Routing for Power Minimization in the Speed Scaling Model

We study network optimization that considers power minimization as an objective. Studies have shown that mechanisms such as speed scaling can significantly reduce the power consumption of telecommunication networks by matching the consumption of each network element to the amount of processing required for its carried traffic. Most existing research on speed scaling focuses on a single network element in isolation. We aim for a network-wide optimization. Specifically, we study a routing problem with the objective of provisioning guaranteed speed/bandwidth for a given demand matrix while minimizing power consumption. Optimizing the routes critically relies on the characteristic of the speed–power curve
$f(s)$
, which is how power is consumed as a function of the processing speed
$s$
. If
$f$
is superadditive, we show that there is no bounded approximation in general for integral routing, i.e., each traffic demand follows a single path. This contrasts with the well-known logarithmic approximation for subadditive functions. However, for common speed–power curves such as polynomials
$f(s) = \mu s^{\alpha}$
, we are able to show a constant approximation via a simple scheme of randomized rounding. We also generalize this rounding approach to handle the case in which a nonzero startup cost
$\sigma$
appears in the speed–power curve, i.e.,
$f(s) = \cases{0, & if $s=0$\cr \sigma + \mu s^{\alpha},& if $s>0$.}$
We present an
$O((\sigma /\mu)^{1/\alpha})$
-approximation, and we discuss why coming up with an approximation ratio independent of the startup cost may be hard. Finally, we provide simulation results to validate our algorithmic approaches.