We analyze a data-processing system with $n$ clients producing jobs which are processed in \textit{batches} by $m$ parallel servers; the system throughput critically depends on the batch size and a corresponding sub-additive speedup function. In practice, throughput optimization relies on numerical searches for the optimal batch size, a process that can take up to multiple days in existing commercial systems. In this paper, we model the system in terms of a closed queueing network; a standard Markovian analysis yields the optimal throughput in $\omega\left(n^4\right)$ time. Our main contribution is a mean-field model of the system for the regime where the system size is large. We show that the mean-field model has a unique, globally attractive stationary point which can be found in closed form and which characterizes the asymptotic throughput of the system as a function of the batch size. Using this expression we find the \textit{asymptotically} optimal throughput in $O(1)$ time. Numerical settings from a large commercial system reveal that this asymptotic optimum is accurate in practical finite regimes.

中文翻译：

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大规模批处理系统中的吞吐量优化

我们分析了一个数据处理系统，它有 $n$ 个客户端生产作业，这些作业由 $m$ 个并行服务器在 \textit{batches} 中处理；系统吞吐量关键取决于批量大小和相应的子加法加速函数。在实践中，吞吐量优化依赖于对最佳批量大小的数值搜索，这一过程在现有商业系统中可能需要多天时间。在本文中，我们根据封闭的排队网络对系统进行建模；标准马尔可夫分析在 $\omega\left(n^4\right)$ 时间内产生最佳吞吐量。我们的主要贡献是系统规模较大的系统的平均场模型。我们表明平均场模型具有独特的，全局吸引驻点，可以在封闭形式中找到，并将系统的渐近吞吐量表征为批量大小的函数。使用这个表达式，我们在 $O(1)$ 时间内找到了 \textit{asymptotically} 最佳吞吐量。来自大型商业系统的数值设置表明，这种渐近最优在实际有限范围内是准确的。