We investigate the properties of a family of polytopes that naturally arise in connection with a problem in distributed data storage, namely service rate region polytopes. The service rate region of a distributed coded system describes the data access requests that the underlying system can support. In this paper, we study the polytope structure of the service rate region with the primary goal of describing its geometric shape and properties. We achieve so by introducing various structural parameters of the service rate region and establishing upper and lower bounds for them. The techniques we apply in this paper range from coding theory to optimization. One of our main results shows that every rational point of the service rate region has a so-called rational allocation, answering an open question in the research area.

翻译：我们研究了一类与分布式数据存储问题自然相关的多面体族的性质，即服务速率区域多面体。分布式编码系统的服务速率区域描述了底层系统能够支持的数据访问请求。本文以描述服务速率区域的几何形状及其性质为主要目标，对其多面体结构进行了研究。为此，我们引入了服务速率区域的多种结构参数，并建立了这些参数的上界和下界。本文所使用的技术涵盖编码理论到优化方法。我们的主要结果之一表明，服务速率区域中的每个有理点都具有所谓的有理分配，从而回答了该研究领域中的一个悬而未决的问题。