I'm writing some software which performs activities using an exponential back-off delay e.g. performs an action at t = 1, 2, 4, 8, 16 etc, assuming a base of 2. I want the base to adjust dynamically depending upon available resources so the activities occur more or less frequently as required.
I have arrived at the following equation:
$a = \sum_{i=1}^n\lfloor\log_b{p+t_i}\rfloor - \lfloor\log_b{t_i}\rfloor $
$a$ is the available resources, $p$ is the period in which to use those resources and $t_i$ is the amount of time since the last action on $i$. It essentially expresses how many actions will be taken on $i$ in the period $p$, then sums all the actions for all items.
How can I solve the above equation for $b$?