Quadratic approximations to the differential cost-to-go function, which yield linear switching curves, have been extensively studied in the literature, In this paper, we provide solutions to the partial differential equations associated with the components of the steady state probability density function of the buffer levels for two part-type, single-machine flexible manufacturing systems under a linear switching curve (LSC) policy, When there are more than two part-types, we derive the probability density function, under a prioritized hedging point (PHP) policy by decomposing the multiple part-type problem into a sequence of two part-type problems, The analytic expression for the steady-state probability density function is independent of the cost function, Therefore, for average cost functions, we can compute the optimal PHP policy or the more general optimal LSC policy for two part-type problems.