# Predicting Traffic Flow in Local Area Networks by the Largest Lyapunov Exponent

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Fundamental Theories

#### 2.1. Phase Space Reconstructions for Time Sequences in Networks

- Suppose there is a time sequence, x(t) = (x(t
_{1}), x(t_{2}), …, x(t_{n})), here x(t) is the sample, n the number of samples, t the time and Δt the sampling interval. The Euclidean subspace with m-dimensions can be constructed by the time sequence as a proper time lag τ = kΔt is chosen, and here k is a positive integer. The first point of the m-dimensional subspace includes m values, in which x(t_{1}) is the first element of the vector, x(t_{1}+ τ) is the next one, and the last one is x(t_{1}+ (m − 1)τ), so the first point in vector form in the phase space is defined as follows:$${Y}_{1}=(x({t}_{1}),\text{}x({t}_{1}+\tau ),\cdots ,\text{}x({t}_{1}+(m-1)\tau ))$$ - Then let x(t
_{2}) be the first element in the vector, the second point in the m-dimensional subspace can be obtained further using the same method, that is:$${Y}_{2}=(x({t}_{2}),\text{}x({t}_{2}+\tau ),\cdots ,\text{}x({t}_{2}+(m-1)\tau ))$$ - The number of points in m-dimensional subspace is N = n − (m − 1)k, and all the points of the subspace are:$$\begin{array}{c}{Y}_{1}=(x({t}_{1}),\text{}x({t}_{1}+\tau ),\cdots ,\text{}x({t}_{1}+(m-1)\tau ))\hfill \\ {Y}_{2}=(x({t}_{2}),\text{}x({t}_{2}+\tau ),\cdots ,\text{}x({t}_{2}+(m-1)\tau ))\hfill \\ \cdots \cdots \hfill \\ {Y}_{N}=(x({t}_{n}-(m-1)\tau ),\text{}x({t}_{n}-(m-2)\tau ),\cdots ,\text{}x({t}_{n}))\hfill \end{array}$$
_{i}can be considered as a point in the reconstructed phase space, including m elements.

_{i}. Obviously, there are N points in m-dimensional phase space, and the lines connecting the phase points can describe the trajectory. In this study, C-C method is used to reconstruct the phase space.

#### 2.2. C-C Method

_{w}, which is obtained by experiments and satisfies τ

_{w}≥ τ

_{p}, where τ

_{p}is the average trajectory period and can be obtained based on the spectra analysis of the time series [13]. Moreover, the relationship between τ

_{w}and τ can be expressed by:

_{w}and τ by the correlation integral method, and then m could be obtained following Equation (1).

#### 2.3. Physical Meaning of the Largest Lyapunov Exponent in Network

- As λ > 0, two trajectories diverge rapidly in phase space, and the long time behaviors of the dynamic system are sensitive to the initial conditions, implying the system is in a chaotic state.
- As λ = 0, two trajectories will not diverge or converge implying there is no chaos.
- As λ < 0, two trajectories in phase space converge, and the long time behaviors of dynamic system are not sensitive to the initial conditions, implying there is no chaos and the system is stable.

_{1}can quantitatively describe the divergence rate of two adjacent trajectories in phase space, that is, it can measure the butterfly-effect, which is the chaotic behavior in time sequences. The butterfly-effect is a visual representation for randomness and uncertainness in dynamics, so λ

_{1}can be used as the quantitative index for the system. Due to the butterfly-effect of the chaotic system, it is difficult to predict the long-term behaviors by λ

_{1}. However, λ

_{1}can be used to predict short-term behaviors. Moreover, the greater the value of λ

_{1}is, the shorter the time is needed for prediction due to the butterfly-effect [15,16].

#### 2.4. Small Data Sets Method

_{1}in this study. First, in the reconstructed phase space, the closest adjacent points on the certain trajectories at initial condition should be found. With a separating limit in short distance, it can be written as:

_{k}is one point in the neighbor of Y

_{j}, and P is the average period of the time sequence.

_{j}(i) is the distance between the closest adjacent pair points j after i steps on the certain trajectories. Then, the largest Lyapunov exponent can be estimated by analyzing the average divergent rate of the closest adjacent points on the certain trajectories:

_{1}(i), for every i, the average value of all lnd

_{j}(i) is calculated, that is:

_{j}(i) which is not zero. The slope of the curve governed by Equation (5) is the largest Lyapunov exponent λ

_{1}, and can be computed numerically by the least-squares approximate method [19].

