Fig. 1. Patches in two consecutive intervals: (k − 1) and k.

Fig. 2. Superposition of the (k − 1)th and kth intervals.

Fig. 3. Concurrent streams in the fourth unit of an interval [1, w + T/N]: (a) one arrival in the second unit, (b) one arrival in the third unit, (c) one arrival in the second and third units.

Fig. 4. P[X_i = k] for the interval [1, w + T/N]: (a) N = 50, T = 1000; (b) N = 100, T = 1000.

Fig. 5. Probability of more than one arrival in a unit of length d.

Fig. 6. Analysis of the value of T: (a) relative error, (b) CCDF of Z.

Fig. 7. Distribution of Z for N = 5 and different values of D.

Fig. 8. Average bandwidth.

Fig. 9. Relative errors for average bandwidth.

Fig. 10. Mean squared error (MSE) values.

Fig. 11. CCDF of the number of concurrent streams for a single object: (a) N = 10, 25, 40; (b) N = 50, 75, 100.

Fig. 12. CCDF of the number of concurrent streams for multiple objects.

Fig. 13. Bandwidth usage distribution for different values of N and the optimal window w defined for N = 75.

Fig. 14. Quality of service as the value of N increases: (a) for N = 10: P[Z > 5] = 10⁻²; P[Z > 7] = 10⁻³; P[Z > 8] = 10⁻⁴, (b) for N = 50: P[Z > 12] = 10⁻²; P[Z > 14] = 10⁻³; P[Z > 17] = 10⁻⁴.

Fig. 15. Distribution of the number of concurrent streams for N = 50 and N = 100 when the threshold window w varies around 20% of the optimal window value.

Fig. 16. Analysis of the multiple object case. (a) P[S > average server bandwidth], (b) % increase in server bandwidth to have P[S > k] = 10⁻², 10⁻³ or 10⁻⁴.

Key parameters

Size of the sub-intervals for different values of N

Values of parameters p and n_w

Bandwidth evaluation for N = 25, 50, 100

Number of objects in each scenario