Several important biological processes are initiated by the binding of a protein to a specific site on the DNA. The strategy adopted by a protein, called transcription factor (TF), for searching its specific binding site on the DNA has been investigated over several decades. In recent times the effects obstacles, like DNA-binding proteins, on the search by TF has begun to receive attention. RNA polymerase (RNAP) motors collectively move along a segment of the DNA during a genomic process called transcription. This RNAP traffic is bound to affect the diffusive scanning of the same segment of the DNA by a TF searching for its binding site. Motivated by this phenomenon, here we develop a kinetic model where a `particle, that represents a TF, searches for a specific site on a one-dimensional lattice. On the same lattice another species of particles, each representing a RNAP, hop from left to right exactly as in a totally asymmetric simple exclusion process (TASEP) which forbids simultaneous occupation of any site by more than one particle, irrespective of their identities. Although the TF is allowed to attach to or detach from any lattice site, the RNAPs can attach only to the first site at the left edge and detach from only the last site on the right edge of the lattice. We formulate the search as a {it first-passage} process; the time taken to reach the target site {it for the first time}, starting from a well defined initial state, is the search time. By approximate analytical calculations and Monte Carlo (MC) computer simulations, we calculate the mean search time. We show that RNAP traffic rectifies the diffusive motion of TF to that of a Brownian ratchet, and the mean time of successful search can be even shorter than that required in the absence of RNAP traffic. Moreover, we show that there is an optimal rate of detachment that corresponds to the shortest mean search time.
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