The C-14 model that was used can't be easily solved for the production rate of C-14, α. If however we assume that the quantity of nitrogen in the atmosphere is approximately constant then we can get a solution if the other rates and the C-14 equilibrium value are known. We have to use the following state diagram.
It turns out that the solution is α = λBeq/A where A is the amount of nitrogen and Beq the equilibrium value for the C-14 and λ = γ + β. The equilibrium value is then approximately proportional to the production rate and dependent of the rate of removal from the atmosphere.
Supplemental (18 Jul): αA = λB is a steady state condition and is the connection between the two models. Note that the flows along the branches for both models are the same to a high degree of approximation when e-μt is approximately one. One can think of D as a necessary dump.