Summary

File Distribution

• $F$: The size of the file to be distributed (in bits).
• $N$: The number of peers (clients) that need to receive a copy of the file.
• $u_s$​: The upload capacity of the server (bits per second).
• $d_i$: The download capacity of the $i$-th peer.
• $u_i$: The upload capacity of the $i$-th peer.
• $d_{min}$: The minimum download capacity among all peers ($\min \{d_1,d_2,...,d_n\}$).

Client-Server Distribution Formula

In a client-server architecture, the server must send a separate copy of the file to every one of the $N$ peers. The distribution time ($D_{c-s}$​) is limited by two main bottlenecks: $D_{c-s}\ge \max \left\{\frac{NF}{u_s},\frac{F}{d_{\min }}\right\}$

• $\frac{NF}{u_s}$ (Server Bottleneck): The server transmit $N$ copies of file size $F$ with upload rate $u_s$.
• $\frac{F}{d_{\min }}$ (Client Bottleneck): Each individual client must download one full copy of $F$ when the peer with the slowest download rate ($d_{min}$).

Peer-to-peer (p2p) Distribution Formula

In P2P distribution, peers can redistribute chunks of the file to one another. The distribution time ($D_{P2P}$)is limited by 3 factors: $D_{P2P}\ge \max \left\{\frac{F}{u_s},\frac{F}{d_{\min }},\frac{NF}{u_s+{\sum }_{i=1}^N{u}_i}\right\}$

• $\frac{F}{u_s}$ (Server Bottleneck): The server must upload at least one copy of the file to the network, so the peers can begin sharing it.
• $\frac{F}{d_{\min }}$ (Peer Bottleneck): Each peer must download one full copy of the file from the network, so the peer with the slowest download rate ($d_{min}$) limits the distribution time.
• $\frac{NF}{u_s+{\sum }_{i=1}^N{u}_i}$ (Aggregate Upload Bottleneck): The total upload capacity of the server and all peers must be sufficient to distribute $N$ copies of the file. This term accounts for the fact that peers can share the load of uploading the file.