We present a novel mathematical framework for computing the number of maintenance cycles in a system with critical and non-critical components, where "critical" (CR) means that the component's failure is fatal for the system's operation and renders any more repairs inapplicable, whereas "noncritical" (NC) means that the component can undergo corrective maintenance (replacement or minimal repair) whenever it fails, provided that the CR component is still in operation. Whenever the NC component fails, the CR component can optionally be preventively replaced. We extend traditional renewal theory (whether classical or generalized) for various maintenance scenarios for a system composed of one CR and one NC component in order to compute the average number of renewals of NC under the restriction ("bound") necessitated by CR. We also develop approximations in closed form for the proposed "bounded" renewal functions. We validate our formulas by simulations on a variety of component lifetime distributions, including actual lifetime distributions of wind turbine components.

翻译：我们提出了一个新颖的数学框架,用于计算一个系统维护周期的数量,该系统有关键和非关键部件,其中“临界”(CR)是指该部件的故障对系统的运作是致命的,使任何更多的修理无法适用,而“非临界”(NC)是指该部件一旦失灵,只要CR部件仍在运行中,就可以进行纠正性维护(替换或最低限度维修)。每当NC部件失效,CR部件可以被预防性地替换。我们扩展由CR和一个NC部件组成的系统的各种维护假设的传统更新理论(传统或普遍),以便计算CR所必须的限制(“受限制”)下NC平均延长NC的次数。我们还为拟议的“受约束”更新功能开发封闭式近似值。我们通过对各种部件寿命分配的模拟,包括风轮机部件的实际寿命分配,来验证我们的公式。