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My question is this:

I want to express the time it takes for multiple servers to complete tasks for multiple users. I divide the time into $t$ time slots, and each task may occupy multiple time slots. Assuming that a user's task is uploaded in $t$ time slot, how can I express the time slot service after $t$ time slot to allocate corresponding resources for it? How can I express the total time it takes for all users to complete their tasks?

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  • $\begingroup$ Welcome to OR.SE. To me, it sounds like resource-constrained project scheduling. This link may be useful. BTW, if you could provide more information on what types of problems you have faced it would help a lot. $\endgroup$
    – A.Omidi
    May 28, 2023 at 8:48

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I am not sure how you are allocating decision variables but standard MIP will have a binary set of variables $ x_{j,t,s} = 1$ where $ j,t,s$ represent Task $j$ in process/start/occupying time slot $t$ by a server $s$,
$ x_{jts}=0$ if no tasks is assigned/in-process at time slot on the server\

You may add a user $u$ depending upon the model.
Also each of the tasks, $j$ will have predefined service/processing time $T_j$ which can be defined in terms of time slots
Then add set of constraints
$x_{j,t,s} \le \sum_{k=t+1}^{T_j-1}x_{j,k,s} + Tx_{j,t-1,s}$: where $T$ is arbitrary big number like total timespan.

$ \sum_t T_tx_{j,t,s} \le T_j$: where $ T_t$ is span of time slot $t$.

Total time for all users
$ \sum_j\sum_s\sum_t T_tx_{j,t,s}$

If using user as well then
$ \sum_u \sum_j \sum_s \sum_t T_tx_{j,t,s,u}$

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