Lock-Free algorithms under stochastic schedulers

In this work, we consider the following random process, mo- Tivated by the analysis of lock-free concurrent algorithms under high memory contention. In each round, a new scheduling step is allocated to one of n threads, according to a distribution p = (p1; p2; : : : ; pn), where thread i is scheduled with probability pi. When some thread first reaches a set threshold of executed steps, it registers a win, completing its current operation, and resets its step count to 1. At the same time, threads whose step count was close to the threshold also get reset because of the win, but to 0 steps, being penalized for almost winning. We are interested in two questions: how often does some thread complete an operation (system latency), and how often does a specific thread complete an operation (individual latency)? We provide asymptotically tight bounds for the system and individual latency of this general concurrency pattern, for arbitrary scheduling distributions p. Surprisingly, a sim- ple characterization exists: in expectation, the system will complete a new operation every Θ(1/p 2) steps, while thread i will complete a new operation every Θ(1/2=p i ) steps. The proof is interesting in its own right, as it requires a careful analysis of how the higher norms of the vector p inuence the thread step counts and latencies in this random process. Our result offers a simple connection between the scheduling distribution and the average performance of concurrent algorithms, which has several applications.