Study Of Packet Traffic Fluctuations Near Phase Transition Point From Free Flow To Congestion In Data Network Model Anna T. Lawniczak

Pietro P. Lio`

Dept. of Mathematics & Statistics University of Guelph Guelph, Ont., N1G 2W1, Canada email: [email protected]

Dept. of Computer Laboratory University of Cambridge Cambridge, UK email: [email protected]

Shengkun Xie

Jiaying Xu

Dept. of Mathematics & Statistics University of Guelph Guelph, Ont., N1G 2W1, Canada email: [email protected]

Dept. of Mathematics & Statistics University of Guelph Guelph, Ont., N1G 2W1, Canada email: [email protected]

Abstract—Phase transition phenomena between non-congested and congested phases in packet traffic have been observed in many packet switching networks (PSNs). Using the PSN model we investigate the nature of fluctuations in number of packets in transit from their source to their destination, when the mean flow density into the PSN model is close to the phase transition point. A meaningful parameter of PSN behaviour near this critical point is the Hurst exponent that when larger than 0.5 is revealing of a long memory process, i.e. a fractional Brownian motion. In this paper we have used Hurst exponents and long range dependence to analyse PSN model behaviour. We have found that the DFA analysis and several methods for estimating the Hurst exponent suggest the presence of a long memory process for the PSN model using adaptive routing. However, we have not observed this in the case of static routing for the same type of incoming traffic. Thus, the packet traffic is more correlated in PSN model with adaptive routing than the static one. We present our finding, outline the work underway and discuss its expansion. Keywords- OSI Network Layer; packet traffi; Hurst exponent; LRD

I. INTRODUCTION Phase transition phenomena between non-congested and congested states in packet traffic have been observed in empirical studies of data networks of packet switching type and motivated further research (e.g., [1], [2], [3], [4], [5], [6], [7], [8], [9]). The control parameter of the phase transition point is the mean flow density of packets into the packet switching network (PSN). The most famous example of PSN is the Internet. Simulations and observations show that the transmission efficiency becomes the highest at the critical point (e.g., [2], [3], [4], [5], [6], [7]), however the fluctuations become large. Thus, better understanding of these phenomena can lead to the improvement of traffic efficiency and its control in PSNs. We investigate the nature of fluctuations in one of the network performance indicators, i.e. in the number of packets

in transit from their sources to their destinations, in a PSN model ([7], [16]) when the mean flow density of packets into the PSN model is closed to the phase transition point. The PSN model is an abstraction of the Network Layer of the OSI Reference Model. In PSN the Network Layer is responsible for routing packets across the network and for control of congestion, thus, it is the most important layer for our purposes. In our study we consider the minimum hop static routing algorithm and different types of dynamic (adaptive) routing algorithms in which routers cooperate to avoid congested areas of the network. In adaptive routings the routers try to minimize delivery times of packets from their sources to their destinations. For adaptive routings coupled with network connection topology isomorphic to square lattice was observed synchronization in packet traffic dynamics in congested states of the PSN model (e.g., [8], [9], [10]). Such synchronization was not observed for other considered couplings in these works. We investigate the effects of coupling of network connection topology with routing algorithms on the nature of fluctuations in number of packets in transit near phase transition point. In this paper we estimate long range correlations and make use of Hurst exponent to characterize PSN model time series. Estimating the Hurst exponent for a data set provides a measure of whether the data is a pure random walk or has underlying trends. Another way to state this is that a random process with an underlying trend has some degree of autocorrelation. When the autocorrelation has a very long decay this process is sometimes referred to as a long memory process. It allows estimating statistical self-similarity which is related to having infinite self-similar data sets, where any section of the data set would have the same statistical properties as any other. These topics in the context of data networks were investigated by many researchers (e.g., [6] and references there in, [11], [12], [13], [14], [15]).

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.

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We present selected results of our findings particularly those related to the influence of the cooperation of the nodes in adaptive routings, on the nature of the phenomena under investigation and discuss future expansions. II.

