  Reliability is essential for storing files in many applications of
distributed storage systems. To maintain reliability, when a storage node
fails, a new node should be regenerated by a repair process. Most of the
previous results on the repair problem assume perfect (error-free) links in the
networks. However, in practice, especially in a wireless network, the
transmitted packets (for repair) may be lost due to, e.g., link failure or
buffer overflow. We study the repair problem of distributed storage systems in
packet erasure networks, where a packet loss is modeled as an erasure. The
minimum repair-bandwidth, namely the amount of information sent from the
surviving nodes to the new node, is established under the ideal assumption of
infinite number of packet transmissions. We also study the bandwidth-storage
tradeoffs in erasure networks. Then, the use of repairing storage nodes (nodes
with smaller storage space) is proposed to reduce the repair-bandwidth. We
study the minimal storage of repairing storage nodes. For the case of a finite
number of packet transmissions, the probability of successful repairing is
investigated. We show that the repair with a finite number of packet
transmissions may use much larger bandwidth than the minimum repair-bandwidth.
Finally, we propose a combinatorial optimization problem, which results in the
optimal repair-bandwidth for the given packet erasure probability and finite
packet transmissions.
