If I understand this correctly, the 'slow' trains work like a sort of express bus system (or, to be more precise, a streetcar system). They do a slow milk run in the neighbourhood, picking people up. Then, after leaving the slow area, it speeds up to dock with the train, where you transfer over. It's kind of complicated, but I could see it working. So here would be your travel day. Wake up, catch the slow train at the corner, then after a short while, transfer over to the train. Then, when the slow train for your destination docks, transfer to that one. Get off at the stop for work, and walk the rest of the way. The idea here is to cut out the middleman. Instead of having to wait at another bus (train) station for the next high speed train to arrive, you simply transfer directly to the train.
There are, admittedly, a few problems. First off, this would only save time for people who have to regularly make the sort of 'bus-train-bus' connection. Secondly, this doesn't seem very error-proof. If people can't make the transfer fast enough, then you end up being stuck on the slow train until you can make another pass at the next one. Thirdly, you'd need quite a large section to make sure you have enough time to make the transfers.
That being said, this is definitely an interesting idea. I'd like to see someone work all the kinks out, though.