Wavelet Spectral Analysis Of Packet Traffic Near Phase Transition Point From Free Flow To

Congestion In Data Network Model

Anna T. Lawniczak Dept. of Mathematics & Statistics

University of Guelph Guelph, Ont., N1G 2W1, Canada

email: alawnicz@uoguelph.ca

Shengkun Xie Dept. of Mathematics & Statistics

University of Guelph Guelph, Ont., N1G 2W1, Canada

email: sxie@uoguelph.ca

Pietro Lio` Computer Laboratory

University of Cambridge Cambridge, UK

email: pl219@cam.ac.uk

Jiaying Xu Dept. of Mathematics & Statistics

University of Guelph Guelph, Ont., N1G 2W1, Canada

email: jiaying@uoguelph.ca

Abstract —We have applied wavelet statistics to study packet traffic in a data communication network model of the packet switching type. Our focus is on the study of spectral properties of packet traffic near phase transition point (critical point) from free flow to congested states of the network model. We characterize the critical point by the level of packets production at sources in the packet switching network (PSN) model. In our model we consider different dynamic routing cost metrics (i.e., when the costs of transmission of packets from one router to another incorporate the information about how congested the routers are) and static routing cost metric (i.e., when the cost of transmission of packets from one router to another is constant over time). Using wavelets we study spectral properties of number of packets in transit from their sources to their destinations in PSN model for various routing algorithms and network connection topologies when source loads are close to the critical ones. We show that the wavelet power spectra is an important estimator of the changes occurring with the increase of source load values from sub-critical, through critical to super-critical ones and it is dependent on the routing algorithm.

Keywords- OSI Network Layer; continuous wavelet transform; wavelet spectra

I. INTRODUCTION A Packet Switching Network (PSN) is a data

communication network consisting of a number of nodes (i.e., routers and hosts) that are interconnected by communication links. Its purpose is to transmit messages from their sources to their destinations. In a PSN each message is partitioned into smaller units of information called packets that are capsules carrying the information payload. Packets are transmitted across a network from source to destination via different routes. Upon arrival of all packets to the destination, the message is rebuilt. PSNs owe their name to the fact that packets are

individually switched among routers; see [1], and the references therein. Some examples of PSNs are the Internet, wide area networks (WANs), local area networks (LANs), wireless communication systems, ad-hoc networks, and sensors networks. Wired PSNs are described by the ISO (International Standard Organization) OSI (Open Systems Interconnect) 7 layers Reference Model [1].

We focus on the Network Layer because it plays the most important role in the packet traffic dynamics of the network. It is responsible for routing packets from their sources to their destinations and for control of congestion. Phase transition phenomena between non-congested and congested phases in packet traffic have been observed in empirical studies of data networks of packet switching type and motivated further research (e.g., [2], [3], [4], [5], [6]). The control parameter of the phase transition point is the mean flow density of packets into the PSN. The understanding of the nature of the packet traffic near the critical point is important, because observations of data traffic in real networks and in simulations show that throughput is highest at the critical point. Thus, improved understanding may lead to improvements in optimization of network performance. We investigate the nature of fluctuations in number of packets in transit (NPT) from their sources to their destinations in PSN model when the mean flow density of packets into PSN model is closed to the phase transition point. In our model we consider different dynamic routing cost metrics (i.e., when the costs of transmission of packets from one router to another incorporate the information about how congested the routers are) and static routing cost metric (i.e., when the cost of transmission of packets from one router to another is constant over time).

The dynamics of flow of packets in PSN can be very complex. It is not well understood how these dynamics depend on network connection topology coupled with various routing

A.T.L. acknowledges partial financial support from Sharcnet and NSERC of Canada, P.L. from the Univ. of Cambridge, UK and the Royal Soc., UK, S. Xie from the Univ. of Guelph, and J.X. from Sharcnet and the Univ. of Guelph. The authors thank B. Di Stefano and X. Tang for helpful discussions

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algorithms. Therefore state of art methods of signal analysis need to be implemented. Wavelet analysis is becoming a common tool for analyzing localized variations of power within a time series. Using wavelets we study spectral properties of number of packets in transit from their sources to their destinations in PSN model for various routing algorithms and network connection topologies when source loads are close to the critical ones. For a given PSN model set up, the spectral properties of packets in transit change with the increase of source load values from sub-critical, through critical to super-critical ones and they are dependent on the routing algorithm. In the next section we provide a justification for the appropriateness of using wavelets in our study.

