  We perform a theoretical study of non-equilibrium effects in charge transport
through a hybrid single-electron transistor based on a small normal metal (N)
island with the gate-controlled number of electrons, tunnel-coupled to
voltage-biased superconducting (S) electrodes (SINIS). Focusing on the
turnstile mode of the transistor operation with the gate voltage driven
periodically, and electrons on the island being out of equilibrium, we find
that the current quantization accuracy is a non-monotonic function of the
relaxation rate $\Gamma_{\mathcal{F}}$ of the distribution function
$\mathcal{F}(\epsilon)$ on the island due to tunneling, as compared to the
drive frequency $f$, electron-electron $1/\tau_{ee}$ and electron-phonon
$1/\tau_{eph}$ relaxation rates. Surprisingly, in the strongly non-equilibrium
regime, $f\gg \Gamma_{\mathcal{F}}\gg\tau_{ee}^{-1},\tau_{eph}^{-1}$, the
turnstile current plateau is recovered, similarly to the ideal equilibrium
regime, $\tau_{eph}^{-1}\gg \Gamma_{\mathcal{F}}$. The plateau is destroyed in
the quasiequilibrium regime when the electron-electron relaxation is faster
than tunneling.
