# Task Scheduler Optimization active GPUs

Incorporating the concept of Quality of Service (QoS) parameters, the task allocation problem can be reformulated to optimize both computation time and resource allocation efficiency, introducing a multi-objective optimization problem:

Minimize: $$Z = \alpha \sum\_{j=1}^{m} \sum\_{i=1}^{n} X\_{ij} \cdot T\_{ij} + \beta \sum\_{i=1}^{n} (1 - U\_i)$$

Subject to:

$$
\sum\_{j=1}^{m} X\_{ij} \cdot D\_j \leq C\_i, \ \forall i
$$

$$
\sum\_{i=1}^{n} X\_{ij} = 1, \ \forall j
$$

$$
U\_i = \frac{\sum\_{j=1}^{m} X\_{ij} \cdot D\_j}{C\_i}, \ \forall i
$$

Where:

*  $$U\_i$$represents the utilization rate of node $$i$$,
* $$\alpha$$ and $$\beta$$ are coefficients balancing the importance between minimizing computation time and maximizing resource utilization.


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