  In a distributed storage system serving hot data, the data recovery
performance becomes important, captured e.g. by the service rate. We give
partial evidence for it being hardest to serve a sequence of equal user
requests (as in PIR coding regime) both for concrete and random user requests
and server contents.
  We prove that a constant request sequence is locally hardest to serve: If
enough copies of each vector are stored in servers, then if a request sequence
with all requests equal can be served then we can still serve it if a few
requests are changed.
  For random iid server contents, with number of data symbols constant (for
simplicity) and the number of servers growing, we show that the maximum number
of user requests we can serve divided by the number of servers we need
approaches a limit almost surely. For uniform server contents, we show this
limit is 1/2, both for sequences of copies of a fixed request and of any
requests, so it is at least as hard to serve equal requests as any requests.
For iid requests independent from the uniform server contents the limit is at
least 1/2 and equal to 1/2 if requests are all equal to a fixed request almost
surely, confirming the same.
  As a building block, we deduce from a 1952 result of Marshall Hall, Jr. on
abelian groups, that any collection of half as many requests as coded symbols
in the doubled binary simplex code can be served by this code. This implies the
fractional version of the Functional Batch Code Conjecture that allows
half-servers.
