Rapid development and wide application of cyber-physical systems (CPS) have created the possibility of remotely monitoring product working cycles. This indicates that CPS can provide the technological support required for designing manufacturers’ warranties integrated with working cycle and for modelling users' (or consumers') random maintenance policies sustaining the reliability of the product through warranty. In this study, by designing limited working cycles as a warranty limit, we have proposed manufacturer's two-dimensional warranty first and last to ensure the product's reliability during the warranty coverage period. Subsequently, we have incorporated working cycle into the traditional maintenance policies and modelled users' random age replacement first and last to sustain the reliability of the product through its warranty. By analysing parameter values, we have offered some classic models, i.e., the expected long-run cost rates that represent certain special problems. Finally, sensitivities of both the proposed warranties and the modelled random age replacements have been analysed through numerical experiments. It has been demonstrated that whichever takes place first or last influences if the warranty cost of the proposed two-dimensional warranty is higher than that of the traditional one-dimensional warranty. Users can get a longer post-warranty period when the warranty limit m, i.e., the limited random working cycles, is lower.

信息物理系统（CPS）的快速发展和广泛应用为远程监控产品工作周期创造了可能性。这表明 CPS 可以提供设计制造商保修与工作周期相结合以及建模用户（或消费者）随机维护策略以通过保修维持产品可靠性所需的技术支持。在本研究中，通过设计有限的工作周期作为保修限制，我们提出了制造商的二维保修优先和最后保证，以确保产品在保修范围内的可靠性。随后，我们将工作周期纳入到传统的维护策略中，并为用户建模。首先和最后随机更换年龄，以在保修期内维持产品的可靠性。通过分析参数值，我们提供了一些经典模型，即代表某些特殊问题的预期长期成本率。最后，通过数值实验分析了提议的保证和建模的随机年龄替换的敏感性。已经证明，如果提议的二维保修的保修成本高于传统的一维保修，无论先发生还是最后发生都会影响。保修期满时，用户可以获得更长的保修期 已通过数值实验分析了提议的保证和模拟的随机年龄替换的敏感性。已经证明，如果提议的二维保修的保修成本高于传统的一维保修，无论先发生还是最后发生都会影响。保修期满时，用户可以获得更长的保修期 已通过数值实验分析了提议的保证和模拟的随机年龄替换的敏感性。已经证明，如果提议的二维保修的保修成本高于传统的一维保修，无论先发生还是最后发生都会影响。保修期满时，用户可以获得更长的保修期m，即有限的随机工作周期，较低。