Time of appearance of a large gap in a dynamic Poisson point process

We study the distribution of the 'gap time', the first time that a large gap appears, in the spatial birth and death point process on $[0,1]$ in which particles are added uniformly in space at rate $\lambda$ and are removed independently at rate $1$, as a function of the parameter $\lambda$ and the specified gap size function $w_\lambda$ as $\lambda\to\infty$. If $w_\lambda$ is a large enough multiple of the typical largest gap $(\log(\lambda)+O(1))/\lambda$ and the initial distribution has a high enough local density of particles and not too many particles in total, then the gap time, scaled by its expected value, converges in distribution to exponential with mean $1$. If in addition $\limsup_\lambda w_\lambda < 1$ then the expected time scales like $e^{\lambda w_\lambda}/(\lambda^2 w_\lambda(1-w_\lambda))$.