This research presents a novel stochastic framework for proactive cybersecurity defense timing under a single attack scenario. The approach models the defense process as a continuous observation mechanism in which the defense instant and the subsequent observation slot follow independent exponential distributions. Laplace-Carson transforms combined with first-excess theory yield the joint detection function that brackets the attack moment. Marginalization under Markovian Poisson arrivals then produces the probability density of the defense moment and conditional expectations of pre-attack and post-attack observation times. These closed-form results enable quantitative assessment of defense timing sensitivity to threat intensity and support precise calibration of observation parameters for low-latency proactive measures. Major contributions include the explicit derivation of marginal distributions and expected values, visualization of defense moment density, and the bridging of stochastic duel methodology with practical cybersecurity applications.

翻译：本研究提出一种新颖的随机框架，用于在单次攻击场景下优化主动网络安全防御时机。该方法将防御过程建模为连续观测机制，其中防御时刻及后续观测时隙均服从独立指数分布。利用拉普拉斯-卡尔森变换结合首次超越理论，推导出可覆盖攻击时刻的联合探测函数。基于马尔可夫泊松到达过程的边缘化处理，得到防御时刻的概率密度函数以及攻击前后观测时长的条件期望。这些闭式解析结果能够定量评估防御时机对威胁强度的敏感性，并为低延迟主动防御措施的观测参数精确标定提供理论支持。主要贡献包括：边缘分布与期望值的显式推导、防御时刻密度的可视化呈现，以及随机对决方法论与网络安全实际应用之间的桥梁构建。