Probability that an L bits long frame will contain error(s), given that the single bit error probability is p_e is

Erroneous frames will be sent again. The number of transmissions required is the sum of the series

The portion available for useful bits – excluding retransmissions and administrative overhead – is

The optimum length of the frame is a zero of the derivative

The first part of the formula can not poosibly be zero, so the look for the zeroes of the latter part. We multiply that part of the formula by L² and the result is the following second-degree equation

the zeroes of which are

i.e.

Since

for small values e we may further approximate

The minus sign in front of the square root may be dropped because L can´t be negative. We make sure that the value is actually a maximum (not a minimum) by looking at the formula of h when L is very small or very large.