  We address the experimentally relevant situation, where a non-equilibrium
state is created at the edge of a quantum Hall system by injecting charge
current into a chiral edge state with the help of a quantum point contact,
quantum dots, or mesoscopic Ohmic contact. We show that the commonly accepted
picture of the full equilibration of a non-equilibrium state at finite
distances longer than a characteristic length scale contradicts to the charge
conservation requirement. We use a phenomenological transmission line model to
account for the local equilibration process and the charge and energy
conserving dynamics of the collective mode. By solving this model in the limit
of long distances $L$ from the injection point, we demonstrate that the
correction of the electron distribution function to its eventual equilibrium
form scales down slowly as $1/\sqrt{L}$.
