MHB Optimal time for scheduling the tasks

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The discussion revolves around scheduling tasks with specific execution times and prerequisites using three processors, aiming for an execution time (ω) that does not exceed 5/3 of the optimal time. The user calculated ω to be 20 but seeks clarification on determining the optimal execution time. Responses suggest that the optimal time likely refers to the scenario where all processors operate at full capacity without idle time or potentially with an infinite number of processors. The exact definition of optimal time is not provided in the user's materials, leading to uncertainty. Overall, the conversation emphasizes the importance of understanding optimal execution parameters in task scheduling.
mathmari
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Hey! 😊

We have the following tasks with the respective times of execution:
\begin{equation*}\begin{matrix}T_1/5 & T_2/4 & T_3/8 & T_4/3 & T_5/4 & T_6/6 & T_7/8 & T_8/3 & T_9/5 & T_{10}/3\end{matrix}\end{equation*} The tasks $T_3,T_4$ have as prerequisite the task $T_2$, the tasks $T_5, T_6, T_7$ have as prerequisite the task $T_4$ and the tasks $T_8, T_9$ have as prerequisite the task $T_7$. Make the schedule of these tasks using three processors so that the time of execution of the tasks $\omega$ is not more than $5/3$ of the optimal time for the execution of the tasks. First I made a diagramm of the tasks in respect of the prerequisites:

tasks.JPG
Then using 3 processors I got the following:

scheduling.JPG
Is this correct? This means that the time needed for the execution of the tasks is $\omega=20$, right?

But which is the optimal time for the execution of the tasks, since $\omega$ has to less or equal to $5/3$ of that optimal time?

Or is the way I did the above wrong? :unsure:
 
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mathmari said:
This means that the time needed for the execution of the tasks is $\omega=20$, right?
Hey mathmari!

Looks correct to me. (Nod)

mathmari said:
But which is the optimal time for the execution of the tasks, since $\omega$ has to less or equal to $5/3$ of that optimal time?
That depends on how the 'optimal time for the execution of the tasks' is defined.
Is that somewhere in your texts? I can't seem to easily find it. (Wondering)

If I had to guess given the context, I'd guess that it is the time needed if all CPU's run at 100% without wasting any time waiting.
Alternatively it might be the time needed if we had an infinite number of processors. 🤔
 
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