Distributed systems use randomized work stealing to improve performance and resource utilization. In most prior analytical studies of randomized work stealing, jobs are considered to be sequential and are executed as a whole on a single server. In this article, we consider a homogeneous system of servers where parent jobs spawn child jobs that can feasibly be executed in parallel. When an idle server probes a busy server in an attempt to steal work, it may either steal a parent job or multiple child jobs.

To approximate the performance of this system, we introduce a Quasi-Birth-Death Markov chain and express the performance measures of interest via its unique steady state. We perform simulation experiments that suggest that the approximation error tends to zero as the number of servers in the system becomes large. To further support this observation, we introduce a mean field model and show that its unique fixed point corresponds to the steady state of the Quasi-Birth-Death Markov chain. Using numerical experiments, we compare the performance of various simple stealing strategies as well as optimized strategies.

分布式系统使用随机工作窃取来提高性能和资源利用率。在大多数先前的随机工作窃取分析研究中，作业被认为是连续的，并且作为一个整体在单个服务器上执行。在本文中，我们考虑一个同质的服务器系统，其中父作业生成可以并行执行的子作业。当空闲服务器探测繁忙服务器以尝试窃取工作时，它可能会窃取一个父作业或多个子作业。

为了近似该系统的性能，我们引入了准生死马尔可夫链，并通过其独特的稳态来表达感兴趣的性能指标。我们进行的模拟实验表明，随着系统中服务器数量的增加，近似误差趋于零。为了进一步支持这一观察，我们引入了平均场模型，并表明其独特的不动点对应于准生死马尔可夫链的稳态。通过数值实验，我们比较了各种简单窃取策略以及优化策略的性能。