Let D be the time the first gap of length \ell starts in an inhomogeneous Poisson process with rate function \mu(t). We give an integral test for D to be finite a.s., which in particular shows that the critical rate of increase of \mu(t) is \ell\log t. Asymptotic properties of the tail P(D>t) are studied and compared to the exponential decay in the homogeneous case.

The discrete time analogue of the setting is runs of length \ell of ones in Bernoulli 0-1 sequence, the study of which is a classical topic in the i.i.d. case but for which time inhomogeneity seems little developed.