#### 2.5. Longest Predictable Duration

_{0}. For a chaotic system, the largest Lyapunov exponent λ

_{1}describes the average divergent distance between Y

_{j+n}and Y

_{k+n}, which evolve after n iterations from the two adjacent phase points Y

_{j}and Y

_{k}, that is:

_{0}, that is:

#### 2.6. Prediction Method Based on the Largest Lyaponov Exponent

_{j}in phase space will be assumed as the predicted center point, and Y

_{k}is the closest point with a distance d

_{j}(0) from Y

_{j}. Then, there exists a relation as follows:

_{j}and Y

_{k}evolve into Y

_{j}

_{+1}and Y

_{k}

_{+1}, respectively. Based on the physical meaning of the largest Lyapunov exponent mentioned above, the following equation can be obtained as:

_{n}

_{+1}), the last component of the phase point Y

_{j}

_{+1}, is the only unknown and could be obtained from the following equation:

## 3. Numerical Examples and Analysis

#### 3.1. Largest Lyapunov Exponent of LAN Traffic

**Figure 1.**Real LAN traffic data in the week. (

**a**) Real network traffic; (

**b**) Average daily traffic sequence.

**Figure 2.**Real LAN traffic data on Monday. (

**a**) Real network traffic data; (

**b**) Average traffic sequence per hour.

_{w}are computed numerically and shown in Table 1. The Small data sets method is used to compute the largest Lyapunov exponent of the network traffic sequence in the working periods. The changes of y(i) versus i in working period are shown in Figure 3, and Figure 4 shows the approximate linear part of Figure 3. The slope of the approximate linear part is computed by the least square method and its value is λ

_{1}shown in Table 1.

Working Periods | 8:00–9:00 | 9:00–10:00 | 10:00–11:00 | 11:00–12:00 | 14:00–15:00 | 15:00–16:00 | 16:00–17:00 | 17:00–18:00 |
---|---|---|---|---|---|---|---|---|

Average traffic (byte/s) | 772,602 | 1,881,664 | 8,721,630 | 2,313,709 | 2,731,960 | 1,504,641 | 985,395 | 1,555,366 |

τ_{w} | 32 | 48 | 99 | 40 | 100 | 51 | 72 | 42 |

τ | 4 | 4 | 1 | 1 | 2 | 3 | 4 | 3 |

m | 9 | 13 | 100 | 41 | 51 | 18 | 19 | 15 |

λ_{1} | 0.0169 | 0.0171 | 0.0147 | 0.0386 | 0.0065 | 0.0103 | 0.0076 | 0.0178 |

#### 3.2. Prediction

#### 3.2.1. Dot Prediction

_{1}the real traffic data and d

_{2}the predicted traffic data. We have:

_{1}be the number of the network traffic data as f is less than 10%, and n

_{2}the number of the network traffic data as f is less than 30%. Then, f

_{1}and f

_{2}are defined clearly as the probability density as follows:

Date | Monday | Tuesday | Wednesday | Thursday | Friday | Saturday | Sunday |
---|---|---|---|---|---|---|---|

rate of flow (byte/s) | 1.86538 × 10^{6} | 1.40022 × 10^{6} | 7.35079 × 10^{5} | 1.39503 × 10^{6} | 1.42701 × 10^{6} | 1.80545 × 10^{5} | 1.55155 × 10^{5} |

τ | 4 | 2 | 3 | 7 | 2 | 2 | 4 |

m | 6 | 12 | 3 | 3 | 19 | 27 | 8 |

λ_{1} | 0.1848 | 0.0475 | 0.146 | 0.1562 | 0.044 | 0.1557 | 0.0606 |

n | 5 | 21 | 7 | 6 | 23 | 6 | 17 |

f_{1}(%) | 60 | 71 | 57 | 67 | 52 | 83 | 41 |

f_{2}(%) | 80 | 90 | 86 | 100 | 91 | 100 | 71 |

#### 3.2.2. Interval Prediction

_{3}is the number of the network traffic data which are included in the predicting interval. Then f

_{3}is defined as follows:

_{j+}

_{1}(m) in Equation (13) should be limited to the neighbor of Y

_{k+}

_{1}(m), the largest deviation value is selected in the 5% of Y

_{k+}

_{1}(m) as follows:

Date | Monday | Tuesday | Wednesday | Thursday | Friday | Saturday | Sunday |
---|---|---|---|---|---|---|---|

