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# Mean-field analysis of a scaling MAC radio protocol

• * Corresponding author

This work is supported by the OTKA 123914 project and the TUDFO/51757/2019-ITM grants

• We examine the transient behavior of a positioning system with a large number of tags trying to connect to the infrastructure with an exponential backoff policy in case of unsuccessful connection. Using a classic mean-field approach, we derive a system of differential equations whose solution approximates the original process. Analysis of the solution shows that both the solution and the original system exhibits an unusual log-periodic behavior in the mean-field limit, along with other interesting patterns of behavior. We also perform numerical optimization for the backoff policy.

Mathematics Subject Classification: Primary: 68M20, 60J20; Secondary: 90B20.

 Citation: • • Figure 1.  State transitions of a single user; $p_i$ are constant, $c_i$ depend on other users

Figure 2.  Convergence of $w_0(t)$ and $w_1(t)$ when $\alpha$ is fixed and $L\to\infty$

Figure 3.  Simulation for $N_{L+i}(Nt)/N$ versus numerical solution for $z_i(t)$ for $i = 0, 1, 2$ (parameters are $N = 2^{10}, \gamma = 2, L = 10, \alpha = 0$)

Figure 4.  Early rapid transition: $z_i(t)$ for values of $i$ considerably smaller than 0 ($\alpha = 0$ and $\gamma = 2$)

Figure 5.  The functions $z(\gamma, \alpha, t)$ for $\gamma = 20$ and $\alpha = 0$ (thick line), $1/10, \dots, 9/10$

Figure 6.  The functions $z(\gamma, \alpha, t)$ for $\gamma = 2$ and $\alpha = 0, 1/10, \dots, 9/10$

Figure 7.  The values $z_i(2, \alpha, 1)$ for $\alpha = 0, 1/20, \dots, 19/20$

Figure 8.  The values $z_i(20, \alpha, 1)$ for $\alpha = 0, 1/20, \dots, 19/20$

Figure 9.  Mean of the scaled connection time for $\gamma=20$

Figure 10.  Mean of the scaled connection time for $\gamma=2$

Figure 11.  Mean of the scaled connection time as a function of $\gamma$

Figure 12.  Simulation for $1-\bar N_0(Nt)/N$ (red line) versus $\bar z(t)$ (dashed blue line); parameters are $N=2^{10},\gamma=2,L=10,\alpha=0,t_0=0.5$

Figure 13.  $z(t)$ (no switching, black line) versus $\bar z(t)$ (switching at time $t_0=0.72$, optimal for $m_z$, dotted red line) versus $\bar z'(t)$ (switching at time $t_0=0.39$, optimal for the 99.9% quantile, dashed blue line). Parameters are $\gamma=2,L=10,\alpha=0$

Table 1.  Optimization of the switching time for a prescribed quantile ($\alpha = 0$)

 switching mean time quantile $\gamma$ time $t_0$ to connect 0.9 0.95 0.99 0.999 2 $\infty$ 2.722 5.306 7.171 12.91 25.47 2 0.718 2.198 3.738 4.522 6.791 11.57 2 0.607 2.230 3.687 4.369 6.328 10.44 2 0.534 2.321 3.732 4.344 6.089 9.730 2 0.453 2.561 3.954 4.486 5.983 9.094 2 0.387 3.019 4.448 4.912 6.201 8.877 1.65 $\infty$ 2.628 4.746 6.050 9.776 17.20 1.65 1.008 2.321 3.782 4.439 6.213 9.634 1.65 0.838 2.361 3.748 4.313 5.825 8.729 1.65 0.777 2.408 3.775 4.307 5.719 8.428 1.65 0.677 2.563 3.916 4.390 5.637 8.017 1.65 0.573 2.940 4.325 4.737 5.805 7.833
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