Mean-field analysis of a scaling MAC radio protocol

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 $