I am trying to solve a scheduling problem with linear programming. I have N disks that each have a capacity of constant C. At each time interval t_i, a set of write requests with different sizes arrive that we need to store their data on the disks. Each disk has a different write speed. we can delay writes, but after writing data to disk, we need to keep data in the disk during the t_i -> t_i+1 interval and after that, we might drop the data. If we keep the data on disk we can save time if another write request with the same data arrives in the future (like caching). The goal is to minimize the writing time.
My question is how to keep track of time progress?
I know how to solve problems like this:
$$ C_i = \Sigma a_ix_{i} $$ $$ subject\space to: $$ $$ x_i \leq b_i $$
But here I think the model should be like this:
$$ C_i = \Sigma a_ix_{i} $$ $$ subject\space to: $$ $$ t_i < t < t_{i+1} $$ $$ etc. $$
Any suggestions are appreciated.
p.s. I have asked this question on StackOverflow but I think here is more relevant.
