Unlike in the regular ALOHA protocolregular ALOHA protocol, where other nodes can send messages that interfere with ours at any time, in the Slotted ALOHA protocol, the only other time a message can be sent to interfere with ours is if it's sent at the exact time ours is sent (since in Slotted ALOHA messages can only be sent at specific intervals, like every 5 seconds for example)
Therefore, the probability that our message will be the only one, and have no interference is the probability (in a Poisson Process) that only 1 message on the system (ours!) is sent on the mark
$$P[(N(t + 1) - N(t)) = 1] = \frac{{e^{ - \lambda\tau } (\lambda\tau) ^ 1 }}{{1!}} = \lambda\tau e^{ - \lambda\tau} = \lambda e^{ - \lambda}$$
This gives us a function of the throughputs for all $\lambda$s.
To find the $\lambda$ with the highest throughput (that is, what expected rate gives us the highest throughput) we take the derivative and set to zero:
$$e^{-\lambda} - \lambda e^{-\lambda} = e^{-\lambda} (1 - \lambda) = 0 $$
$$\therefore \lambda = 1$$
In other words, when $\lambda$ is $1$, our throughput is highest. How high exactly? Just plug it back into the function we came up with, $\lambda e^{ - \lambda}$ and get:
$$ 1 * e^-1 = \frac{1}{e} = 0.36787....$$
Here's nice graph comparing the two ALOHAs, made by Reuven Cohen.
