For service life tests, a shifted Weibull distribution, also known as the translated or three-parameter Weibull distribution, is commonly used. The shifted Weibull distribution promises completely fault-free operation until time t = L_0, meaning that the process is deterministic in the early stage. Only after this phase does the distribution allow random behavior, i.e. from the time t = L_0 on, the process is stochastic. This model, which is based on two consecutive time periods of quite different nature, is at odds with the idea of a continuously progressing fatigue, wear, or decay process as long as there are no influences from outside. To replace this arguably inconsistent model, variants of the Weibull distribution of purely stochastic nature are proposed and investigated that start with a reduced probability of failure before transitioning to normal Weibull behavior. [The german version of this paper can be found under https://doi.org/10.15488/11468]