We begin this dissertation by establishing a Transient Little Law, which is a time dependent analogue of the well-known Little's law L = λW in queueing theory. Our law expresses the expected queue length in a general input-output system in terms of transient Palm probabilities of the waiting times and the expected value of the input process. Next, we use the Transient Little Law to establish an integro-differential equation for the expected WIP in a production system, and develop an approximation method to solve it numerically. Our numerical experiments show that the approximation is quite accurate. Finally, we present an optimization-based production planning model that determines when to release material into a plant so that the resulting production output closely matches prescribed requirements for a finite horizon.; The main contribution is an approximation of the transient behavior of general multi-class queueing networks, which includes a characterization of the random lead time to produce products that depend dynamically on the time-dependent WIP throughout the network. Our approximation is designed to be suitable for linear programming optimization of inputs to stochastic processing networks. Our model complements existing approximation models, such as QNA (Queueing Network Analyzer), which are designed for equilibrium analysis of networks.
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