Predicting Traffic Flow in Local Area Networks by the Largest Lyapunov Exponent
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(t1), x(t2), …, x(tn)), 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(t1) is the first element of the vector, x(t1 + τ) is the next one, and the last one is x(t1 + (m − 1)τ), so the first point in vector form in the phase space is defined as follows:
- Then let x(t2) 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:
- The number of points in m-dimensional subspace is N = n − (m − 1)k, and all the points of the subspace are:
2.2. C-C Method
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.
2.4. Small Data Sets Method
2.5. Longest Predictable Duration
2.6. Prediction Method Based on the Largest Lyaponov Exponent
3. Numerical Examples and Analysis
3.1. Largest Lyapunov Exponent of LAN Traffic
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 |
τw | 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 |
λ1 | 0.0169 | 0.0171 | 0.0147 | 0.0386 | 0.0065 | 0.0103 | 0.0076 | 0.0178 |
3.2. Prediction
3.2.1. Dot Prediction
Date | Monday | Tuesday | Wednesday | Thursday | Friday | Saturday | Sunday |
---|---|---|---|---|---|---|---|
rate of flow (byte/s) | 1.86538 × 106 | 1.40022 × 106 | 7.35079 × 105 | 1.39503 × 106 | 1.42701 × 106 | 1.80545 × 105 | 1.55155 × 105 |
τ | 4 | 2 | 3 | 7 | 2 | 2 | 4 |
m | 6 | 12 | 3 | 3 | 19 | 27 | 8 |
λ1 | 0.1848 | 0.0475 | 0.146 | 0.1562 | 0.044 | 0.1557 | 0.0606 |
n | 5 | 21 | 7 | 6 | 23 | 6 | 17 |
f1(%) | 60 | 71 | 57 | 67 | 52 | 83 | 41 |
f2(%) | 80 | 90 | 86 | 100 | 91 | 100 | 71 |
3.2.2. Interval Prediction
Date | Monday | Tuesday | Wednesday | Thursday | Friday | Saturday | Sunday |
---|---|---|---|---|---|---|---|
n3 | 3 | 16 | 18 | 4 | 18 | 6 | 15 |
f3(%) | 60 | 76 | 75 | 67 | 78 | 100 | 88 |
4. Concluding Remarks
Acknowledgments
Author Contributions
Conflicts of Interest
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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
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 StyleLiu, 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