I see that we were lacking in complete documentation of PLL transfer function and noise analysis. I made this document for the same.

https://git.ligo.org/cit-ctnlab/ctn_noisebudget/blob/master/technicalNotes/Beatnote_PLL_Loop_Study.pdf

Edit Fri May 17 14:02:14 2019:

Above link is no longer maintained. Now there is a DCC doc (see LIGO-T1900263) for this analysis.