Randomization is a key concept in distributed computing to tackle impossibility results. This also holds for self-stabilization in anonymous networks where coin flips are often used to break symmetry. Although the use of randomization in self-stabilizing algorithms is rather common, it is unclear what the optimal coin bias is so as to minimize the expected convergence time. This paper proposes a technique to automatically synthesize this optimal coin bias. Our algorithm is based on a parameter synthesis approach from the field of probabilistic model checking. It over- and under-approximates a given parameter region and iteratively refines the regions with minimal convergence time up to the desired accuracy. We describe the technique in detail and present a simple parallelization that gives an almost linear speed-up. We show the applicability of our technique to determine the optimal bias for the well-known Herman’s self-stabilizing token ring algorithm. Our synthesis obtains that for small rings, a fair coin is optimal, whereas for larger rings a biased coin is optimal where the bias grows with the ring size. We also analyze a variant of Herman’s algorithm that coincides with the original algorithm but deviates for biased coins. Finally, we show how using speed reducers in Herman’s protocol improve the expected convergence time.

随机化是分布式计算中解决不可能结果的关键概念。这也适用于自稳定在匿名网络中，经常使用硬币翻转来破坏对称性。尽管自稳定算法中随机化的使用相当普遍，但尚不清楚最佳硬币偏差是什么，以最小化预期的收敛时间。本文提出了一种自动合成这种最优硬币偏差的技术。我们的算法基于概率模型检查领域的参数合成方法。它对给定的参数区域进行过近似和欠近似，并以最小的收敛时间迭代地细化这些区域，直至达到所需的精度。我们详细描述了这项技术，并提出了一种简单的并行化，它可以提供几乎线性的加速。我们展示了我们的技术在确定众所周知的 Herman 自稳定令牌环算法的最佳偏差方面的适用性。我们的综合得出，对于小戒指，公平的硬币是最佳的，而对于较大的戒指，有偏差的硬币是最佳的，其中偏差随着戒指的大小而增长。我们还分析了 Herman 算法的一个变体，该变体与原始算法一致，但偏离了有偏见的硬币。最后，我们展示了如何使用Herman 协议中的减速器改善了预期的收敛时间。