#### Capacity bounds for distributed storage

##### Michael Luby

One of the primary objectives of a distributed storage system is to reliably store large amounts of source data for long durations using a large number $N$ of unreliable storage nodes, each with $c$ bits of storage capacity. Storage nodes fail randomly over time and are replaced with nodes of equal capacity initialized to zeroes, and thus bits are erased at some rate $E$. To maintain recoverability of the source data, a repairer continually reads data over a network from nodes at a rate $R$, and generates and writes data to nodes based on the read data. The distributed storage source data capacity is the maximum amount of source data that can be reliably stored for long periods of time. We prove the distributed storage source data capacity asymptotically approaches $\left(1-\frac{E}{2 \cdot R}\right) \cdot N \cdot c$ as $N$ and $R$ grow. This equation expresses a fundamental trade-off between network traffic and storage overhead to reliably store source data.

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