We study the problem of how to coordinate the actions of independent agents in a distributed system where message arrival times are unbounded, but are determined by an exponential probability distribution. Asynchronous protocols executed in such a model are guaranteed to succeed with probability 1. We demonstrate a case in which the best asynchronous protocol can be improved on significantly. Specifically, we focus on the task of performing actions by different agents in a linear temporal order -- a problem known in the literature as Ordered Response. In asynchronous systems, ensuring such an ordering requires the construction of a message chain that passes through each acting agent, in order. Solving Ordered Response in this way in our model will terminate in time that grows linearly in the number of participating agents $n$, in expectation. We show that relaxing the specification slightly allows for a significant saving in time. Namely, if Ordered Response should be guaranteed with high probability (arbitrarily close to 1), it is possible to significantly shorten the expected execution time of the protocol. We present two protocols that adhere to the relaxed specification. One of our protocols executes exponentially faster than a message chain, when the number of participating agents $n$ is large, while the other is roughly quadratically faster. For small values of $n$, it is also possible to achieve similar results by using a hybrid protocol.

翻译：我们研究了分布式系统中独立代理如何协调行动的问题，其中消息到达时间无界，但由指数概率分布决定。在此类模型中执行的异步协议能以概率1保证成功。我们展示了一个案例，其中最佳异步协议可以得到显著改进。具体而言，我们关注不同代理按线性时间顺序执行行动的任务——该问题在文献中称为“有序响应”。在异步系统中，确保此类顺序需要构建一条依次经过每个行动代理的消息链。在我们的模型中，以此方式解决有序响应问题的预期终止时间随参与代理数n线性增长。我们证明，略微放宽规范条件可大幅节省时间。即，若需以高概率（任意接近1）保证有序响应，则可显著缩短协议的预期执行时间。我们提出了两种符合放宽规范的协议。当参与代理数n较大时，其中一种协议的执行速度比消息链呈指数级提升，另一种则大致呈平方级提升。对于较小的n值，亦可采用混合协议实现类似效果。