Lnp†(16,ONE), λc=0.085 for Lnp†(16,QS), λc= 0.085 for Lnp†(16, QSPO). 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.

PACKET SWITCHING NETWORK MODEL

We use the PSN model, developed in [7], [16], and a C++ simulator called Netzwerk-1 [7], [17]. The PSN model, like in real networks is concerned primarily with packets and their routings; it is scalable, distributed in space, and time discrete. It avoids the overhead of protocol details present in many PSN simulators designed with different aims in mind. As a case study we consider the PSN model with a network connection topology that is isomorphic to Lnp†(16), that is a two-dimensional non-periodic square lattice with 16 nodes in the horizontal and vertical directions. 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 called 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. When we want to specify what type of an edge cost function (ecf) (and additionally what value of source load λ) the PSN model set-up is using we used the following convention: Lp†(16, ecf) (Lp†(16, ecf, λ)), where ecf = ONE, or QS, or QSPO. The ecf ONE assigns a value of “one” to each edge in the lattice Lnp†(16). Thus, this results in a static routing. The ecf QS assigns to each edge in the lattice Lnp†(16) 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 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. At time k = 0, the set-up of the PSN model is initialized with empty queues and the routing tables are computed. The time-discrete, synchronous and spatially distributed PSN model algorithm consists of the sequence of five operations advancing the simulation time from k to k + 1. These operations are: (1) Update routing tables, (2) Create and route packets, (3) Process incoming queue, (4) Evaluate network state, (5) Update simulation time. The detailed description of this algorithm is provided in [7], [16]. 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 subcritical 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. Details about how we estimate the critical source load are in [7], [16]. 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=0.045 for

III.

NUMBER OF PACKETS IN TRANSIT FLUCTUATIONS ANALYSIS Number of packets in transit (NPT), N(k), is one of the networks performance indicators providing direct measure of how heavy is the network traffic. N(k) is given by the total number of packets in the network at time k, i.e. the sum of the number of packets in the out-going queue at time k at each network node. In congested states of PSN model N(k) fluctuates around an increasing trend with an increase of time (see, right figure in Figure 1 (a) and Figure 2 (a) ), in noncongested states N(k) fluctuates around some constant value after an initial transient time (see, left figure in Figure 1 (a) and Figure 2 (a) ).

(a)

(b)

(c) Figure 1. Time series plots of NPT, DNPT and INPT for the PSN model np set-up L (16, ONE, λ) ( = NS(16, ONE, λ) in the legend box) for † λ=SUBCSL in the left column and λ=SUPCSL in the right column. a) NPT time series, the vertical lines at k=2000 mark the cut off points of truncation, the red solid graphs are the sample mean (left figure) and the power trend (right figure) fitted to the data of NPT after truncating, and two red dotted graphs refer to the 95% CI band of the sample mean (left figure) and of the fitted trend (right figure). b) DNPT time series in the time window (2000, 8000]. c) INPT time series in the time window (2000, 8000]. Notice, the scale ranges on the vertical axis in the right column figures are larger than the ones in the left column figures.

After removing the deterministic trend of the NPT signals and discarding the first 2000 points, the residuals are saved and are named as DNPT signals. Their graphs are in Figure 1 (b) and Figure 2 (b) for selected PSN model set-ups.

(a)

(b)

packets routes, from their sources to their destinations, are included in the routing costs. To determine the nature of the dependency in NPT we estimated Hurst parameter using several methods, i.e. the aggregated variance (A. Var.) method, the difference aggregated variance (Diffvar.) method, the periodogram method, the discrete wavelet transform method, and their routines in R. We applied these methods to estimate Hurst parameter of the incremental NPT (INPT) time series, that is defined by N(k+1)-N(k). The estimates of the Hurst parameter are given in Table 1. The selected graphs of INPT time series describing local fluctuations in NPT time series are plotted in Figure 1 (c) and Figure 2 (c). Visually these time series look very similar. We also applied detrended fluctuation analysis (DFA) introduced by Peng et al. in [18], [19] to calculate the scaling exponents. The advantage of this method is that it provides means to detect long range dependence (LRD) for non-stationary time series. Based on the results of Table 1 and Table 2 we may conclude that for PSN model set-up with ecf ONE and source loads SUBCSL, CSL, SUPCSL we do not detect LRD in our data but we detect LRD in data of the PSN model set-ups with ecf QS or QSPO. However, this requires further investigation due to differences in the estimates. Our results confirm that for PSN model set-ups with ecf QS or QSPO correlations are much stronger than in PSN models with ecf ONE as [8], [9], [10] suggested.

