Make-to-stock systems with backorders: IPA gradients

We consider single-stage, single-product make-to-stock systems with random demand and random service (production) rate, where demand shortages at the inventory facility are backordered. The make-to-stock system is modeled as a stochastic fluid model (SFM), in which the traditional discrete arrival, service and departure stochastic processes are replaced by corresponding stochastic fluid-flow rate processes. IPA (infinitesimal perturbation analysis) gradients of various performance metrics are derived with respect to parameters of interest (here, base-stock level and production rate), and are showed to be unbiased and easy to compute. The (random) IPA gradients are obtained via sample path analysis under very mild assumptions, and are inherently nonparametric in the sense that no specific probability law need be postulated. The formulas derived can be used in simulation, as well as in real-life systems.