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Research on characterization of wireless LANs traffic
Commun Nonlinear Sci Numer Simulat 16 (2011) 3179–3187
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Commun Nonlinear Sci Numer Simulat journal homepage: www.elsevier.com/locate/cnsns
Research on characterization of wireless LANs traffic Huifang Feng a,⇑, Yantai Shu b, Oliver W.W. Yang c a b c
College of Mathematics and Information Science, Northwest Normal University, Lanzhou 730070, China School of Computer Science and Technology, Tianjin University, Tianjin 300072, China School of Information Technology and Engineering, University of Ottawa, Ottawa, Canada K1N 6N5
a r t i c l e
i n f o
Article history: Received 21 July 2010 Received in revised form 8 October 2010 Accepted 15 October 2010 Available online 23 October 2010 Keywords: Wireless LANs traffic Fourier power spectrum Correlation dimension The largest Lyapunov exponent Principal components analysis
a b s t r a c t In this paper, we employ actual wireless data that draw from well known archives of network traffic traces and investigate the characterization of the wireless LANs traffic. Firstly, useful preliminary information regarding the general wireless traffic dynamics is obtained using one standard statistical technique named Fourier power spectrum. Then the estimation of the parameters, such as the correlation dimension, the largest Lyapunov exponent and the principal components analysis indicate the existence of low-dimensional deterministic chaos in wireless traffic time series. Our results also show that the parameters selection of the phase space reconstruction influence the value of the correlation dimension and the largest Lyapunov exponent, but can not influence on diagnosis of chaotic nature of wireless traffic. Ó 2010 Elsevier B.V. All rights reserved.
1. Introduction Recently the IEEE 802.11 wireless LANs (WLAN) [1] have been widely deployed and more and more mobile computers are being equipped with the IEEE 802.11 compatible wireless network devices. WLAN is a shared-medium communication network that transmits information over wireless links. It is the most deployed wireless networks in the world and is high likely to play a major role in wireless home networks and next-generation wireless communications. However, wireless networks face new problems when compared to the wired networks such as the error-prone radio resource, signal interference, mobility of users, and the need for fair sharing method in the current carrier sense multiple access/collision avoidance (CSMA/CA) access mechanism. As the popularity of WLAN grows at an unprecedented rate, the performance of the wireless service in WLAN becomes an important issue. The performance of various mechanisms and policies, which have been proposed to achieve good performance, to various extents, depends on the network’s traffic characteristics. Thus, accurate models for the traffic and an understanding of the impact of various factors on the traffic characteristics are necessary for improving the capability of WLAN in general and developing efficient schedulers, admission policies, etc., in particular. There is rich literature on traffic characterization in wired networks. A number of recent empirical studies of traffic measurements from wired LANs have convincingly demonstrated that network traffic is self-similar or long-range dependent (LRD) in nature [2–4]. Recently, it was found that the Internet congestion control systems could exhibit complex nonlinear dynamical behaviours as demonstrated by the TCP-Drop tail congestion control system in certain circumstances [5]. The Refs. [6,7] have performed chaotic time series analysis of the average queue length in TCP-RED (Random Early Detection). They compared the performance of RED and adaptive RED via ordinary network performance indices of throughput and packet loss rate, and the nonlinear dynamics indices of the largest Lyapunov exponent and Hurst parameter. Ranjan et al. [8] proposed a discrete time model of TCP-RED and found that border collision bifurcation could lead to chaos in this model. ⇑ Corresponding author. Tel.: +86 931 7972487. E-mail addresses: [email protected], [email protected] (H. Feng). 1007-5704/$ - see front matter Ó 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.cnsns.2010.10.022
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The Refs. [9] applied a nonlinear time series analysis approach to the traffic measurements obtained at the input of a medium size local area network. There is significantly less work of the same detail for WLAN [10–18]. Researchers have only recently focused on this problem. Traffic trace from WLAN has been collected and analyzed to answer specific questions related to mobility and related client behaviors [10,11], protocol operate [12], etc. The Ref. [13] analyzes the syslog and tcpdump traces from 31 access points (AP) in campus WLAN to model flow arrivals, and proposed a Weibull distribution, and capture the non-stationarity of traffic in the variation of its scale parameter, which is estimated via weibull regression. It has been observed that the traffic in ad hoc wireless networks and WLAN has shown its self-similarity [14–17]. Tickoo and Sikdar [17] investigated the impact of the 802.11 media access control (MAC) on the second order scaling of the traffic by simulation. Their simulation results show that while individual TCP sources in wireless networks show evidence of self-similarity, the aggregate traffic shows multi-fractal properties. Although there is work done for the understanding of wireless traffic, there is much to be learned and understood of their characteristics. Besides self-similarity, the other important feature of WLAN traffic should be address. We propose to use the Chaos Theory to more accurately analyze the characteristics of wireless traffic. The main and creative contributions include the following aspects: (1) we employ three actual wireless traces and investigate the characterization of the WLAN traffic. One trace was collected from the WLAN at Tianjin University. The other two traces draw from well known archives of network traffic traces. (2) the general wireless traffic dynamics is obtained using one standard statistical technique named Fourier power spectrum. The comparison of power spectrum for three wireless traffic, noise, periodic, and chaotic time series (Chua’s Circuit) is discussed. (3) the correlation dimension, the largest Lyapunov exponent and the principal components analysis (PCA) method are employed for comprehensive characterization of the wireless traffic dynamics. (4) comparing to the autocorrelation function method and Takens’ embedding theorem, the mutual information method and Cao’s algorithm is employed to select the optimal embedding delay and the embedding dimension respectively. The results show that these parameter selections influence the value of the correlation dimension and the largest Lyapunov exponent, but can not influence on diagnosis of chaotic nature of wireless traffic. The remainder of this paper is organized as follows. In the Section 2 we present a brief review of the methods employed in the present study, both preliminary and comprehensive, for characterizing the wireless traffic dynamics. Details of the wireless traffic traces considered for investigation are provided in Section 3. The analysis of the wireless traffic and the results are discussed in this section. Important conclusions drawn from the present study are presented in Section 4. 