Timing-based servo (TBS) is a powerful technology widely used in state-of-the-art tape systems for track following. To optimize TBS systems for future operating points with aggressive track-density scaling, an accurate model is needed to characterize how the servo read head geometry, servo format parameters, and magnetic media properties affect the readback signal. In this paper, we derive new analytical closed-form expressions for the readback signal of shielded magneto-resistive (MR) heads scanning over the magnetic transitions of TBS patterns. The expressions, derived via the reciprocity principle, are applicable to arctan-shaped transitions between arbitrarily magnetized areas of a thin-film magnetic medium and are given in the displacement and wavevector domain. We apply these new expressions to analyze isolated transitions as well as multi-transition TBS patterns on longitudinal, non-oriented or perpendicular media. Comparison between experimentally captured servo readback signals and the analytical expressions show an excellent match. By adding electronics and transition jitter noise, the readback signal expressions can be extended to a servo channel model suitable for TBS performance prediction by means of Monte Carlo simulations and/or bounds on the variance of the position-error signal.