When you fall off into space and can't get back up you have a problem. This problem is worse if you are fat, weak, or have a skinny rope. The old way or normal way is to grab the rope with both hands and do a pull-up and let go, if your belayer then takes in you gain 6 inches to a foot. Its actually hard to do this and with ropes that are 9.4 and less its really really hard. If you only have a short way to go, you can put your heel on the rope and layback up the rope, this can get you up to where you can touch the rock and then climb up the other rope.
If you are way down the rope you can layback up the rope and then just hand over hand the rope, if you are very strong you might make it, probably not. If you keep lay backing in series and your belayer takes in it's less strenuous than pull up/ pumping the rope.
If you have one prussic knot and one crab you layback up the rope clip in to the knot and then keep lay backing the rope. This will get you up the rope slowly and securely if you keep shorting the rope with knots.
The little bit of rope for prussicing is better if its not too thin or too thick. Too thin and it sticks, and two fat it doesn't grip. One modern gadget like a rope man or a such will work much better. Two little bits of rope sling or two light gadgets will work really very well.
The purpose of this was to show you it's possible with one bit of sling and one carabiner. In an emergency you should have one bit of string somewhere and one crab.
Here's another little tip. The figure of 8 is my chosen knot for tying in, and it's this because I never get it wrong, I can do it in the dark etc.
If you tie it like this, which is with the end pushed thru a loop so it makes it neater and less likely to grab an end to clip the rope thru runners, its also stronger. It's also stronger which maybe makes it the strongest knot for tying in, this is an important consideration as the bowline which is dangerous in my opinion was always touted as the strongest.
This final photo is with end double threaded thru. Try to understand what is happening here before you try and use it like this, or don't use it. If I fall on this knot it will take impact force out of the system during a fall and so a few things happen, there is less force on everything which is always good, the knot becomes even stronger, you can untie the rope by pulling on the end and its easier, and the rope and system take less bang. If you are on marginal trad gear, like rp 1s and 2s, loose rock, expanding flakes you will make them a few percent stronger, in other words they might hold.
There you go that's my Xmass present to you all.