2. Methodologies If the mathematical formulation of the system is available, then recognizing chaotic behaviour is relatively straightforward. Since the system evolution is deterministic, broadband noise spectra, for example, would be sufficient to identify the presence of chaos. Furthermore, since the number of variables is known, the reconstruction of the phase-space and the attractor and the estimation of the various invariants are straightforward. However, when one deals with controlled experiments, where one cannot record all the variables, and/or with observables from an uncontrolled system (like the atmospheric or hydrologic system), whose mathematical formulation and total number of variables may not be known exactly, the problem of identifying chaos becomes complicated. In such cases, Fourier analysis alone, for instance, cannot be used to identify the presence of chaos, since the observable might be a random variable. The difficulty in using Fourier power spectrum analysis, or any other single (statistical) method, to identify chaos in a phenomenon necessitated formulation of new methods and modification of existing ones. This resulted in the emergence of a wide variety of methods, popular among them are correlation dimension method, the largest Lyapunov exponent method, Kolmogorov entropy method, surrogate data method, nonlinear prediction and PCA method. It is important to note that, inspite of the advantages they possess, none of these methods can provide an infallible distinction between a chaotic and a stochastic system. The advantages and limitations of each of these methods have already been made available in the literature [19–21] and, therefore, are not reported herein. As none of these methods can provide an infallible distinction between a chaotic and a stochastic system, it is necessary to employ diverse techniques to provide at least convincing, if not conclusive, evidence regarding the presence/absence of chaotic dynamics in a phenomenon. In view of this, in the present study, a total of four methods are employed to investigate the dynamics of the wireless traffic. The investigation is carried out in two steps. In the first step, useful preliminary information regarding the general wireless traffic dynamics is obtained using one standard statistical technique named Fourier power spectrum. This is followed by the application of three specific chaos identification methods that could provide comprehensive characterization of the wireless traffic dynamics: (1) the correlation dimension method; (2) the largest Lyapunov exponent method; and (3) the principal components analysis method. These four methods are described next. 2.1. Fourier power spectrum The Fourier power spectrum is particularly useful for characterizing the regularities/irregularities in observed signals (or time series). In general, for a purely random process, the power spectrum oscillates randomly about a constant value, indicating that no frequency explains anymore of the variance of the sequence than anyother frequency. For a periodic or
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quasi-periodic sequence, only peaks at certain frequencies exist; measurement noise adds a continuous floor to the spectrum. Thus, in the spectrum, signal and noise are readily distinguished. Chaotic signals may also have sharp spectral lines but even in the absence of noise there will be continuous part (broadband) of the spectrum. 2.2. Reconstructing the attractor An important first step in any (specific) chaos identification technique is the reconstruction of the phase space of the time series under investigation and, hence, its attractor (a geometric object which characterizes the long-term behaviour of a system in the phase space). Such a reconstruction approach uses the concept of embedding a single-variable series in a multidimensional phase space to represent the underlying dynamics. Packard et al. [19] proposed a method of reconstructing the phase space system from a time series by using a time delay (i.e. embedding delay) variable to construct a time-delay vector. The reconstructed trajectory, Y, can be expressed as a matrix where each row is a phase space vector. That is,
Y ¼ ½Y 1 ; Y 2 ; . . . ; Y N T ;
ð1Þ
where Yi is the state of the system at discrete time i. For a time series xi (i = 1, 2, . . . , n), each Yi is given by
Y i ¼ ½xi ; xiþs ; . . . ; xiþðm1Þs :
ð2Þ
This vector constructs an m-dimensional reconstructed phase space. Here s is embedding delay; m is embedding-dimension, i.e. the coordinate number of the phase space. Y is a N m matrix, and the constants m, N, n and s are related as N = n (m 1)s. The choice of s is essential for the phase space reconstruction, since only an optimum s gives the best separation of neighboring trajectories within the minimum embedding phase space. The problem of selection of s has received a lot of attention not only from nonlinear dynamicists, but also from researchers in various natural and physical sciences. Research in this direction thus far has resulted in the formulation of a large number of methods and recommendations. Popular among these are the autocorrelation function method and the mutual information method [22–24]. Fraser and Swinney [22] proposed to use the first minimum of mutual information as the optimal embedding delay. The mutual information can be defined as follows:
IðsÞ ¼
X i;j
Pij ðsÞ ln
Pij ðsÞ ; Pi Pj
ð3Þ
where Pi is the probability to find a time series value in the ith interval, and Pij(s) is the joint probability that an observation is located in the ith interval and the observation time t later is located in the jth interval. In general, we are interested in minima of I(s). At the first minimum of I(s), xi+s adds the largest amount of information to the information we already have by knowing xi, without completely losing the correlation between them. Hence, the first minimum of I(s) marks the optimal choices for the embedding delay. In this paper, we use a heuristic method based on the autocorrelation function. The embedding delay s is chosen such that the autocorrelation function attains the value of 1 1/e (e is 2.718) [25]. Takens [26] and Mane [27] proved in mathematics that if m P 2D2 + 1 (D2 is the correlation dimension of the system), the reconstructed phase space and original phase space are diffeomorphically equivalent. Therefore, we can investigate the original phase space through studying the reconstructed phase space. The correlation dimension of the attractor is calculated for each embedding dimension, and the embedding dimension is chosen to be when the correlation dimension saturates. We also can use Cao’s algorithm to evaluate the embedding dimension [28]. We define the following quantity:
aði; mÞ ¼
Y i ðm þ 1Þ Y nði;mÞ ðm þ 1Þ ; Y i ðmÞ Y nði;mÞ ðmÞ
i 2 f1; 2; . . . ; n msg;
ð4Þ
where kk is the maximum norm and Yi(m + 1) is the ith reconstructed vector with embedding dimension m + 1. (1 6 n(i, m) 6 n ms) is an integer such that Yn(i,m)(m) is the nearest neighbour of Yi(m) in the m -dimensional reconstructed space. Given the function:
EðmÞ ¼
nm Xs 1 aði; mÞ: n ms i¼1
ð5Þ
We evaluate its variation from m to m + 1:
E1ðmÞ ¼ Eðm þ 1Þ=EðmÞ
ð6Þ
and consider as an estimate of the embedding dimension the first value m at which the above function is approximately constant.
