The mean dwell time is usually defined as the probability of finding a particle inside the barrier region divided by the incoming flux and averaged over the initial momentum distribution. This probabilistic definition results in approximate values for the mean dwell time, which holds true only for high over-barrier energies when the interference effects are neglected. In the present work, a rigorous method is developed for the mean dwell time which provides exact values for the initial wave packets of arbitrary shape and length. Their propagation is systematically investigated through an asymmetric potential barrier. The results are applied for two cases: (i) the initial wave packet has been prepared in the distant past, and (ii) wave packet evolution is considered for the future time with the wave packet prepared at the instant time equal to zero. The behavior of these tunneling times is studied analytically within the limit of the small sub-barrier and high over-barrier energies of incident particles. All the tunneling times investigated tend to the classical time of flight in the high-energy limit.