Why does a download start slow and then speed up?

Nothing is warming up, and nobody is throttling you. A sender has no measurement of the path in front of it, so it begins with a deliberately small amount of data outstanding, about fourteen kilobytes, and doubles that amount at the end of every round trip in which nothing was lost. What you are watching is a measurement being taken.

Here $b$ is how many bytes the sender will allow to be unacknowledged at once, $i$ is the starting allowance of roughly fourteen kilobytes, and $k$ is the number of round trips completed without loss. The sender cannot look at its own link and learn anything useful, because the path to you passes through whichever link is narrowest and that link tells it nothing. So fourteen kilobytes go out, they are acknowledged a round trip later, and only then does the allowance become twenty eight.

The tempting picture is of a fixed start-up cost, a second of nothing while something negotiates, or a server being stingy with a connection it could easily fill. Neither survives two tests. A fixed delay would not end where this ends, at the first lost packet, and it would not be remembered: the next response on a connection that has already carried traffic starts at the rate the previous one finished at, because the sender already holds its estimate. A throttle would be a rate the sender held itself to, whereas what is observable is a rate that climbs and stops climbing exactly when loss appears, doubling while nothing is dropped and cut sharply the moment something is. That is a probe, not a tariff. It also explains the shape of a slow start in the literal sense: it is not slow because the sender is being careful in a vague way, it is slow because each increase costs a full round trip to discover.

Because the doubling happens once per round trip, its cost is counted in round trips rather than bytes, and widening the pipe does not shorten it. On a hundred megabit link with an 80 millisecond round trip the sender needs 1,000,000 bytes outstanding to fill it: 12,500,000 bytes a second times 0.08 seconds. Running the doublings out from 14: 28, 56, 112, 224, 448, 896, 1792. The allowance passes a megabyte only at the seventh doubling, so the path is filled after seven round trips, which is 560 milliseconds, and across those the sender delivered at most $14(2^{7}-1) = 1778$ kilobytes. That is 14,565,376 bits over 0.56 seconds, an average of about 26 megabits per second on a link that calls itself a hundred. Give the same path a gigabit and the crossing point moves from the seventh doubling to the tenth: three round trips bought by ten times the bandwidth.

Two limits end the picture, and they are easy to confuse. The doubling lasts only until the first loss; after that the allowance is halved and growth becomes far slower, which is what lets many senders share one path roughly fairly. Separately, the receiver advertises its own buffer limit in every acknowledgement, and the smaller of the two numbers governs. If the receiver's buffer is 64 kilobytes while the path could carry 512, the transfer sits flat at 64 and nothing done at the sending end moves it. That flatness is the diagnostic: a rate that climbs and collapses and climbs again is the network estimate doing its job, while a rate that is identical on a fast path and a slow one is a receiver cap. Once neither limit binds, the ramp is finished and the link rate governs for the remainder, which on anything past a few hundred kilobytes is most of the transfer.