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2.3. Determining correlation dimension A deterministic chaotic system usually converges to a strange attractor with non-integer dimensions after a long time evolution. The strange attractor has fractal geometric characteristics. The dimension is one of the most important parameters in characterizing a chaotic attractor. It represents the complexity of a nonlinear system quantitatively. The bigger the dimension is, the more complexity the system is. Among many definitions of dimension, correlation dimension is widely used due to its relative simplicity, fast convergent rate in numerical calculation and effectiveness in describing the complexity of the system. The method of estimating the correlation dimension D2 from a limited time series was first put forward by Berliner [20] and Grassberger et al. [29]. It is the most commonly employed method. Suppose we have a scalar time series xi (i = 1, 2, . . . , n) of a dynamical variable sampled at an equal time interval Dt from which the N vectors Yj (j = 1, 2, . . . , N) in the m-dimensional phase space can be reconstructed using the embedding delay technique. Then the correlation dimension D2 is defined and calculated as:
D2 ¼ lim r!0
ln C m ðrÞ ; ln r
ð7Þ
where Cm(r) is known as the correlation integral and can be computed as:
C m ðrÞ lim
N!1
N N X X 2 Hðr kY i Y j kÞ; NðN 1Þ i¼1 j¼iþ1
ð8Þ
where close pairs are counted via the Heaviside function, H (x), which is equal to zero for negative argument and one otherwise. r is an arbitrary distance measure, the correlation distance kYi Yjk is the distance between the vectors Yi and Yj. In our study, for computational efficiency, the distance between two points is defined as:
kY i Y j k ¼ max jxiþls xjþls j : 0 6 l 6 m 1 :
ð9Þ
Thus, the correlation dimension formula for finite time series is as follows:
D2 ¼ lim r!0
ln C m ðrÞ ln r
ð10Þ
and using the calculation method of local slope:
D2 ¼
d ln C m ðrÞ : d ln r
ð11Þ
In some scale, the curve of the local slope is relatively smooth. This interval is called scaling region. We outline the procedure of calculating the correlation dimension from time series as follows: Plot the local slope D2 ¼ d lnd ClnmrðrÞ versus lnr for each m. When m greater than some value, a plateau is will arise, and D2 will converge asymptotically to the real correlation dimension of the system. The value is the real correlation dimension of the system. For a chaotic attractor, the correlation dimension D2 is non-integer, the value of which determines whether the system is low- or high-dimensional. In the above method, the size of data set is required greater than 10D2 =2 , i.e. n > 10D2 =2 [21]. 2.4. Lyapunov exponent Lyapunov exponents are defined as the logarithms of the absolute value of the eigenvalues of the linearized dynamics averaged over the attractor. It gives an additional feature that describes deterministic processes. These exponents measure the exponential divergence or convergence of nearby orbits. Points close together in phase space are nearly identical states; points with separating orbits become unaligned with each other. The rate of separation (becoming unaligned) expresses the sensitivity of the system’s dynamic behaviour to small differences in their initial states. The sensitivity to initial conditions is characteristic of the chaotic behaviour because it leads to the exponential divergence of nearby orbits. The exponential divergence of the initially nearby orbits is measured by at least one positive Lyapunov exponent, which can be used to characterize a chaotic system quantitatively [25,30]. The Lyapunov exponent measure how predictable or unpredictable the system is. The positive Lyapunov exponent is necessary conditions for deterministic chaos but not sufficient. The small data sets algorithm [23] is easy to implement, and robust to changes in the following quantities: embedding dimension, size of data set, reconstruction delay, and noise level. We use this approach to calculate the largest Lyapunov exponent. We summarize in the following small data sets algorithm for a time series. Step Step Step Step
1 2 3 4
: Calculating mean period P can be accomplished using the fast Fourier transform (FFT). : Calculating correlation dimension and determining embedding dimension using Takens’ theorem, i.e. m P 2D2 + 1. : Reconstructing the phase space. : After reconstructing the dynamics system, the algorithm locates the nearest neighbor of each point on the trajectory. The nearest neighbor, Y^j , is found by searching for the point that minimizes the distance to the particular reference point, Yj. This is expressed as:
H. Feng et al. / Commun Nonlinear Sci Numer Simulat 16 (2011) 3179–3187
dj ð0Þ ¼ min kY j Y^j k; jj ^jj > P: ^j
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ð12Þ
Step 5 : The jth pair of nearest neighbors diverges at a rate approximately given by the largest Lyapunov exponent:
dj ðiÞ dj ð0Þek1 i :
ð13Þ
Taking the logarithm of both sides of Eq. (13) gives
ln dj ðiÞ ln dj ð0Þ þ k1 i:
ð14Þ
Eq. (14) represents a set of approximately parallel lines, each with a slope roughly proportional to k1 Step 6 : The largest Lyapunov exponent is estimated using a least-squares fit, with a constant, to the average line defined by:
yðiÞ ¼
1 < ln dj ðiÞ >; Dt
ð15Þ
where < > denotes the average over all values of j. 2.5. Principal components analysis Principal component analysis (also known as empirical orthogonal function analysis) is widely used to condense the amount of data and to extract important features or patterns from the data. PCA is another method for detecting chaos from time series. According to Takens’ embedding theorem [26,29], we can use time series xi (i = 1, 2, . . . , n) to construct a trajectory matrix YNm as well as the covariance matrix YTY. We can compute the covariance matrix YTY and the corresponding the eigenvectors and eigenvalues. The eigenvalues is also called singular values. Because the eigenvalues of a symmetric, nonnegative definite matrix are all real and nonnegative, we can arrange the eigenvalues in decreasing order. By comparing the values of eigenvalues to the total sum of eigenvalues, we can get an idea how much weight is concentrated along the particular eigenvector. 3. Analysis, results and discussion 3.1. Data set used We use three traces to verify the proposed model. One trace (t030801) was collected from the WLAN at the Network Research Laboratory at Tianjin University. They were conducted on an AP at the Network Research Laboratory, which carried all WLAN traffic to access the Internet. Another one (final.anon) is from the Mobile Computing Group at Stanford University [31]. Their wireless network is a WaveLAN network with WavePoint II access points acting as bridges between the wireless and wired networks. Ref. [11] describes the network, tracing methodology, and the characteristics of the trace of wireless traffic in detail. The last one is the trace (trace.pcap) of the intranet traffic on the ACM SIGCOMM’01 conference held in the U.C. San Diego in August 2001 [32]. The trace consists of two parts. The first part is a record of performance monitoring data sampled from wireless APs serving the conference. And the second part consists of anonymous packet headers of all wireless traffic. Both parts of the trace spanned the three days of the conference, capturing the workload of 300,000 flows from 195 users. The auditorium was covered by four ORiNOCO™ AP-1000 wireless APs. The subnet of APs was connected to a Cisco Catalyst 2924 switch over a 100BaseT link, which connected to the venue’s intranet, then the campus gigabit backbone, and finally to the Internet. The configuration of the wireless network, methodology for trace collection and analysis was described in [10]. 3.2. Preliminary characterization of wireless traffic dynamics Fig. 1 shows the comparison of power spectrum for three wireless traffic, noise, periodic and chaotic time series (Chua’s Circuit) [33]. The power spectrum of a chaotic time series is easy to distinguish from Gaussian noise or periodic time series. The power spectrum of the noise is flat for all frequencies. The power spectrum of a period one orbit is dominated by one central peak. Chaotic time series has a broad-band power spectrum with a rich spectral structure. The broad-band nature of the chaotic power spectrum indicates the existence of a continuum of frequencies. As can be seen, the power spectrum of wireless traffic not only exhibits sharp spectral lines (or peaks) at certain frequencies but also is somewhat continuous (broadband) which are regarded as a possible indicator of chaos in the wireless traffic. 3.3. Comprehensive characterization of wireless traffic dynamics First the embedding delay of the phase space reconstruction is determined by autocorrelation function method. The correlation integrals and the dimensions for wireless traffic are computed using the Grassberger–Procaccia algorithm. In Fig. 2
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Fig. 1. Comparison of power spectrum for wireless traffic, noise, periodic and chaotic time series.
