  We revisit the problem of two-terminal transport of non-interacting Fermi
particles in a mesoscopic device. First, we generalize the problem by including
into consideration relaxation processes in contacts (which are characterized by
the contact self-thermalization rate $\gamma$) and then solve it by using the
master equation approach. In the limit $\gamma\rightarrow0$ the obtained
results are shown to reproduce those of the Landauer-B\"uttiker theory. Thus,
the presented analysis proves analytical correspondence between the
Landauer-B\"uttiker and master-equation approaches to quantum transport, -- the
problem which resisted solution for decades.
