  This paper proposes an analytical framework for peer-to-peer (P2P) networks
and introduces schemes for building P2P networks to approach the minimum
weighted average download time (WADT). In the considered P2P framework, the
server, which has the information of all the download bandwidths and upload
bandwidths of the peers, minimizes the weighted average download time by
determining the optimal transmission rate from the server to the peers and from
the peers to the other peers. This paper first defines the static P2P network,
the hierarchical P2P network and the strictly hierarchical P2P network. Any
static P2P network can be decomposed into an equivalent network of sub-peers
that is strictly hierarchical. This paper shows that convex optimization can
minimize the WADT for P2P networks by equivalently minimizing the WADT for
strictly hierarchical networks of sub-peers. This paper then gives an upper
bound for minimizing WADT by constructing a hierarchical P2P network, and lower
bound by weakening the constraints of the convex problem. Both the upper bound
and the lower bound are very tight. This paper also provides several suboptimal
solutions for minimizing the WADT for strictly hierarchical networks, in which
peer selection algorithms and chunk selection algorithm can be locally
designed.
