A renewal theoretic analysis of a class of manufacturing systems

The authors present the partial differential equations (PDEs) describing the transients of the probability density functions (PDFs) characterizing the statistical evolution of a manufacturing system producing a single product under hedging-point control policies. The authors demonstrate the Markov renewal nature of the dynamics of the controlled process and use the system of PDEs to compute the transition kernel of that renewal process. This Markov renewal viewpoint is particularly useful in discussing ergodicity in view of the abundant literature on the asymptotic behavior of Markov renewal processes. Moreover, besides allowing direct determination of system steady state, when it exists, it permits the computation of various statistics, as well as, in some cases, the derivation of bounds on the speed of convergence to steady state