Consider the production of a single consumable product that is fabricated in a process of ¤ stages that is subject to an uncertain environment. There are a number of workstations on each stage that have different operating parameters. The workstations are subject to the discrete events of repair, failure, and preventive maintenance that generate a jump in the state of the system. Between each stage of the manufacturing process is a finite buffer that holds pieces before they can be processed by

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... the next stage. If a buffer is full, then the preceding stage cannot produce pieces since there will be no place for them to go. This formulation of a manufacturing system is a hybrid system consisting of the meeting the global production demand while locally managing the operational status of the workstations. This formulation of the optimal scheduling of production is a quasi-LQGP problem whose jumps are generated by State Dependent Poisson Processes (SDPP). A numerical example is presented to illustrate the model. was formulated as a quasi-LQGP problem. On each stage, there were a number of workstations that were subject to the events of failure, repair, and preventive maintenance. These event transitions, jumps in the value of the state, are modeled using state dependent Poisson processes (SDPP) whose coefficients were parameterized by the current value of the state. This manufacturing system is a hybrid system in the sense that the continuous model for meeting the production goal is that of an MMS, with local considerations for the discrete event transitions of the individual workstations on each stage that cause jumps in the value of the state, which is similar to an FMS. In this paper we reformulate the optimal production scheduling model presented in [14] to include finite buffers between the stages. In [14] , the stages are assumed to have infinite buffers between stages. This causes a decoupling of the stages and additional handling is needed to clear the buffers. In this paper, all of the stages are coupled and the production of pieces for stage © is restricted to be less than or equal to the total capacity, production plus buffer capacity, of stage © ; see Figure 1. This new formulation greatly changes the characteristics of the manufacturing Stage i Stage i+1 Buffer Stage i Figure 1: In between the stages there is a buffer with finite capacity to hold pieces that have completed © stages but can not be processed immediately by stage © . system. The state space decomposition of the LQGP and quasi-LQGP problem allow for larger dynamical systems to be modeled since they are not subject to the Curse of Dimensionality [1] of dynamic programming in state space. The model formulation here is similar to that of Sethi, Zhang, and Zhang [10] and Sethi and Zhang [11] of the dynamical or -machine flowshop. In these models, each stage only consists of one machine and no maintenance is performed. The models are analyzed using asymptotics for the hierarchal production planning for the flowshop with machines that can fail and be repaired. In the model presented here, each stage can have a number of different machines and therefore the constraints are more complex and change with the value of the state, which is subject to a larger number of discrete jumps. The use of the quasi-LQGP problem allows for the ability to model larger and more complex manufacturing systems than those presented in [11, 10] . Motivation for the manufacturing model presented here as well as that of [14] can be found in the fabrication of semiconductors (see the online tutorials [3, 8, 7] ). The fabrication process can be viewed globally as two operations that are commonly referred to as the front-and back-end. The front-end is where the actual wafers are produced and the back-end is where the wafer processed and packaged into individual integrated circuits. The overall fabrication with two stages would be modeled as that of paradigm presented in [14] . However, the back-end and subtasks of the front-end would best be modeled as the manufacturing system paradigm presented here with finite buffers between the processing stages. The determination of which paradigm to use depends on the relative processing times, ability to store excess pieces, and on the physical parameters and layout of the workstations in the various stages.