This problem recurs in my business frequently, and it would be really handy to have a closed-form formula for it.
Assume we have a known number of items "X(t)" items at some time t. The lifespan of an item is "L", after which it must be replaced. We want to grow the total population of items at some constant rate "G" per interval, i.e. X(t+1) = G * X(t).
I want to calculate the number of new items "N(t)" required as a function of t. The number of new items is that number which both accomplishes a growth of G after replacing the items which were added at time t-L. So, A(t) = G*X(t-1) + A(t-L).
Any help would be much appreciated.
Let $A(t)$ be the number of items added at time $t$. Let $X(t)$ be the total number of items in stock.
Start with $X(0)=0$. Then $X(1)=A(1)$, $X(2)=A(2)$, etc until you get to the point where items reach their end of life.
For a life of $L$ this means that the first time we notice something different is $X(L)=X(L-1)+A(L)-A(0)$
For future times we have $X(t)=X(t-1)+A(t)-A(t-L)$
You want $X(t)=GX(t-1)$
So $GX(t-1)=X(t-1)+A(t)-A(t-L)$
Or $(G-1)X(t-1)=A(t)-A(t-L)$
Moving on one step we get $(G-1)X(t)=A(t+1)-A(t+1-L)$
But $X(t)=GX(t-1)$, so $(G-1)GX(t-1)=A(t+1)-A(t+1-L)$
Multiplying our earlier equation by $G$ gives $(G-1)GX(t-1)=GA(t)-GA(t-L)$
Putting them together gives $A(t+1)-A(t+1-L)=GA(t)-GA(t-L)$
This gives the recurrence relationship $A(t+1)-GA(t)-A(t+1-L)+GA(t-L)=0$
This can be solved to give you the formula you need for $A(t)$
Step the times up by $L$ to give: $A(t+L+1)-GA(t+L)-A(t+1)+GA(t)=0$
This recurrence relationship has the characteristic equation $\lambda^{L+1}-G\lambda^L-\lambda+G=0$
We find the roots of this equation:
$\lambda^L(\lambda-G)-(\lambda-G)=0$
$(\lambda^L-1)(\lambda-G)=0$
$\lambda^L=1$ or $\lambda=G$
$\lambda=G$ or $\lambda=1$ or $\lambda=\cos (\frac{2k\pi}L)+i\sin (\frac{2k\pi}L)$ for $k \in \lbrace 1, 2, ..., L-1\rbrace$ ($L$th roots of unity)
This gives us $A(t)=aG^t+b+\Sigma \lbrace c_k \cos (\frac{2kt\pi}L)+d_k\sin (\frac{2kt\pi}L)\rbrace$
Realistically the $\cos$ and $\sin$ terms are going to be small compared to $G^t$, so you can say that:
$A(t) \approx aG^t+b$