II. WAVELETS STATISTICAL INFERENCE OF NETWORKS In short, a wavelet is an oscillation that decays quickly and

that can be used as type of building block for constructing functions. More precisely wavelets form bases for function spaces such as L2(R). Wavelets differ from Fourier methods in that they allow the localization of a signal in both time and frequency. Wavelet transform typically outperforms Fourier transform when the signal under consideration contains discontinuities and sharp spikes. A wavelet transform leads to an additive decomposition of a signal into a series of different components describing smooth and rough features of the signal. For sake of space, we refer the reader to [8], [9], [10], and [11] for a general statistical introduction to wavelets. In this paper, only the continuous wavelet transform (CWT) is used, although comparable results have been obtained using the DWT.

Recent theoretical and applied works have demonstrated the appropriateness of wavelets for the analysis of signals containing non-stationarity, unsteadiness and non-Markovian features, see [12] and [13]. The key property of the wavelet transform that makes it useful for studying non-stationarity is that a time series is broken down into coefficients that reflect changes at various scales and at particular times. For non-stationary processes, the wavelet coefficients for a given scale are approximately uncorrelated. The consideration that non-stationarity and unsteadiness are common peculiarities of networks systems has suggested us to explore wavelet statistical inference for PSN analysis.

III. OUR PSN MODEL FRAMEWORK We use the PSN model, developed in [4], and a C++

simulator called Netzwerk-1 [4], [7]. The PSN model, like in real networks is concerned primarily with packets and their routings; it is scalable, distributed in space, and time discrete.

We use the following naming convention for various considered PSN model set-ups. We denote the network model set-up parameters by Lα (L, ecf, λ), where α = p (periodic) or np (non-periodic) stands for periodicity of network connection topology L isomorphic to two-dimensional square lattice with L nodes in the horizontal and vertical directions, the case considered here. The parameter λ stands for the source load value. The parameter ecf represents an edge cost function. The PSN model set-up is using ecf = ONE, or QS, or

QSPO; the ecf ONE assigns a value of “one” to each edge in the lattice L. Thus, this results in a static routing. The ecf QS assigns to each edge in the lattice L a value equal to the length of the outgoing queue at the node from which the edge originates. The ecf QSPO assigns a value that is the sum of the ecfs ONE and QS. The routing decisions made using ecf QS or QSPO imply adaptive or dynamic routing where packets have the ability to avoid congested nodes during the PSN model simulation. In the PSN model each node performs the functions of a host and a router and maintains one incoming and one outgoing queue which is of unlimited length and operates according to a first-in, first-out policy. At each node, independently of the other nodes, packets are created randomly with probability λ corresponding to the source load. In the PSN model all messages are restricted to one packet carrying only the following information: time of creation, destination address, and number of hops taken.

In the PSN model time is discrete and we observe its state at the discrete times k = 0, 1, 2, …, T, where T is the final simulation time. In this paper, T=8000. The set-up of the PSN model is initialized with empty queues and the routing tables are computed. At each simulation step, the time-discrete, synchronous and spatially distributed PSN model algorithm consists of the sequence of five operations: (1) Update routing tables, (2) Create and route packets, (3) Process incoming queue, (4) Evaluate network state, (5) Update simulation time; see [4] for details.

