= Solution
Only times up to $\tau$ contribute to the <cumulative incidence function> for type $B$, because its <cause-specific hazard> vanishes afterwards. With $q=\theta_A+\theta_B>0$,
$$
F_B(t)=\int_0^{\min(t,\tau)}\theta_B e^{-qu}\,du,
$$
so
$$
\boxed{F_B(t)=\frac{\theta_B}{\theta_A+\theta_B}\left(1-e^{-(\theta_A+\theta_B)\min(t,\tau)}\right).}
$$
This 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.
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