Fig. 2. Correlation integral Cm(r) versus correlation distance r for embedding dimension m = 2, 3 . . . 17.
we present the results of the correlation integral analysis for embedding dimensions m = 2 up to 17. The relationship between lnCm(r) and lnr, shown in Fig. 2, indicates clear scaling regions for all the embedding dimensions used, allowing fairly accurate estimates of the correlation dimensions. The correlation dimensions, D2 against the corresponding m values are presented in Fig. 3. Fig. 3 shows an increase in the correlation dimension with the embedding dimension up to a certain point, and saturation beyond this point. Such saturation may be an indication of the deterministic dynamics in the wireless traffic.
8 final.anon trace.pcap
D2
6
t030801
4
2
0
0
5
m
10
15
Fig. 3. Correlation dimension D2 versus embedding dimension m.
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The saturation value of the correlation dimension is about 4.87119 (when m = 10), 2.23556 (when m = 9), and 2.7707 (when m = 10), respectively. The obtained low correlation dimension seems to suggest the possible presence of chaotic behaviour in the wireless traffic. In order to confirm our estimation we estimated the largest Lyapunov exponent. Fig. 4 shows a typical plot of y(i) versus iDt. There is a long linear region that can be used to extract the largest Lyapunov exponent. We obtain that the largest Lyapunov exponent of wireless traffic trace is around 0.0979, 0.1522, and 0.1339. respectively. These positive value indicates the existence of chaotic behaviour and is consistent with the estimation of a low-dimensional attractor from the correlation dimension method. Fig. 5 shows the PCA results for the three wireless traffic traces and Gauss white noise. For noise (which is generally regarded as Gauss white noise with mean value 0 and variance 1 in practical systems), synplectic geometry spectrums of this noise give the even distribution of its total energy. We can see that the intrinsic information of the analyzed trace is accumulated in the first and few lower-index components. Thus, it further shows that the wireless traffic exhibits intrinsic nonlinear characteristics. From Fig. 5, we can see that the reconstructed phase space can be captured mainly by 4- or 5dimensional data. This indicates that the majority of network dynamics can be expressed by a system with 4 or 5 state variables. 3.4. Parameter influence on diagnosis of chaotic nature of wireless traffic The choice of embedding delay and embedding dimension is very important to the phase space reconstruction of any chaotic time series. The further question we need to answer here is whether these parameter selections have influence on diagnosis of chaotic nature of wireless traffic. Firstly, the proper embedding delay can be determined by using the mutual information method. In Fig. 6 is represented the graphics of mutual information for wireless traffic respectively. As can be seen from Fig. 7 the first minimum of these curve are obviously at an embedding delay of 8, 8 and 4. We determine embedding dimension using Takens’ theorem and reconstruct phase space. Based on forenamed methods, the saturation value of the correlation dimension is about 4.1688, 3.0868 and 2.7194, respectively, and the largest Lyapunov exponent of wireless traffic traces is 0.0967, 0.1236 and 0.1345 respectively. In despite of values are different from that of above method, these results also indicate that wireless traffic exhibited chaotic. Secondly, the embedding dimension is evaluated by Cao’s algorithm. In Fig. 7, we can see the plot of the Cao’s number E1 versus its embedding dimension m. We determine m using Cao’s algorithm, yielding m = 12, 9, 12 for the t030801, final.anon, and trace.pcap, respectively. According to the forenamed methods, we evaluate the embedding delay and calculate the largest Lyapunov exponent. Lyapunov exponent of wireless traffic trace is the value is 0.0938, 0.1138, and 0.0923, respectively. These positive values indicate the existence of chaotic behaviour. The above results show that these parameter selections influence the value of the correlation dimension and the largest Lyapunov exponent, but can not influence on diagnosis of chaotic nature of wireless traffic. 4. Conclusion and future work In this work, the several real traffic traces collected from actual WLAN, was examined for the existence of low-dimensional chaos. First the power-law decay of the Fourier power spectrum suggested the possible existence of a low-dimensional attractor. We then estimated the correlation dimension by Grassberger–Procaccia algorithm. It is found out that the correlation dimension is non-integer number. We used the small data sets algorithm to calculate the largest Lyapunov exponent. It appeared that the largest Lyapunov exponent is positive. The principal components analysis showed that the intrinsic information of the traffic is accumulated in the first and few lower-index components. All those results indicated that the wireless traffic is a low dimensional chaotic system. Our results also showed that the selection of parameters for reconstruction phase space influenced the value of the correlation dimension and the largest Lyapunov exponent, but could not influence on diagnosis of chaotic nature of wireless traffic.
Fig. 4. The largest Lyapunov exponent of wireless traffic.
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Fig. 5. The principal components of wireless traffic.
Fig. 6. Estimate of embedding delay by average mutual information.
Fig. 7. Estimate of embedding dimension by Cao’s algorithm.
The shared medium wireless networks have more vulnerabilities and bandwidth and latency constrains than their wired counterparts. The traffic characteristics of wireless and wire network are different. This paper describes an in-depth study on the characteristics of WLAN traffic. Network traffic prediction on small time scale is necessary for congestion control, admission control, bandwidth allocation, and buffer management. This is very important for bandwidth constrained wireless networks. Although we have done some research on WLAN traffic prediction [34], after understanding the characteristics of wireless traffic, we plan to provide an in-depth study on the performance of different prediction models with various trace, seek a perfect prediction models for wireless traffic, and investigate prediction in various time-scales. This is our future work. Acknowledgments This research was supported in part by National Natural Science Foundation of China (NSFC) under Grant No. 61072063, Science and Research Project of the Education Department of Gansu Province (0901-03), and NWNU-KJCXGC-03-52. References [1] IEEE, IEEE Std. 802.11-Wireless LAN Medium Access Control (MAC) and Physical Layer (PHY). Specifications, 1999. [2] Leland W, Taqqu M, Willinger W, Wilson DV. Self-similarity in high-speed packet traffic: analysis and modeling of Ethernet traffic measurements. Stat Sci 1995;10:67–85.
