  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.
