The goal of the T3B experiment is the measurement of the time structure of hadronic showers with nanosecond precision and high spatial resolution together with the CALICE hadron calorimeter prototypes, with a focus on the use of tungsten as absorber medium. The detector consists of a strip of 15 scintillator cells individually read out by silicon photomultipliers (SiPMs) and fast oscilloscopes with a PC-controlled data acquisition system. The data reconstruction uses an iterative subtraction technique which provides a determination of the arrival time of each photon on the light sensor with sub-nanosecond precision. The calibration is based on single photon-equivalent dark pulses constantly recorded during data taking, automatically eliminating the temperature dependence of the SiPM gain. In addition, a statistical correction for SiPM afterpulsing is demonstrated. To provide the tools for a comparison of T3B data with GEANT4 simulations, a digitization routine, which accounts for the detector response to energy deposits in the detector, has been implemented.