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Commun Nonlinear Sci Numer Simulat journal homepage: www.elsevier.com/locate/cnsns
Research on characterization of wireless LANs traffic Huifang Feng a,⇑, Yantai Shu b, Oliver W.W. Yang c a b c
College of Mathematics and Information Science, Northwest Normal University, Lanzhou 730070, China School of Computer Science and Technology, Tianjin University, Tianjin 300072, China School of Information Technology and Engineering, University of Ottawa, Ottawa, Canada K1N 6N5
a r t i c l e
i n f o
Article history: Received 21 July 2010 Received in revised form 8 October 2010 Accepted 15 October 2010 Available online 23 October 2010 Keywords: Wireless LANs traffic Fourier power spectrum Correlation dimension The largest Lyapunov exponent Principal components analysis
a b s t r a c t In this paper, we employ actual wireless data that draw from well known archives of network traffic traces and investigate the characterization of the wireless LANs traffic. Firstly, useful preliminary information regarding the general wireless traffic dynamics is obtained using one standard statistical technique named Fourier power spectrum. Then the estimation of the parameters, such as the correlation dimension, the largest Lyapunov exponent and the principal components analysis indicate the existence of low-dimensional deterministic chaos in wireless traffic time series. Our results also show that the parameters selection of the phase space reconstruction influence the value of the correlation dimension and the largest Lyapunov exponent, but can not influence on diagnosis of chaotic nature of wireless traffic. Ó 2010 Elsevier B.V. All rights reserved.
1. Introduction Recently the IEEE 802.11 wireless LANs (WLAN) [1] have been widely deployed and more and more mobile computers are being equipped with the IEEE 802.11 compatible wireless network devices. WLAN is a shared-medium communication network that transmits information over wireless links. It is the most deployed wireless networks in the world and is high likely to play a major role in wireless home networks and next-generation wireless communications. However, wireless networks face new problems when compared to the wired networks such as the error-prone radio resource, signal interference, mobility of users, and the need for fair sharing method in the current carrier sense multiple access/collision avoidance (CSMA/CA) access mechanism. As the popularity of WLAN grows at an unprecedented rate, the performance of the wireless service in WLAN becomes an important issue. The performance of various mechanisms and policies, which have been proposed to achieve good performance, to various extents, depends on the network’s traffic characteristics. Thus, accurate models for the traffic and an understanding of the impact of various factors on the traffic characteristics are necessary for improving the capability of WLAN in general and developing efficient schedulers, admission policies, etc., in particular. There is rich literature on traffic characterization in wired networks. A number of recent empirical studies of traffic measurements from wired LANs have convincingly demonstrated that network traffic is self-similar or long-range dependent (LRD) in nature [2–4]. Recently, it was found that the Internet congestion control systems could exhibit complex nonlinear dynamical behaviours as demonstrated by the TCP-Drop tail congestion control system in certain circumstances [5]. The Refs. [6,7] have performed chaotic time series analysis of the average queue length in TCP-RED (Random Early Detection). They compared the performance of RED and adaptive RED via ordinary network performance indices of throughput and packet loss rate, and the nonlinear dynamics indices of the largest Lyapunov exponent and Hurst parameter. Ranjan et al. [8] proposed a discrete time model of TCP-RED and found that border collision bifurcation could lead to chaos in this model. ⇑ Corresponding author. Tel.: +86 931 7972487. E-mail addresses: [email protected], [email protected] (H. Feng). 1007-5704/$ - see front matter Ó 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.cnsns.2010.10.022
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The Refs. [9] applied a nonlinear time series analysis approach to the traffic measurements obtained at the input of a medium size local area network. There is significantly less work of the same detail for WLAN [10–18]. Researchers have only recently focused on this problem. Traffic trace from WLAN has been collected and analyzed to answer specific questions related to mobility and related client behaviors [10,11], protocol operate [12], etc. The Ref. [13] analyzes the syslog and tcpdump traces from 31 access points (AP) in campus WLAN to model flow arrivals, and proposed a Weibull distribution, and capture the non-stationarity of traffic in the variation of its scale parameter, which is estimated via weibull regression. It has been observed that the traffic in ad hoc wireless networks and WLAN has shown its self-similarity [14–17]. Tickoo and Sikdar [17] investigated the impact of the 802.11 media access control (MAC) on the second order scaling of the traffic by simulation. Their simulation results show that while individual TCP sources in wireless networks show evidence of self-similarity, the aggregate traffic shows multi-fractal properties. Although there is work done for the understanding of wireless traffic, there is much to be learned and understood of their characteristics. Besides self-similarity, the other important feature of WLAN traffic should be address. We propose to use the Chaos Theory to more accurately analyze the characteristics of wireless traffic. The main and creative contributions include the following aspects: (1) we employ three actual wireless traces and investigate the characterization of the WLAN traffic. One trace was collected from the WLAN at Tianjin University. The other two traces draw from well known archives of network traffic traces. (2) the general wireless traffic dynamics is obtained using one standard statistical technique named Fourier power spectrum. The comparison of power spectrum for three wireless traffic, noise, periodic, and chaotic time series (Chua’s Circuit) is discussed. (3) the correlation dimension, the largest Lyapunov exponent and the principal components analysis (PCA) method are employed for comprehensive characterization of the wireless traffic dynamics. (4) comparing to the autocorrelation function method and Takens’ embedding theorem, the mutual information method and Cao’s algorithm is employed to select the optimal embedding delay and the embedding dimension respectively. The results show that these parameter selections influence the value of the correlation dimension and the largest Lyapunov exponent, but can not influence on diagnosis of chaotic nature of wireless traffic. The remainder of this paper is organized as follows. In the Section 2 we present a brief review of the methods employed in the present study, both preliminary and comprehensive, for characterizing the wireless traffic dynamics. Details of the wireless traffic traces considered for investigation are provided in Section 3. The analysis of the wireless traffic and the results are discussed in this section. Important conclusions drawn from the present study are presented in Section 4. 