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A primary objective of a distributed storage system is to reliably store huge amounts of source data for long periods of time using a large number of high storage capacity nodes. Storage nodes are prone to permanent failures, where such node failures cause all stored data to be permanently lost, and failed nodes are replaced with nodes that initially store no data. To maintain recoverability of the source data as storage nodes fail and are replaced, a total amount of storage capacity that is larger than the source data size is allocated to store the source data, and a repairer continually reads data from and writes data to the storage nodes as they fail and are replaced. We prove information-theoretic lower bounds on the rate at which a repairer reads data from the storage system as a function of the rate at which nodes fail and the amount by which the allocated storage capacity exceeds the source data size. These lower bounds hold for any repairer, i.e., any repairer that does not read data at or above the lower bound rate will provably not maintain recoverability of the source data. The bounds are provably tight asymptotically as the number of storage nodes grows and the allocated storage capacity approaches the source data size. [ABSTRACT FROM AUTHOR]