 Solution

Only times up to $\tau$ contribute to the cumulative incidence function for type $B$, because its cause-specific hazard vanishes afterwards. With $q={\theta }_A+{\theta }_B>0$,soThis increases before $\tau$ and then remains constant; it includes the protection from type $B$ events produced by earlier type $A$ events. If both intensities vanish, $F_B=0$; if ${\theta }_B=0$, it is also identically zero.