Finite horizon control of processing networks via fluid approach: Separated continuous linear programs, infinite virtual buffers and maximum pressure policies

We consider systems in which many items are evolving over time by sharing common resources, and the problem of how to control such systems by allocating resources to various activities which schedule, route and process these items. We represent this by a processing network as defined by Harrison, with the added feature of infinite virtual buffers, which can model exogenous input and output. We address the problem of transient control of such processing networks over a finite time horizon. We use a fluid approach, in which we approximate the processing network by a deterministic continuous linear fluid model and formulate its optimization as a separated continuous linear program. This can be solved by a new simplex type algorithm of Weiss, and the solution consists of piecewise constant allocations of the activities, with a finite number of breakpints. This optimal fluid solution is then used to control the original system. We use the maximum pressure policy of Dai and Lin to track the optimal fluid solution. We prove asymptotic optimality of this fluid approach: As the numbers of items in the system and the processing rates increase, the scaled system converges almost surely to the optimal fluid solution, and the scaled objective value converges to the optimum of the system.