I have a seemingly complicated recurrence relation for which I am having a hard time determining a closed form solution.
Given: $d_0$, $u$, $c$, $s$
$d_n = d_{n-1} - u + c(\lceil \frac{s}{d_{n+1}} \rceil - 1)$
For clarity, this relation is for inventory management purposes. I am using the ceiling function as all $d_n \gt \frac s2$ and when $d_{n+1} \lt s$, $d_n = d_{n-1} - u + c$. Otherwise, if $d_{n+1} \gt s$, $d_n = d_{n-1} - u$.
Any help or direction towards getting to a closed form solution would be appreciated. Also, I'm sure the main inhibitor to finding a closed form solution will be the ceiling function, so perhaps there is another approach that would lend itself better to this application--if so, I'm all ears.
Thanks.