There is very little or no relation between Alexa rank and traffic. Since I have installed the Alexa toolbar I am also visiting an old blog of my on a daily basis. According to GA that blog does barely receive any other visitors.
Just my own visits have driven that blog from "no rank" to around "1,500,000".
Another blog I use as test had little traffic, and no rank before. Now it is on 735,000 (this blog is not part of Yakezie) and improving.
Thus just 10 people who have the Alexa toolbar installed and who visit your site every day, can bring down your Alexa ranking a lot.
That is nice, but does not give you any traffic.
What I am not sure of is if Google takes the Alexa ranking into account when defining how high your site should rank on the search results. If they would do that, a better Alexa ranking could lead to more traffic. But since it is rather easy to get a better Alexa ranking, I do not think Google will use it.