Multi-UAV inspection missions require spare drones to replace active drones during recharging cycles. Existing fleet-sizing approaches often assume steady-state operating conditions that do not apply to finite-horizon missions, or they treat replacement requests as statistically independent events. The latter provides per-request blocking guarantees that fail to translate to mission-level reliability when demands cluster. This paper identifies a structural failure mode where efficient routing assigns similar workloads to each UAV, leading to synchronized battery depletion and replacement bursts that exhaust the spare pool even when average capacity is sufficient. We derive a closed-form sufficient fleet-sizing rule, k = m(ceil(R) + 1), where m is the number of active UAVs and R is the recovery-to-active time ratio. This additive buffer of m spares absorbs worst-case synchronized demand at recovery-cycle boundaries and ensures mission-level reliability even when all UAVs deplete simultaneously. Monte Carlo validation across five scenarios (m in [2, 10], R in [0.87, 3.39], 1000 trials each) shows that Erlang-B sizing with a per-request blocking target epsilon = 0.01 drops to 69.9% mission success at R = 3.39, with 95% of spare exhaustion events concentrated in the top-decile 5-minute demand windows. In contrast, the proposed rule maintains 99.8% success (Wilson 95% lower bound 99.3%) across all tested conditions, including wind variability up to CV = 0.30, while requiring only four additional drones in the most demanding scenario.

翻译：多无人机巡检任务需配备备用无人机，以在充电周期中替换现役无人机。现有机队规模规划方法通常假设稳态运行条件（不适用于有限时长任务），或将更换请求视为统计独立事件。后者虽能提供单次请求的阻塞概率保障，但当需求聚集时，这种保障无法转化为任务级可靠性。本文识别出一种结构性失效模式：高效路径规划为每架无人机分配相近工作量，导致电池耗尽同步化与更换需求爆发，即使平均容量充足也会耗尽备用池。我们推导出闭合形式的充分机队规模规则：k = m(ceil(R) + 1)，其中m为现役无人机数量，R为恢复时间与活动时间之比。该方案通过增加m架备用机形成缓冲，可吸收恢复周期边界处的同步最坏情况需求，即使所有无人机同时耗尽也能确保任务级可靠性。在五类场景（m ∈ [2,10]，R ∈ [0.87,3.39]，每组1000次试验）的蒙特卡洛验证中，采用每请求阻塞目标ε=0.01的Erlang-B规模规划在R=3.39时任务成功率降至69.9%，且95%的备用耗尽事件集中于需求最高的10分钟窗口内。相比之下，所提规则在所有测试条件下（包括风速变异系数达CV=0.30）维持99.8%的成功率（Wilson 95%下限99.3%），在最严苛场景中仅需额外增加四架无人机。