We consider a generic stochastic model of ion transport through a single channel with arbitrary internal structure and kinetic rates of transitions between internal states. This model is also applicable to describe kinetics of a class of enzymes in which turnover events correspond to conversion of substrate into product by a single enzyme molecule. We show that measurement of statistics of single molecule transition time through the channel contains only restricted information about internal structure of the channel. In particular, the most accessible flux fluctuation characteristics, such as the Poisson indicator (P) and the Fano factor (F) as function of solute concentration, depend only on three parameters in addition to the parameters of the Michaelis-Menten curve that characterizes average current through the channel. Nevertheless, measurement of Poisson indicator or Fano factor for such renewal processes can discriminate reactions with multiple intermediate steps as well as provide valuable information about the internal kinetic rates.