n_{3} | 3 | 16 | 18 | 4 | 18 | 6 | 15 |

f_{3}(%) | 60 | 76 | 75 | 67 | 78 | 100 | 88 |

## 4. Concluding Remarks

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

- Liu, Y.; Zhang, J.; Mu, D. Nonlinear Dynamics of the Small-World Networks-Hopf Bifurcation, Sequence of Period-Doubling Bifurcations and Chaos. J. Phys. Conf. Ser.
**2008**, 96, 012061. [Google Scholar] [CrossRef] - Chu, W.; Guan, X.; Cai, Z.; Gao, L. Real-Time Volume Control for Interactive Network Traffic. Comput. Netw.
**2013**, 57, 1611–1629. [Google Scholar] [CrossRef] - Wang, J.; Yuan, J.; Li, Q.; Yuan, R. Correlation Dimension Based Nonlinear Analysis of Network Traffics with Different Application Protocols. Chin. Phys. B
**2011**, 20, 050506. [Google Scholar] [CrossRef] - Feng, H.; Shu, Y.; Yang, O. Nonlinear Analysis of Wireless LAN Traffic. Nonlinear Anal. Real World Appl.
**2009**, 10, 1021–1028. [Google Scholar] [CrossRef] - Nooruzzaman, M.; Koyama, O.; Katsuyama, Y. Congestion Removing Performance of Stackable ROADM in WDM Networks under Dynamic Traffic. Comput. Netw.
**2013**, 57, 2364–2373. [Google Scholar] [CrossRef] - Sergiou, C.; Vassiliou, V.; Paphitis, A. Congestion Control in Wireless Sensor Networks through Dynamic Alternative Path Selection. Comput. Netw.
**2014**, 75, 226–238. [Google Scholar] [CrossRef] - Chen, B.-S.; Yang, Y.-S.; Lee, B.-K.; Lee, T.-H. Fuzzy Adaptive Predictive Flow Control of ATM Network Traffic. IEEE Trans. Fuzzy Syst.
**2003**, 11, 568–581. [Google Scholar] [CrossRef] - Alheraish, A. A Comparison of AR Full Motion Video Traffic Models in B-ISDN. Comput. Electr. Eng.
**2005**, 31, 1–22. [Google Scholar] [CrossRef] - Szeto, W.Y.; Ghosh, B.; Basu, B.; O’Mahony, M. Multivariate Traffic Forecasting Technique Using Cell Transmission Model and SARIMA Model. J. Transp. Eng. ASCE
**2009**, 135, 658–667. [Google Scholar] [CrossRef] - Zhang, N.; Zhang, Y.; Lu, H. Seasonal Autoregressive Integrated Moving Average and Support Vector Machine Models Prediction of Short-Term Traffic Flow on Freeways. Transp. Res. Rec.
**2011**, 2215, 85–92. [Google Scholar] [CrossRef] - Swift, D.K.; Dagli, C.H. A Study on the Network Traffic of Connexion by Boeing: Modeling with Artificial Neural Networks. Eng. Appl. Artif. Intell.
**2008**, 21, 1113–1129. [Google Scholar] [CrossRef] - Takens, F. Detecting Strange Attractors in Turbulence. Lect. Notes Math.
**1981**, 898, 361–381. [Google Scholar] - Kugiumtzis, D. State Space Reconstruction Parameters in the Analysis of Chaotic Time Series. Physica D
**1996**, 95, 13–28. [Google Scholar] [CrossRef] - Kim, H.S.; Eykholt, R.J.; Salas, D. Nonlinear Dynamics, Delay Times, and Embedding Windows. Physica D
**1999**, 127, 48–60. [Google Scholar] [CrossRef] - David, R. Five Turbulent Problems. Physica D
**1983**, 7, 40–44. [Google Scholar] - Boffetta, G.; Paladin, G.; Vulpiani, A. Strong Chaos without the Butterfly Effect in Dynamical Systems with Feedback. J. Phys. A Math. Gen.
**1996**, 29, 2291–2298. [Google Scholar] [CrossRef] - Joach, M.H.; Erner, W.L. Lyapunov Exponents from Time Series of Acoustic Chaos. Phys. Rev. A
**1989**, 39, 2146–2152. [Google Scholar] - Wolf, A.; Swift, J.B.; Swinney, H.L.; Vastano, J.A. Determining Lyapunov Exponent from a Time Series. Physica D
**1985**, 16, 285–317. [Google Scholar] [CrossRef] - Rosenstein, M.T.; Collins, J.J.; Deluca, C.J. A Practical Method for Calculating Largest Lyapunov Exponents from Small Data Sets. Physica D
**1993**, 65, 117–134. [Google Scholar] [CrossRef] - Luo, Y.; Wang, J.F.; Cao, C.X. A Prediction of Network Traffic Flow Based on Lyapunov Exponent. J. Chongqing Univ.
**2004**, 27, 28–30. [Google Scholar]

© 2016 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons by Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Liu, Y.; Zhang, J.
Predicting Traffic Flow in Local Area Networks by the Largest Lyapunov Exponent. *Entropy* **2016**, *18*, 32.
https://doi.org/10.3390/e18010032

**AMA Style**

Liu Y, Zhang J.
Predicting Traffic Flow in Local Area Networks by the Largest Lyapunov Exponent. *Entropy*. 2016; 18(1):32.
https://doi.org/10.3390/e18010032

**Chicago/Turabian Style**

Liu, Yan, and Jiazhong Zhang.
2016. "Predicting Traffic Flow in Local Area Networks by the Largest Lyapunov Exponent" *Entropy* 18, no. 1: 32.
https://doi.org/10.3390/e18010032