Asynchronous power save protocols have been proposed for use in ad hoc networks. In many protocols, nodes independently follow a common periodic wakeup schedule, each with some unknown offset relative to its neighbors. The schedule is defined to ensure deterministic intervals of overlap between nodes, regardless of the distribution of the nodes' wakeup schedules. This paper studies the sensitivity of a simple asynchronous power save protocol to the actual distribution of the nodes' wakeup schedules. In practical terms: For given topology and traffic load, are there particularly "good" or "bad" distributions? We define a simplified model of network operation that allows us to study this question in simulation. The results show that the performance variation has a narrow probability distribution, but with long tails. The variation is shown to derive largely from timing dependencies rather than overall capacity of the system. The result suggests the feasibility of manipulating the wakeup schedule distribution to improve performance. Although the best wakeup distributions often mitigate the performance penalty imposed by the power save protocol, their relative rarity implies that randomized strategies will not be sufficient to obtain maximum advantage.