(c) Figure 2. Time series plots of NPT, DNPT and INPT for the PSN model np set-up L (16, QS, λ) ( = NS(16, QS, λ) in the legend box) for λ=SUBCSL † in the left column and λ=SUPCSL in the right column. a) NPT time series, the vertical lines at k=2000 mark the cut off points of truncation, the red solid graphs are the sample mean (left figure) and the power trend (right figure) fitted to the data of NPT after truncating, and two red dotted graphs refer to the 95% CI band of the sample mean (left figure) and of the fitted trend (right figure). b) DNPT time series in the time window (2000, 8000]. c) INPT time series in the time window (2000, 8000]. Notice, the scale ranges on the vertical axis in the right column figures are larger than the ones in the left column figures.

(a)

(b) The transformed data, the DNPT signals, contain information equivalent to that of fluctuations of the NPT signals. Our simulations show that for the considered PSN model set-ups each sample autocorrelation function (sACF) is not bounded by the 95% confidence band corresponding to a white noise process. This indicates that the observations of DNPT are strongly dependent on their historical lag, see Figure 3. This dependence increases with the increase in the source load values. Furthermore, we observe that for each type of source load SUBCSL, CSL, SUPCSL for PSN model set-up using ecf QS the DNPT time series is more strongly autocorrelated than the DNPT time series for PSN model set-up using ecf ONE. The same holds true if instead of ecf QS PSN model is set-up with ecf QSPO. Thus, the autocorrelation in DNPT time series increases when the queuing (delay) times on

(c) Figure 3. Plots of sample autocorrelation functions (sACF) of DNPT signals for PSN model set-up Lnp† (16, ecf, λ) for ecf = ONE in the left column, ecf = QS in the right column, and λ=SUBCSL in (a), λ=CSL in (b), and λ=SUPCSL in (c). The two red lines in each plot show the 95% confidence interval of sACF corresponding to a white noise process.

TABLE I.

ESTIMATES OF H URST PARAMETER H USING ROUTINES IN R np

L † (16, ecf, λ ) ONE, SUBCSL ONE, CSL ONE, SUPCSL QS, SUBCSL QS, CSL QS, SUPCSL QSPO, SUBCSL QSPO, CSL QSPO, SUPCSL

A.Var. method 0.21 0.29 0.34 0.35 0.39 0.41 0.41 0.41 0.43

Diffvar. method 0.37 0.37 0.49 0.47 0.57 0.48 0.53 0.51 0.56

Per. method 0.45 0.47 0.47 0.49 0.51 0.50 0.51 0.48 0.48

Wavelet method 0.29 0.44 0.43 0.52 0.52 0.52 0.53 0.51 0.47

REFERENCES [1] [2] [3]

[4] [5]

[6] [7]

TABLE II.

CLASSIFICATION BASED ON SCALING EXPONENT OBTAINED BY DFA METHOD

Lnp† (16, ecf, λ ) ONE, SUBCSL ONE, CSL ONE, SUBCSL QS, SUBCSL QS, CSL QS, SUPCSL QSPO, SUBCSL QSPO, CSL QSPO, SUPCSL

IV.

