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In this paper, we investigate quasi-ballistic transport in nanowire field-effect transistors (NW-FETs). In order to do so, we address the 1-D Boltzmann transport equation (BTE) and find its exact analytical solution for any potential profile with the constraint of dominant elastic scattering. A simulation code implementing a self-consistent Schrodinger-Poisson solver in the transverse direction and the present BTE solution in the longitudinal direction is worked out, providing the I-V characteristics of the NW-FET. Such characteristics are compared with those computed using a numerical BTE solver accounting for both inelastic and elastic collisions, and the two of them turn out to agree very nicely. From this comparison, it may be concluded that inelastic scattering plays a minor role for small-diameter FETs with device lengths in the decananometer range. Next, a methodology for the calculation of the transmission and backscattering coefficients is worked out for the first time starting from the scattering probabilities. The aforementioned coefficients turn out to be functions of the ratio between the carrier transit time and a suitably averaged momentum-relaxation time. Therefore, one of the main conclusions of this paper is that, so long as inelastic collisions are negligible, the so-called kT layer plays no role in 1-D quasi-ballistic carrier transport.