  We analyze the statistics of gaps ($\Delta H$) between successive avalanches
in one dimensional random field Ising models (RFIMs) in an external field $H$
at zero temperature. In the first part of the paper we study the
nearest-neighbour ferromagnetic RFIM. We map the sequence of avalanches in this
system to a non-homogeneous Poisson process with an $H$-dependent rate
$\rho(H)$. We use this to analytically compute the distribution of gaps
$P(\Delta H)$ between avalanches as the field is increased monotonically from
$-\infty$ to $+\infty$. We show that $P(\Delta H)$ tends to a constant
$\mathcal{C}(R)$ as $\Delta H \to 0^+$, which displays a non-trivial behaviour
with the strength of disorder $R$. We verify our predictions with numerical
simulations. In the second part of the paper, motivated by avalanche gap
distributions in driven disordered amorphous solids, we study a long-range
antiferromagnetic RFIM. This model displays a gapped behaviour $P(\Delta H) =
0$ up to a system size dependent offset value $\Delta H_{\text{off}}$, and
$P(\Delta H) \sim (\Delta H - \Delta H_{\text{off}})^{\theta}$ as $\Delta H \to
H_{\text{off}}^+$. We perform numerical simulations on this model and determine
$\theta \approx 0.95(5)$. We also discuss mechanisms which would lead to a
non-zero exponent $\theta$ for general spin models with quenched random fields.