DFA method 0.38 0.42 0.44 0.53 0.53 0.51 0.58 0.56 0.53

Classification SRD SRD SRD LRD LRD LRD LRD LRD LRD

CONCLUSIONS

In this paper we have used Hurst exponents and long range dependence to analyse PSN model behaviour. We have found that the DFA analysis and most of the methods for estimating the Hurst exponent (A.Var, Diffvar, Per) suggest the presence of a certain quantity of fractional Brownian motion (H>0.5) for PSN with ecf QS or QSPO, when the incoming traffic was generated by random variables of Bernoulli type. An immediate interpretation is that packet traffic in PSN model set-ups with ecf QS or QSPO is more correlated than in PSN models with ecf ONE, because queuing times at the routers are part of the transmission cost. Routing decisions are defined in such a way that routing costs are minimized. We intend to study how the autocorrelation in the PSN model will change if other types of incoming traffic are considered. ACKNOWLEDGMENT The authors acknowledge the prior work of A. Gerisch, X. Tang, and B. N. Di Stefano with A.T.Lawniczak. The authors thank B. Di Stefano and X. Tang for helpful discussions.

[8]

[9]

[10]

[11]

[12]

[13]

[14]

[15]

[16]

[17]

[18]

[19]

A.Y. Tretyakov, H. Takayasu, M. Takayasu, “Phase Transition in a Computer Network Model, Physica A, 253, pp. 315-322, 1998. T. Ohira, R. Sawatari, “Phase Transition in a Computer Network Traffic Model”, Phys. Rev. E 58, pp. 193-195, 1998. H. Fukś and A.T. Lawniczak, “Performance of Data Networks with Random Links”, Mathematics and Computers in Simulation 1726, pp. 117, 1999. R.V. Solè and S. Valverde, “Information Transfer and Phase Transition in a Model of Internet Traffic”, Physica A 289, pp. 595-605, 2001. M. Woolf, D.K. Arrowsmith, R.J. Mondragόn and and J.M. Pitts, “Optimization and Transition in a Chaotic Model of Data Traffic”, Physical Rev. E 66, p. 056106, 2002. L. Kocarev, G. Vattay, Eds., Complex Dynamics in Communication Networks. New York, Springer-Verlag, 2005. A.T. Lawniczak, A. Gerisch, and 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. 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. 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. A.T. Lawniczak and K.P. Maxie, A. Gerisch, “From individual to collective behaviour in CA like models of data communication networks”, Springer-Verlag, LNCS 3305, pp. 325-334, 2004. M.W. Garrett and W. Willinger, “Analysis, Modeling and Generation of Self-similar Video Traffic”, Proc. of the ACM SIGCOMM’94, London, UK, 269-280. W. Willinger, W. Taqqu, M.S. Sherman, and D. Wilson, “Self-similarity Through High Variability: Statistical Analysis of Ethernet LAN Traffic at the Source Level”, IEEE?ACM Transactions on Networking 5, pp.7186, 1997. M.E. Crovella and A. Bestavros, “Self-similarity in World Wide Web Traffic Evidence and possible Causes”, Proc. of the ACM SIGMETRICS 96, pp. 160169, Philadelphia, PA, 1996. A. Feldman, A. Gilbert, and W. Willinger, Data Networks as Cascades: Investigating the Multifractal Nauture of internet WAN Traffic, Comp. Communication Rev., Proc. of the ACM/SICCOMM’98, 28, 42-55, 1998. M. Roughan, D. Veitch, “Measuring Long-Range Dependence under Changing Traffic Conditions”, IEEE INFOCOM’99, pp. 1513-1521, 1999. A.T. Lawniczak, A. Gerisch, and B. Di Stefano, “Development and Performance of Cellular Automaton Model of OSI Network Layer of Packet-Switching Networks”, Proc. IEEE CCECE 2003- CCGEI 2003, Montreal, Quebec, Canada (May/mai 2003), pp. 001-004, 2003. A. Gerisch, A.T. Lawniczak, and 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. C.-K. Peng, S.V. Buldyrev, S. Havlin, M. Simons, H.E. Stanley, and A.L. Goldberger, “Mozaic Organization of DNA Nucleatides”, Physical Review E, 49(2), pp. 1685-1689, 1994. C.-K. Peng, S. Havlin, H.E. Stanely, and A.L. Goldberger, “Quantification of Scaling Exponents and Crossover Phenomena in Nonstationary Heartbeat Time Series”, Chaos 5, pp. 82-87, 1995.