Abstract: We study the influence of the inelastic relaxation time τ̃E of the quasiparticle distribution function f(E) on the phase slip process in quasi-one-dimensional superconducting rings at a temperature close to the critical temperature Tc. We find that the initial time of growth of the order parameter |Δ| in the phase slip core after the phase slip is a nonmonotonic function of τ̃E which has a maximum at τ̃E≃τ̃GL=πℏ/8kB(Tc−T) and has a tendency to saturate for large τ̃E⪢τ̃GL. The effective heating of the electron subsystem due to the increase in |Δ| in the phase slip center together with the above effect result in a nonmonotonic dependence of the number of subsequent phase slips on τ̃E in rings of relatively large radius (in which each phase slip reduces the current density to a small fraction of its initial value). During the phase slip process the order parameter distribution has two peaks near the phase slip core due to the diffusion of the nonequilibrium quasiparticles from that region.