400001~500000 packets 1.3244 0.6699 33.7254 149.0870 500001~600000 packets 1.5603 0.5527 63.6562 244.4920 600001~700000 packets 1.4265 1.0000 51.9600 134.7900 700001~800000 packets 2.0000 48.7472 92.0000 800001~900000 packets 2.0000 26.2082 26.0000 10 900001~1000000 packets 1.5480 20.9367 1.8193 1.0000 In order to demonstrate the predominance of the new network traffic, the simulated trace of the new model, the model through experiments, the fifth data set of table.1 is simulated trace of the S4 model and the simulated trace of stochastically selected as the target for simulation For the FBM model By comparing the traces in Fig.1, it is convenience in comparing, the FBM model, the S4 model shown that the FBM model trace can’t describe the and the new model are respectively used to generate the burstiness of network traffic, however the S4 model trace simulation data Fig.1 includes the trace of the actual and the new model trace can capture the burstiness Although the S4 model can capture the burstiness, the of the mathematic model can provide the theoretic scale of the burstiness of the S4 model is about 10000 to foundation of assigning network resource, improving 30000 The scale of the burstiness of the new model is traffic efficiency and guaranteeing Quality of Service about 1000 to 2000 The scale of the burstiness of the [6][7] actual trace is about 1000 So the error of the S4 model is This paper introduces the definition and properties of larger than the new model Therefore it is said that the new the alpha-stable distribution, and then a new model is model is better than the other models in fitting the actual advanced based on the LFSN processes In terms of packets of network traffic So the new model can provide comparing the simulation traces of the three models with great advantages in the future research the actual network data, it is shown that the new model is better than the other models In the future we will research CONCLUSION The significance of network traffic modeling is to design a mathematic model, which can mimic the trends a method of prediction based on the new model for assigning network resource observed in measured data Consequently, the prediction 1200 1000 packet 800 600 400 200 0 100 200 300 400 500 400 500 (A) time (10second) 1800 packet 1300 800 300 -200 100 200 300 (B) time (10second) 30000 packet 25000 20000 15000 10000 5000 0 100 200 (C) 300 400 time (10second) 500 400 packet 300 200 100 101 201 301 (D) time(10seconds) 401 501 Fig.1 The (A) plot is the actual trace; the (B) plot is the trace generated by the new model; the (C) plot is the trace generated by the S4 model; the (D) plot is the trace generated by the FBM model REFERENCES [1] Orenstein, P.; Kim, H.; Lau, C.L “Bandwidth [6] Narasimha, R.; Seungsin Lee; Rao, R “Discrete-time allocation for self-similar traffic consisting of multiple scale invariant systems: relation to long-range dependence traffic classes with distinct characteristics”, Global and FARIMA models, Acoustics, Speech, and Signal Telecommunications Conference, 2001 Processing”, 2002 IEEE International Conference on, [2] W E Leland, M S Taqqu, W Willinger, and D V Volume: 4, 2002 Wilson, “On the self-similar nature of ethernet traffic” [7] Nolan, J.P., Panorska, A K and McCulloch J H (extended version), IEEE/ACM Trans Networking, vol 2, “Estimation of stable spectral 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and the mean or median of the distribution alpha- stable random measure on R with Lebesque A random variable X that follows an alpha- stable control measure The new. .. new network traffic model advanced distribution with the above parameters is denoted by X ~ Sα (σ , β , µ ) in this paper is based on the Linear Fractional Stable Noise scale and location parameters ,... is said that the new the alpha- stable distribution, and then a new model is model is better than the other models in fitting the actual advanced based on the LFSN processes In terms of packets