The mean time between failures (MTBF) and mean time to repair (MTTR) of manufacturing equipment (e.g., machines) are used in every quantitative method for production systems performance analysis, continuous improvement, and design. Unfortunately, the literature offers no methods for evaluating the smallest number of up- and downtime measurements necessary and sufficient to calculate reliable estimates of these equipment characteristics. This article is intended to provide such a method. The approach is based on introducing the notion of (α,β)(\alpha, \beta) -precise estimates, where α\alpha characterizes the estimate’s accuracy and β\beta its probability. Using this notion, this article evaluates the critical number, n∗(α,β)n^{*} (\alpha,\beta) , of up- and downtime measurements necessary and sufficient to calculate (α,β)(\alpha, \beta) -precise estimates of MTBF and MTTR. In addition, this article derives a probabilistic upper bound of the observation time required to collect n∗(α,β)n^{*} (\alpha,\beta) measurements. Note to Practitioners—To evaluate and predict production systems behavior, managers of manufacturing operations need to know equipment reliability characteristics. Quantifying the equipment status by MTBF and MTTR, this article provides answers to the following questions: Q1: How many measurements of machines up- and downtime are required to obtain reliable estimates of MTBF and MTTR? Q2: How long the observation period must be to collect the desired number of measurements? The answer to Q1 is provided by a rule (formula), which is based on the desired estimate accuracy (characterized by α\alpha ) and its likelihood (quantified by β\beta ). The answer to Q2 consists of selecting a small number of initial measurements, which can be used to calculate an upper bound of the total observation time.

中文翻译：

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(α,β)-MTBF 和 MTTR 的精确估计：定义、计算和观察时间


制造设备（例如机器）的平均故障间隔时间 (MTBF) 和平均修复时间 (MTTR) 用于生产系统性能分析、持续改进和设计的每种定量方法。不幸的是，文献没有提供评估最少数量的正常运行和停机测量的方法，这些测量是计算这些设备特性的可靠估计所必需且充分的。本文旨在提供这样一种方法。该方法基于引入 (α,β)(\alpha, \beta) 精确估计的概念，其中 α\alpha 表征估计的准确性，β\beta 表征估计的概率。使用这个概念，本文评估了计算 (α,β)(\alpha) 所必需且充分的正常运行时间和停机时间测量的临界数量 n*(α,β)n^{*} (\alpha,\beta) , β) - MTBF 和 MTTR 的精确估计。此外，本文还推导了收集 n*(α,β)n^{*} (\alpha,\beta) 测量值所需的观测时间的概率上限。从业者须知——为了评估和预测生产系统行为，制造运营经理需要了解设备可靠性特征。本文通过 MTBF 和 MTTR 量化设备状态，回答了以下问题： Q1：需要对机器正常运行和停机时间进行多少次测量才能获得 MTBF 和 MTTR 的可靠估计？ Q2：观察期必须多长才能收集所需数量的测量结果？ Q1 的答案由规则（公式）提供，该规则基于所需的估计精度（由 α\alpha 表征）及其可能性（由 β\beta 量化）。 Q2 的答案包括选择少量初始测量值，这些初始测量值可用于计算总观察时间的上限。