2. Methodologies If the mathematical formulation of the system is available, then recognizing chaotic behaviour is relatively straightforward. Since the system evolution is deterministic, broadband noise spectra, for example, would be sufficient to identify the presence of chaos. Furthermore, since the number of variables is known, the reconstruction of the phase-space and the attractor and the estimation of the various invariants are straightforward. However, when one deals with controlled experiments, where one cannot record all the variables, and/or with observables from an uncontrolled system (like the atmospheric or hydrologic system), whose mathematical formulation and total number of variables may not be known exactly, the problem of identifying chaos becomes complicated. In such cases, Fourier analysis alone, for instance, cannot be used to identify the presence of chaos, since the observable might be a random variable. The difficulty in using Fourier power spectrum analysis, or any other single (statistical) method, to identify chaos in a phenomenon necessitated formulation of new methods and modification of existing ones. This resulted in the emergence of a wide variety of methods, popular among them are correlation dimension method, the largest Lyapunov exponent method, Kolmogorov entropy method, surrogate data method, nonlinear prediction and PCA method. It is important to note that, inspite of the advantages they possess, none of these methods can provide an infallible distinction between a chaotic and a stochastic system. The advantages and limitations of each of these methods have already been made available in the literature [19–21] and, therefore, are not reported herein. As none of these methods can provide an infallible distinction between a chaotic and a stochastic system, it is necessary to employ diverse techniques to provide at least convincing, if not conclusive, evidence regarding the presence/absence of chaotic dynamics in a phenomenon. In view of this, in the present study, a total of four methods are employed to investigate the dynamics of the wireless traffic. The investigation is carried out in two steps. In the first step, useful preliminary information regarding the general wireless traffic dynamics is obtained using one standard statistical technique named Fourier power spectrum. This is followed by the application of three specific chaos identification methods that could provide comprehensive characterization of the wireless traffic dynamics: (1) the correlation dimension method; (2) the largest Lyapunov exponent method; and (3) the principal components analysis method. These four methods are described next. 2.1. Fourier power spectrum The Fourier power spectrum is particularly useful for characterizing the regularities/irregularities in observed signals (or time series). In general, for a purely random process, the power spectrum oscillates randomly about a constant value, indicating that no frequency explains anymore of the variance of the sequence than anyother frequency. For a periodic or
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quasi-periodic sequence, only peaks at certain frequencies exist; measurement noise adds a continuous floor to the spectrum. Thus, in the spectrum, signal and noise are readily distinguished. Chaotic signals may also have sharp spectral lines but even in the absence of noise there will be continuous part (broadband) of the spectrum. 2.2. Reconstructing the attractor An important first step in any (specific) chaos identification technique is the reconstruction of the phase space of the time series under investigation and, hence, its attractor (a geometric object which characterizes the long-term behaviour of a system in the phase space). Such a reconstruction approach uses the concept of embedding a single-variable series in a multidimensional phase space to represent the underlying dynamics. Packard et al. [19] proposed a method of reconstructing the phase space system from a time series by using a time delay (i.e. embedding delay) variable to construct a time-delay vector. The reconstructed trajectory, Y, can be expressed as a matrix where each row is a phase space vector. That is,
Y ¼ ½Y 1 ; Y 2 ; . . . ; Y N T ;
ð1Þ
where Yi is the state of the system at discrete time i. For a time series xi (i = 1, 2, . . . , n), each Yi is given by
Y i ¼ ½xi ; xiþs ; . . . ; xiþðm1Þs :
ð2Þ
This vector constructs an m-dimensional reconstructed phase space. Here s is embedding delay; m is embedding-dimension, i.e. the coordinate number of the phase space. Y is a N m matrix, and the constants m, N, n and s are related as N = n (m 1)s. The choice of s is essential for the phase space reconstruction, since only an optimum s gives the best separation of neighboring trajectories within the minimum embedding phase space. The problem of selection of s has received a lot of attention not only from nonlinear dynamicists, but also from researchers in various natural and physical sciences. Research in this direction thus far has resulted in the formulation of a large number of methods and recommendations. Popular among these are the autocorrelation function method and the mutual information method [22–24]. Fraser and Swinney [22] proposed to use the first minimum of mutual information as the optimal embedding delay. The mutual information can be defined as follows:
IðsÞ ¼
X i;j
Pij ðsÞ ln
Pij ðsÞ ; Pi Pj
ð3Þ
where Pi is the probability to find a time series value in the ith interval, and Pij(s) is the joint probability that an observation is located in the ith interval and the observation time t later is located in the jth interval. In general, we are interested in minima of I(s). At the first minimum of I(s), xi+s adds the largest amount of information to the information we already have by knowing xi, without completely losing the correlation between them. Hence, the first minimum of I(s) marks the optimal choices for the embedding delay. In this paper, we use a heuristic method based on the autocorrelation function. The embedding delay s is chosen such that the autocorrelation function attains the value of 1 1/e (e is 2.718) [25]. Takens [26] and Mane [27] proved in mathematics that if m P 2D2 + 1 (D2 is the correlation dimension of the system), the reconstructed phase space and original phase space are diffeomorphically equivalent. Therefore, we can investigate the original phase space through studying the reconstructed phase space. The correlation dimension of the attractor is calculated for each embedding dimension, and the embedding dimension is chosen to be when the correlation dimension saturates. We also can use Cao’s algorithm to evaluate the embedding dimension [28]. We define the following quantity:
aði; mÞ ¼
Y i ðm þ 1Þ Y nði;mÞ ðm þ 1Þ ; Y i ðmÞ Y nði;mÞ ðmÞ
i 2 f1; 2; . . . ; n msg;
ð4Þ
where kk is the maximum norm and Yi(m + 1) is the ith reconstructed vector with embedding dimension m + 1. (1 6 n(i, m) 6 n ms) is an integer such that Yn(i,m)(m) is the nearest neighbour of Yi(m) in the m -dimensional reconstructed space. Given the function:
EðmÞ ¼
nm Xs 1 aði; mÞ: n ms i¼1
ð5Þ
We evaluate its variation from m to m + 1:
E1ðmÞ ¼ Eðm þ 1Þ=EðmÞ
ð6Þ
and consider as an estimate of the embedding dimension the first value m at which the above function is approximately constant.
