Palm's Theorem for Nonstationary Processes

by Gordon Crawford

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Like most models for calculating stock requirements, the models used by the Air Force to calculate requirements and allocations have traditionally assumed that the failure process generates arrivals approximating a steady-state Poisson arrival process. Although many real-world arrival processes are approximately Poisson, few exhibit steady-state behavior in the long run. To understand capabilities and requirements in real-world scenarios, we must model the transition from peacetime to wartime failure rates and know the distribution of the number of parts in the repair pipeline. Classically these models have used a steady-state result known as Palm's Theorem to model the number of spare parts in the repair pipeline. The research reported here generalizes this theory and allows the precise calculation of the distribution of the number of parts in the repair pipeline at any time during a time-varying or dynamic scenario, which may include abrupt transitions in the level of activity.

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