Turbulent jet ignition (TJI) is a promising combustion technology for burning highly diluted air-fuel mixtures. Computationally efficient models to assess the effect of the operating conditions and design parameters on the ignition propensity and timing are of paramount importance for the development of combustion systems employing TJI. To this end, a one-dimensional (1-D) jet model, which is based on the solution of the section integrated mass and momentum conservation equations, is derived in the present study. The model is extended with two additional transport equations for the turbulence intensity and the ignition precursor/tracer, that marks the ignition event. One-dimensional transient flamelet calculations are performed to generate tables for the ignition precursor source term that account for the turbulence and chemistry interaction. Further simplification of the model is carried out to obtain a novel penetration correlation and a computationally inexpensive Lagrangian ignition model. The extended jet model is hierarchically validated using available literature data for non-reactive and reactive jets, as well as experiments conducted in a state-of-the-art optically accessible prechamber. The derived model is able to reproduce both flow-related quantities (velocity and turbulence profiles, jet penetration) and the ignition delay time under different variations. This study also illustrates how numerical simulations in canonical configurations (one-dimensional flamelet) can be used in practical applications of TJI.