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2.3. Determining correlation dimension A deterministic chaotic system usually converges to a strange attractor with non-integer dimensions after a long time evolution. The strange attractor has fractal geometric characteristics. The dimension is one of the most important parameters in characterizing a chaotic attractor. It represents the complexity of a nonlinear system quantitatively. The bigger the dimension is, the more complexity the system is. Among many definitions of dimension, correlation dimension is widely used due to its relative simplicity, fast convergent rate in numerical calculation and effectiveness in describing the complexity of the system. The method of estimating the correlation dimension D2 from a limited time series was first put forward by Berliner [20] and Grassberger et al. [29]. It is the most commonly employed method. Suppose we have a scalar time series xi (i = 1, 2, . . . , n) of a dynamical variable sampled at an equal time interval Dt from which the N vectors Yj (j = 1, 2, . . . , N) in the m-dimensional phase space can be reconstructed using the embedding delay technique. Then the correlation dimension D2 is defined and calculated as:
D2 ¼ lim r!0
ln C m ðrÞ ; ln r
ð7Þ
where Cm(r) is known as the correlation integral and can be computed as:
C m ðrÞ lim
N!1
N N X X 2 Hðr kY i Y j kÞ; NðN 1Þ i¼1 j¼iþ1
ð8Þ
where close pairs are counted via the Heaviside function, H (x), which is equal to zero for negative argument and one otherwise. r is an arbitrary distance measure, the correlation distance kYi Yjk is the distance between the vectors Yi and Yj. In our study, for computational efficiency, the distance between two points is defined as:
kY i Y j k ¼ max jxiþls xjþls j : 0 6 l 6 m 1 :
ð9Þ
Thus, the correlation dimension formula for finite time series is as follows:
D2 ¼ lim r!0
ln C m ðrÞ ln r
ð10Þ
and using the calculation method of local slope:
D2 ¼
d ln C m ðrÞ : d ln r
ð11Þ
In some scale, the curve of the local slope is relatively smooth. This interval is called scaling region. We outline the procedure of calculating the correlation dimension from time series as follows: Plot the local slope D2 ¼ d lnd ClnmrðrÞ versus lnr for each m. When m greater than some value, a plateau is will arise, and D2 will converge asymptotically to the real correlation dimension of the system. The value is the real correlation dimension of the system. For a chaotic attractor, the correlation dimension D2 is non-integer, the value of which determines whether the system is low- or high-dimensional. In the above method, the size of data set is required greater than 10D2 =2 , i.e. n > 10D2 =2 [21]. 2.4. Lyapunov exponent Lyapunov exponents are defined as the logarithms of the absolute value of the eigenvalues of the linearized dynamics averaged over the attractor. It gives an additional feature that describes deterministic processes. These exponents measure the exponential divergence or convergence of nearby orbits. Points close together in phase space are nearly identical states; points with separating orbits become unaligned with each other. The rate of separation (becoming unaligned) expresses the sensitivity of the system’s dynamic behaviour to small differences in their initial states. The sensitivity to initial conditions is characteristic of the chaotic behaviour because it leads to the exponential divergence of nearby orbits. The exponential divergence of the initially nearby orbits is measured by at least one positive Lyapunov exponent, which can be used to characterize a chaotic system quantitatively [25,30]. The Lyapunov exponent measure how predictable or unpredictable the system is. The positive Lyapunov exponent is necessary conditions for deterministic chaos but not sufficient. The small data sets algorithm [23] is easy to implement, and robust to changes in the following quantities: embedding dimension, size of data set, reconstruction delay, and noise level. We use this approach to calculate the largest Lyapunov exponent. We summarize in the following small data sets algorithm for a time series. Step Step Step Step
1 2 3 4
: Calculating mean period P can be accomplished using the fast Fourier transform (FFT). : Calculating correlation dimension and determining embedding dimension using Takens’ theorem, i.e. m P 2D2 + 1. : Reconstructing the phase space. : After reconstructing the dynamics system, the algorithm locates the nearest neighbor of each point on the trajectory. The nearest neighbor, Y^j , is found by searching for the point that minimizes the distance to the particular reference point, Yj. This is expressed as:
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dj ð0Þ ¼ min kY j Y^j k; jj ^jj > P: ^j
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ð12Þ
Step 5 : The jth pair of nearest neighbors diverges at a rate approximately given by the largest Lyapunov exponent:
dj ðiÞ dj ð0Þek1 i :
ð13Þ
Taking the logarithm of both sides of Eq. (13) gives
ln dj ðiÞ ln dj ð0Þ þ k1 i:
ð14Þ
Eq. (14) represents a set of approximately parallel lines, each with a slope roughly proportional to k1 Step 6 : The largest Lyapunov exponent is estimated using a least-squares fit, with a constant, to the average line defined by:
yðiÞ ¼
1 < ln dj ðiÞ >; Dt
ð15Þ
where < > denotes the average over all values of j. 2.5. Principal components analysis Principal component analysis (also known as empirical orthogonal function analysis) is widely used to condense the amount of data and to extract important features or patterns from the data. PCA is another method for detecting chaos from time series. According to Takens’ embedding theorem [26,29], we can use time series xi (i = 1, 2, . . . , n) to construct a trajectory matrix YNm as well as the covariance matrix YTY. We can compute the covariance matrix YTY and the corresponding the eigenvectors and eigenvalues. The eigenvalues is also called singular values. Because the eigenvalues of a symmetric, nonnegative definite matrix are all real and nonnegative, we can arrange the eigenvalues in decreasing order. By comparing the values of eigenvalues to the total sum of eigenvalues, we can get an idea how much weight is concentrated along the particular eigenvector. 3. Analysis, results and discussion 3.1. Data set used We use three traces to verify the proposed model. One trace (t030801) was collected from the WLAN at the Network Research Laboratory at Tianjin University. They were conducted on an AP at the Network Research Laboratory, which carried all WLAN traffic to access the Internet. Another one (final.anon) is from the Mobile Computing Group at Stanford University [31]. Their wireless network is a WaveLAN network with WavePoint II access points acting as bridges between the wireless and wired networks. Ref. [11] describes the network, tracing methodology, and the characteristics of the trace of wireless traffic in detail. The last one is the trace (trace.pcap) of the intranet traffic on the ACM SIGCOMM’01 conference held in the U.C. San Diego in August 2001 [32]. The trace consists of two parts. The first part is a record of performance monitoring data sampled from wireless APs serving the conference. And the second part consists of anonymous packet headers of all wireless traffic. Both parts of the trace spanned the three days of the conference, capturing the workload of 300,000 flows from 195 users. The auditorium was covered by four ORiNOCO™ AP-1000 wireless APs. The subnet of APs was connected to a Cisco Catalyst 2924 switch over a 100BaseT link, which connected to the venue’s intranet, then the campus gigabit backbone, and finally to the Internet. The configuration of the wireless network, methodology for trace collection and analysis was described in [10]. 3.2. Preliminary characterization of wireless traffic dynamics Fig. 1 shows the comparison of power spectrum for three wireless traffic, noise, periodic and chaotic time series (Chua’s Circuit) [33]. The power spectrum of a chaotic time series is easy to distinguish from Gaussian noise or periodic time series. The power spectrum of the noise is flat for all frequencies. The power spectrum of a period one orbit is dominated by one central peak. Chaotic time series has a broad-band power spectrum with a rich spectral structure. The broad-band nature of the chaotic power spectrum indicates the existence of a continuum of frequencies. As can be seen, the power spectrum of wireless traffic not only exhibits sharp spectral lines (or peaks) at certain frequencies but also is somewhat continuous (broadband) which are regarded as a possible indicator of chaos in the wireless traffic. 3.3. Comprehensive characterization of wireless traffic dynamics First the embedding delay of the phase space reconstruction is determined by autocorrelation function method. The correlation integrals and the dimensions for wireless traffic are computed using the Grassberger–Procaccia algorithm. In Fig. 2
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Fig. 1. Comparison of power spectrum for wireless traffic, noise, periodic and chaotic time series.