In the PSN model, for each family of network set-ups, which differ only in the value of the source load λ, values of λsub-c for which packet traffic is congestion-free are called sub-critical source loads, while values λsup-c for which traffic is congested are called super-critical source loads. The critical source load λc is the largest sub-critical source load. It is the maximum source load, at which the PSN network traffic is free from congestion. Thus, λc is a phase transition point from free flow to congested state of a network. For the PSN model set-ups considered here the estimated critical source load (CSL) values are, respectively, λc=0.115 for Lp (16,ONE), λc=0.120 for Lp (16,QS), λc= 0.120 for Lp (16, QSPO), λc=0.045 for Lnp (16,ONE). In what follows, we will discuss the simulation results for source load values SUBCSL=CSL-0.005, CSL and SUPCSL=CSL+0.005, thus, near phase transition point from free flow to congested state of the network.

IV. RESULTS AND DISCUSSIONS This article proposes the use of wavelets to represent an

explanatory time series of PSN with different traffic load, topology, and routing characteristics. Figure 1 shows the time plots of the NPT signals for the PSN model set-up Lp (16, ONE, λ) with different source load values. The figure shows a remarkable difference between the NPT graphs for SUPCSL (red line) and SUBCSL (blue line), CSL (green line). The differences between the NPT graphs for SUPCSL, and SUBCSL and CSL are even larger for the PSN model set-ups with ecf=QS and QSPO (not shown).

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Figure 1The time plots of the NPT signals for the PSN model set-ups Lp (16, ONE, λ)where λ=SUBCSL (blue graphs), CSL (green graphs), or SUPCSL

(red graphs) in the time window (0, 8000].

After removing the deterministic trend of the NPT signals and discarding the first 2000 points, the residuals are saved and renamed as DNPT signals. We have carried out exploratory analysis of different detrend functions (not shown). Figure 2a,b demonstrate how the DNTP is a better signal than NTP for the PSN model set-up Lp (16, ONE, SUBCSL).

(a)

(b)

Figure 2. Time series plots of NPT and DNPT for the PSN model set-up Lp

(16, ONE, SUBCSL); a) NPT time series, the vertical line at k=2000 marks the cut off point of truncation, the red solid line is the sample mean of NPT after truncating, and two red dotted lines refer to the 95% CI band of the sample mean b) DNPT time series

We have carried out a CWT comparative analysis of the static and adaptive routing with respect to different network load and topology. We exemplify in the four plots of Figure 3

the CWT methodology we have used. The CWT results in an information-rich graph (plot a and d) which allows to easily detect differences between different PSN model set-ups but may result difficult for quick comparative analyses. In this work, in order to reduce the dimensionality of the information, we simply focus on the global and scaled wavelet power spectra (plot b and c). For sake of space, we show in Figure 4 (plots a-k) the scaled wavelet spectra of few of the complete combinations of PSN model parameterization that we have considered: a) Lp (16, ONE, CSL), b) Lp (16, ONE, SUPCSL), c) Lp (16, QS, SUBCSL), d) Lp (16, QS, CSL), e) Lp (16, QS, SUPCSL), f) Lp (16, QSPO, SUBCSL), g) Lp (16, QSPO, CSL), h) Lp (16, QSPO, SUPCSL), j) Lnp (16, ONE, SUBCSL) k) Lnp (16, ONE, CSL). Our exhaustive analysis has shown that for a given PSN model set-up, the spectral properties of number of packets in transit change with the increase of source load values from sub-critical, through critical to super-critical ones and they are dependent on the routing algorithm.

(a) (b)

(c) (d)

Figure 3. CWT implementations of DNPT for the PSN model set-up Lp (16, ONE, SUBCSL). a) The filled wavelet power spectra contour plot. The cone shaped curve corresponds to the cone of influence (COI). The colormap shows log2(power) from dark blue (lowest value) to dark red (highest value). b) Global wavelet power spectra, c) Scaled wavelet power spectra corresponding to local maximum of global wavelet power spectra within cone of influence (COI), d) The filled contour plot for real part of the wavelet complex coefficients. The colormap shows log2(real(wave)) from dark blue (lowest value) to dark red (highest value).

