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So if I had google fiber (1GBPS plan) and were to upload a 1 GB file via FTP server, would there be any delay or would it be limited to the download speed of a server?

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No, for several reasons:

  1. It's a 1 gigabit per second service and you're uploading a 1 gigabyte file.

  2. You're also limited by the server's available inbound bandwidth.

  3. The advertised speed is a raw line speed, not a true data speed. Address and control information also has to pass over the line.

  4. The TCP protocol that FTP uses can't perfectly fill a line.

Real world reports of people with Google fiber are that they can upload a 1 GB file to a well-connected server in just under 11 seconds.

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1 Gbps actually transmits 119.21 MB/s, so no.

And even if it was 1GB/s, it would depend on a couple of factors.

Lag and packet loss have a important effect over the maximum speed of a TCP connection, because each packet containing a couple KBytes of data transmitted have a receivement confirmation. It is very optimized, but depending on the speed and the lag it will most likely always cause delays.

Your bandwidth won't be 100% used for transmitting the file data. Packet headers are a substantial portion of the transmitted traffic, and they will consume part of the bandwidth.

Also, your hard disk cannot read 1GB of data in 1 second, and the receiver probably can't write 1 GB of data in 1 second either.

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