Fig. 2. Correlation integral Cm(r) versus correlation distance r for embedding dimension m = 2, 3 . . . 17.
we present the results of the correlation integral analysis for embedding dimensions m = 2 up to 17. The relationship between lnCm(r) and lnr, shown in Fig. 2, indicates clear scaling regions for all the embedding dimensions used, allowing fairly accurate estimates of the correlation dimensions. The correlation dimensions, D2 against the corresponding m values are presented in Fig. 3. Fig. 3 shows an increase in the correlation dimension with the embedding dimension up to a certain point, and saturation beyond this point. Such saturation may be an indication of the deterministic dynamics in the wireless traffic.
8 final.anon trace.pcap
D2
6
t030801
4
2
0
0
5
m
10
15
Fig. 3. Correlation dimension D2 versus embedding dimension m.
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The saturation value of the correlation dimension is about 4.87119 (when m = 10), 2.23556 (when m = 9), and 2.7707 (when m = 10), respectively. The obtained low correlation dimension seems to suggest the possible presence of chaotic behaviour in the wireless traffic. In order to confirm our estimation we estimated the largest Lyapunov exponent. Fig. 4 shows a typical plot of y(i) versus iDt. There is a long linear region that can be used to extract the largest Lyapunov exponent. We obtain that the largest Lyapunov exponent of wireless traffic trace is around 0.0979, 0.1522, and 0.1339. respectively. These positive value indicates the existence of chaotic behaviour and is consistent with the estimation of a low-dimensional attractor from the correlation dimension method. Fig. 5 shows the PCA results for the three wireless traffic traces and Gauss white noise. For noise (which is generally regarded as Gauss white noise with mean value 0 and variance 1 in practical systems), synplectic geometry spectrums of this noise give the even distribution of its total energy. We can see that the intrinsic information of the analyzed trace is accumulated in the first and few lower-index components. Thus, it further shows that the wireless traffic exhibits intrinsic nonlinear characteristics. From Fig. 5, we can see that the reconstructed phase space can be captured mainly by 4- or 5dimensional data. This indicates that the majority of network dynamics can be expressed by a system with 4 or 5 state variables. 3.4. Parameter influence on diagnosis of chaotic nature of wireless traffic The choice of embedding delay and embedding dimension is very important to the phase space reconstruction of any chaotic time series. The further question we need to answer here is whether these parameter selections have influence on diagnosis of chaotic nature of wireless traffic. Firstly, the proper embedding delay can be determined by using the mutual information method. In Fig. 6 is represented the graphics of mutual information for wireless traffic respectively. As can be seen from Fig. 7 the first minimum of these curve are obviously at an embedding delay of 8, 8 and 4. We determine embedding dimension using Takens’ theorem and reconstruct phase space. Based on forenamed methods, the saturation value of the correlation dimension is about 4.1688, 3.0868 and 2.7194, respectively, and the largest Lyapunov exponent of wireless traffic traces is 0.0967, 0.1236 and 0.1345 respectively. In despite of values are different from that of above method, these results also indicate that wireless traffic exhibited chaotic. Secondly, the embedding dimension is evaluated by Cao’s algorithm. In Fig. 7, we can see the plot of the Cao’s number E1 versus its embedding dimension m. We determine m using Cao’s algorithm, yielding m = 12, 9, 12 for the t030801, final.anon, and trace.pcap, respectively. According to the forenamed methods, we evaluate the embedding delay and calculate the largest Lyapunov exponent. Lyapunov exponent of wireless traffic trace is the value is 0.0938, 0.1138, and 0.0923, respectively. These positive values indicate the existence of chaotic behaviour. The above results show that these parameter selections influence the value of the correlation dimension and the largest Lyapunov exponent, but can not influence on diagnosis of chaotic nature of wireless traffic. 4. Conclusion and future work In this work, the several real traffic traces collected from actual WLAN, was examined for the existence of low-dimensional chaos. First the power-law decay of the Fourier power spectrum suggested the possible existence of a low-dimensional attractor. We then estimated the correlation dimension by Grassberger–Procaccia algorithm. It is found out that the correlation dimension is non-integer number. We used the small data sets algorithm to calculate the largest Lyapunov exponent. It appeared that the largest Lyapunov exponent is positive. The principal components analysis showed that the intrinsic information of the traffic is accumulated in the first and few lower-index components. All those results indicated that the wireless traffic is a low dimensional chaotic system. Our results also showed that the selection of parameters for reconstruction phase space influenced the value of the correlation dimension and the largest Lyapunov exponent, but could not influence on diagnosis of chaotic nature of wireless traffic.
Fig. 4. The largest Lyapunov exponent of wireless traffic.
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Fig. 5. The principal components of wireless traffic.
Fig. 6. Estimate of embedding delay by average mutual information.
Fig. 7. Estimate of embedding dimension by Cao’s algorithm.
The shared medium wireless networks have more vulnerabilities and bandwidth and latency constrains than their wired counterparts. The traffic characteristics of wireless and wire network are different. This paper describes an in-depth study on the characteristics of WLAN traffic. Network traffic prediction on small time scale is necessary for congestion control, admission control, bandwidth allocation, and buffer management. This is very important for bandwidth constrained wireless networks. Although we have done some research on WLAN traffic prediction [34], after understanding the characteristics of wireless traffic, we plan to provide an in-depth study on the performance of different prediction models with various trace, seek a perfect prediction models for wireless traffic, and investigate prediction in various time-scales. This is our future work. Acknowledgments This research was supported in part by National Natural Science Foundation of China (NSFC) under Grant No. 61072063, Science and Research Project of the Education Department of Gansu Province (0901-03), and NWNU-KJCXGC-03-52. References [1] IEEE, IEEE Std. 802.11-Wireless LAN Medium Access Control (MAC) and Physical Layer (PHY). Specifications, 1999. [2] Leland W, Taqqu M, Willinger W, Wilson DV. Self-similarity in high-speed packet traffic: analysis and modeling of Ethernet traffic measurements. Stat Sci 1995;10:67–85.
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