Algorithm 1 Exhaustive search algorithm

1:procedureFind Tx period($v_0$, $a_0$, $x_{i-1}$, $v_{i-1}$, $a_{i-1}$, $x_i$, $v_i$, $a_i$, $d_d$, $d_{Th}$, $T_d$, $T_{\tau }$, $t_{Th}$, $f(.)$) 2:                ▹$x_i$—long. position of vehicle i, $v_i$—velocity of vehicle i, $a_i$—acceleration of vehicle i, $d_d$—target inter-vehicle distance, $d_{Th}$—minimum distance constraint, $T_d$—set of possible transmission periods, $T_{\tau }$—set of possible initial time offsets, $t_{Th}$—maximum prediction time span, $f(.)$—controller function 3:    for each $\mathsf{\Delta }{t}_l\in T_d$, $l=1,\dots ,\left|T\right|$ do: 4:        for each ${\tau }_m\in T_{\tau }$, $m=1,\dots ,|T_{\tau }|$ do: 5:           Calculate: 6:           $x_{i-1}^{*}=x_{i-1}+v_{i-1}{\tau }_m+a_{i-1}\frac{{\tau }_m^2}{2}$, $v_{i-1}^{*}=v_{i-1}+a_{i-1}{\tau }_m$ 7:           $x_i^{*}=x_i+v_i{\tau }_m+a_i\frac{{\tau }_m^2}{2}$, $v_i^{*}=v_i+a_i{\tau }_m$, 8:           $v_0^{*}=v_0+a_0{\tau }_m$, 9:           $a_i^{*}=f(v_0^{*},a_0,x_{i-1}^{*},v_{i-1}^{*},a_{i-1},x_i^{*},v_i^{*},a_i^{*},d_d)$, 10:          $d^{*}=x_{i-1}^{*}-x_i^{*}$, $t=\tau$ 11:           while $d^{*}>d_{Th}$ & $t<t_{Th}$ & $v_i^{*}>0$ do: 12:               Calculate: 13:               $x_{i-1}^{*}=x_{i-1}^{*}+v_{i-1}^{*}{\mathsf{\Delta }t}_l+a_{i-1}\frac{{\mathsf{\Delta }t}_l^2}{2}$, $v_{i-1}^{*}=v_{i-1}^{*}+a_{i-1}{\mathsf{\Delta }t}_l$ 14:               $x_i^{*}=x_i^{*}+v_i^{*}{\mathsf{\Delta }t}_l+a_i^{*}\frac{{\mathsf{\Delta }t}_l^2}{2}$, $v_i^{*}=v_i^{*}+a_i^{*}{\mathsf{\Delta }t}_l$, 15:               $v_0^{*}=v_0^{*}+a_0{\mathsf{\Delta }t}_l$ 16:               $d^{*}=x_{i-1}^{*}-x_i^{*}$, $t=t+{\mathsf{\Delta }t}_l$, 17:               $a_i^{*}=f(v_0^{*},a_0,x_{i-1}^{*},v_{i-1}^{*},a_{i-1},x_i^{*},v_i^{*},a_i^{*},d_d)$ 18:               if $a_{i-1}^{*}-a_i^{*}>0$ & $v_{i-1}^{*}-v_i^{*}>0$ & $d^{*}>d_{Th}$ then: 19:                   $t=\infty$ 20:                   break 21:               end if 22:           end while 23:           Set: $t_m^{*}=t$ 24:        end for 25:        Select: $m^{*}=arg{max}_m{t}_m^{*}$ 26:        Set: ${\tau }_l^{*}={\tau }_{m^{*}}$ 27:        Set: $t_l^{**}=t_{m^{*}}^{*}$ 28:    end for 29:    Select: $l^{*}=arg{max}_l{t}_l^{**}$ 30:    Set: ${\tau }^{\left(sel\right)}={\tau }_{l^{*}}^{*}$ 31:    Set: ${\mathsf{\Delta }{t}_{Tx}}^{\left(sel\right)}={\mathsf{\Delta }t}_{l^{*}}$ return$({\tau }^{\left(sel\right)},{\mathsf{\Delta }{t}_{Tx}}^{\left(sel\right)})$ 32:end procedure