  We revisit the non-preemptive speed-scaling problem, in which a set of jobs
have to be executed on a single or a set of parallel speed-scalable
processor(s) between their release dates and deadlines so that the energy
consumption to be minimized. We adopt the speed-scaling mechanism first
introduced in [Yao et al., FOCS 1995] according to which the power dissipated
is a convex function of the processor's speed. Intuitively, the higher is the
speed of a processor, the higher is the energy consumption. For the
single-processor case, we improve the best known approximation algorithm by
providing a $(1+\epsilon)^{\alpha}\tilde{B}_{\alpha}$-approximation algorithm,
where $\tilde{B}_{\alpha}$ is a generalization of the Bell number. For the
multiprocessor case, we present an approximation algorithm of ratio
$\tilde{B}_{\alpha}((1+\epsilon)(1+\frac{w_{\max}}{w_{\min}}))^{\alpha}$
improving the best known result by a factor of
$(\frac{5}{2})^{\alpha-1}(\frac{w_{\max}}{w_{\min}})^{\alpha}$. Notice that our
result holds for the fully heterogeneous environment while the previous known
result holds only in the more restricted case of parallel processors with
identical power functions.
