  Totally asymmetric simple exclusion process (TASEP) was originally introduced
as a model for the traffic-like collective movement of ribosomes on a messenger
RNA (mRNA) that serves as the track for the motor-like forward stepping of
individual ribosomes. In each step, a ribosome elongates a protein by a single
unit using the track also as a template for protein synthesis. But,
pre-fabricated, functionally competent, ribosomes are not available to begin
synthesis of protein; a subunit directionally scans the mRNA in search of the
pre-designated site where it is supposed to bind with the other subunit and
begin the synthesis of the corresponding protein. However, because of `leaky'
scanning, a fraction of the scanning subunits miss the target site and continue
their search beyond the first target. Sometimes such scanners successfully
identify the site that marks the site for initiation of the synthesis of a
different protein. In this paper, we develop an exclusion model, with three
interconvertible species of hard rods, to capture some of the key features of
these biological phenomena and study the effects of the interference of the
flow of the different species of rods on the same lattice. More specifically,
we identify the meantime for the initiation of protein synthesis as appropriate
mean {\it first-passage} time that we calculate analytically using the
formalism of backward master equations. In spite of the approximations made,
our analytical predictions are in reasonably good agreement with the numerical
data that we obtain by performing Monte Carlo simulations. We also compare our
results with a few experimental facts reported in the literature and propose
new experiments for testing some of our new quantitative predictions.
