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, $fgg Gamma_{mathcal{F}}ggtau_{ee}^{-1},tau_{eph}^{-1}$, 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.
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