The run is on concrete. So I'm hoping a deep litter method, the pine shavings seem to be a good method, and the coop and run are well covered from the elements, so I probably won't have too much to worry about.
As others have already mentioned, if your run is sitting on concrete, then you don't have a true deep litter system which is organic material placed on top of soil, and you also need moisture to get the composting up and running. If your run has a roof overhead and no rain water gets on the litter, then it will not really compost like you seem to want.
Essentially, what you have is a dry deep bedding system both inside the coop and in the covered run. That is not bad, and you will probably find that you don't have to clean it out very often. I only clean out my coop dry deep bedding twice a year, and I could easily get by with maybe only once a year cleaning. I throw out all my old coop bedding into my chicken run compost system, which is open to rainfall and sits on top of soil.