Matrix analytic methods are developed to compute the probability distribution of response times (i.e. data access times) in distributed
storage systems protected by erasure coding, which is implemented by sharding a data object into N fragments, only K < N of which are required to reconstruct the object. This leads to a partial-fork-join model with a choice of canceling policies for the redundant N − K tasks. The accuracy of the analytical model is supported by tests against simulation in a broad range of setups. At increasing workload intensities, numerical results show the extent to which increasing the redundancy level reduces the mean response time of storage reads and significantly flattens the tail of their distribution; this is demonstrated at medium-high quantiles, up to the 99th. The quantitative reduction in response time achieved by two policies for canceling redundant tasks is also shown: for cancel-at-finish and cancel-at-start, which limits the additional load introduced whilst losing the benefit of selectivity amongst fragment service times.