Using a semi-phenomenological model of the imbalanced Alfvénic turbulence, we study the turbulence transition from magnetohydrodynamic to sub-ion kinetic scales in the conditions typical for the fast solar wind. We show that the transition takes place in a wide wave number interval where the energy spectrum does not follow any power law. For larger turbulence imbalances, both boundaries of the transition wave number interval shift to larger scales. The energy exchange between dominant and subdominant components of the turbulence is essential in the model reducing the imbalance at kinetic scales; however, the total balance is never reached. Our results are compatible with recent solar wind observations at ion scales and numerical simulations of imbalanced turbulence.