Datasets:

Monketoo
/

math-docs-dataset

First we detail the optimization problem in Section 3.1. Then we describe the four energy models in Section 3.2. Finally, we illustrate the models and motivate the problem with an example in Section 3.3.

3.1. Optimization problem

Consider an application task graph $\mathcal{G} = (V, \mathcal{E})$, with $n = |V|$ tasks denoted as $V = {T_1, T_2, \ldots, T_n}$, and where the set $\mathcal{E}$ denotes the precedence edges between tasks. Task $T_i$ has a cost $w_i$ for $1 \le i \le n$. We assume that the tasks in $\mathcal{G}$ have been allocated onto a parallel platform made up of identical processors. We define the execution graph generated by this allocation as the graph $\mathcal{G} = (V, E)$, with the following augmented set of edges:

• $\mathcal{E} \subseteq E$: if an edge exists in the precedence graph, it also exists in the execution graph;

• if $T_1$ and $T_2$ are executed successively, in this order, on the same processor, then $(T_1, T_2) \in E$.

The goal is to the minimize the energy consumed during the execution while enforcing a deadline D on the execution time. We formalize the optimization problem in the simpler case where each task is executed at constant speed. This strategy is optimal for the CONTINUOUS model (by a convexity argument) and for the DISCRETE and INCREMENTAL models (by definition). For the VDD-HOPPING model, we reformulate the problem in Section 5.1. For each task $T_i \in V$, $b_i$ is the starting time of its execution, $d_i$ is the duration of its execution, and $s_i$ is the speed at which it is executed. We obtain the following formulation of the MINENERGY(G, D) problem, given an execution graph G = (V, E) and a deadline D; the $s_i$ values are variables, whose values are constrained by the energy model (see Section 3.2).

$\begin{array}{ll} \text{Minimize} & \displaystyle\sum_{i=1}^{n} s_i^3 \times d_i \\ \text{subject to} & (i) \quad w_i = s_i \times d_i \quad \text{for each task } T_i \in V \\ & (ii) \quad b_i + d_i \leq b_j \quad \text{for each edge } (T_i, T_j) \in E \\ & (iii) \quad b_i + d_i \leq D \quad \text{for each task } T_i \in V \\ & (iv) \quad b_i \geq 0 \quad \text{for each task } T_i \in V \end{array} \tag{1}$

Constraint (i) states that the whole task can be executed in time $d_i$ using speed $s_i$. Constraint (ii) accounts for all dependencies, and constraint (iii) ensures that the execution time does not exceed the deadline $D$. Finally, constraint (iv) enforces that starting times are non-negative. The energy consumed throughout the execution is the objective function. It is the sum, for each task, of the energy consumed by this task, as we detail in the next section. Note that $d_i = w_i/s_i$, and therefore the objective function can also be expressed as $\sum_{i=1}^{n} s_i^2 \times w_i$.

Note that, whatever the energy model, there is a maximum speed that cannot be exceeded, denoted $s_{max}$. We point out that there is a solution to the minimization problem if and only if there is a solution with $s_i = s_{max}$ for all $1 \le i \le n$. Such a solution would correspond to executing each task as early as possible (according to constraints (ii) and (iv)) and as fast as possible. The optimal solution then slows down tasks to save as much energy as possible, while enforcing the deadline constraint. There is no guarantee on the uniqueness of the solution,