The present paper concerns a new methodology for the adaptive optimization of piecewise deterministic non Markovian systems via a simple example of interest in manufacturing, This methodology takes into account the fact that piecewise deterministic systems are rarely Markovian and that, consequently, classical control theory based on dynamic programming of Markovian systems cannot provide quantitative answers in most realistic situations, although it does provide useful insights. The example considered in this paper, which allows us to push the analysis sufficiently far and with enough details to understand the foundations and to appreciate the scope and limitations of the method, is that of a single-machine/single-part production system, first studied from the point of view of control theory by Kimemia and Gershwin [23]. We obtain stochastic gradient estimates via the perturbation analysis of Ho and Cao [18] by the "regenerative" method of Konstantopoulos and Zazanis [24], in view of stochastic optimization, Its mathematical justification requires a careful study of the "regenerative" structure of the process, In particular, we give the necessary and sufficient conditions of stability of this system via the method of Loynes [29].