V. CONCLUSIONS AND FUTURE DIRECTIONS In our work we explore packet traffic dynamics near phase

transition point from free flow to congestion. We study how this dynamics is affected by the coupling of network connection topology with routing algorithms. We have shown that wavelets power spectra are important estimators of the desirable characteristics of a PSN. Clearly wavelet-based statistics opens a new direction in the analysis of networks signals.

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(a) (b) (c) (d)

(e) (f)

(g) (h)

(j) (k)

Figure 4. Scaled wavelet power spectra for different PSN models (see text): a) Lp (16, ONE, CSL). b) Lp (16, ONE, SUPCSL), c) Lp (16, QS, SUBCSL), d) Lp (16, QS, CSL); e) Lp (16, QS, SUPCSL); f) Lp (16, QSPO, SUBCSL); g) Lp (16, QSPO, CSL); h) Lp (16, QSPO, SUPCSL); j) Lnp (16, ONE, SUBCSL) k) Lnp (16, ONE, CSL).

Future work will focus on multiscale properties of wavelet analysis, and on recent results on wavelet shrinkage for irregularly spaced spatial data [12]. These theoretical developments may allow investigate dynamical properties of networks generated from combinations of motifs, i.e. basic patterns of interconnections (small connected subgraphs). We believe that this approach may also lead to insights on designing modular and self-organized networks and identify hidden network communities and unstable nodes.

In conclusion, we report that wavelet statistical analysis of networks has two important benefits: the improved understanding of the topological, load and routing constraints that shape the network layer level and the introduction of new classes of statistical estimators of network congestion.

ACKNOWLEDGMENT The authors acknowledge the prior work of A. Gerisch, X.

Tang, and B. N. Di Stefano with A.T.Lawniczak.

REFERENCES [1] A. Leon-Garcia and I. Widjaja, Communication Networks. Boston,

McGraw-Hill, 2000. [2] A.Y. Tretyakov, H. Takayasu, M. Takayasu, “Phase Transition in a

Computer Network Model, Physica A, 253, pp. 315-322, 1998. [3] L. Kocarev, G. Vattay, Complex Dynamics in Communication

Networks. New York, Springer-Verlag, 2005. [4] A.T. Lawniczak, A. Gerisch, B. Di Stefano, “OSI Network-layer

Abstraction: Analysis of Simulation Dynamics and Performance Indicators”, Science of Complex Networks. J. F. Mendes, Ed., AIP Conference Proc., vol. 776, pp. 166-200, 2005.

[5] A.T. Lawniczak and X. Tang, “Packet Traffic Dynamics Near Onset of Congestion in Data Communication Network Model”, Acta Physica Polonica B, Vol. 37 (5), pp. 1579-1604, 2006.

[6] A.T. Lawniczak and X. Tang, “Network Traffic Behaviour near Phase Transition Point”, The European Physical Journal B - Condensed Matter, 50 (1-2), 231-236, 2006.

[7] A. Gerisch, A.T. Lawniczak, B. Di Stefano, “Building Blocks of a Simulation Environment of the OSI Network Layer of Packet Switching Networks”, Proc. IEEE CCECE 2003-CCGEI 2003, Montreal, Quebec, Canada (May/mai 2003), pp. 001-004, 2003.

[8] B. Vidakovic, Statistical Modeling by Wavelets. New York: John Wiley, 1999.

[9] A. Bruce, & H.-Y. Gao, Applied Wavelet Analysis with S-Plus. New York, Springer-Verlag, 1996.

[10] C. K.Chui, An Introduction to Wavelets. New York, Academic Press, 1992

[11] I. Daubechies, Ten Lectures on Wavelets. Philadelphia, SIAM, 1992. [12] G. P. Nason and T. Sapatinas, “Wavelet packet transfer function

modelling of nonstationary time series”, Statistics and Computing 12: 45-56, 2002.

[13] A. Serroukh, A. T. Walden and D. B. Percival, “Statistical Properties and Uses of the Wavelet Variance Estimator for the Scale Analysis of Time Series”, Journal of the American Statistical Association, 95, no. 449, pp 184 – 196, 2000.

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