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We present and study a new model for energy-aware and profit-orientedscheduling on a single processor. The processor features dynamic speed scalingas well as suspension to a sleep mode. Jobs arrive over time, are preemptable,and have different sizes, values, and deadlines. On the arrival of a new job,the scheduler may either accept or reject the job. Accepted jobs need a certainenergy investment to be finished in time, while rejected jobs cause costs equalto their values. Here, power consumption at speed $s$ is given by$P(s)=s^{\alpha}+\beta$ and the energy investment is power integrated overtime. Additionally, the scheduler may decide to suspend the processor to asleep mode in which no energy is consumed, though awaking entails fixedtransition costs $\gamma$. The objective is to minimize the total value ofrejected jobs plus the total energy.  Our model combines aspects from advanced energy conservation techniques(namely speed scaling and sleep states) and profit-oriented scheduling models.We show that \emph{rejection-oblivious} schedulers (whose rejection decisionsare not based on former decisions) have -- in contrast to the model withoutsleep states -- an unbounded competitive ratio w.r.t\text{.} the processorparameters $\alpha$ and $\beta$. It turns out that the worst-case performanceof such schedulers depends linearly on the jobs' value densities (the ratiobetween a job's value and its work). We give an algorithm whose competitivenessnearly matches this lower bound. If the maximum value density is not too large,the competitiveness becomes $\alpha^{\alpha}+2e\alpha$. Also, we show that itsuffices to restrict the value density of low-value jobs only. Using atechnique from \cite{Chan:2010} we transfer our results to processors with afixed maximum speed.

Slow Down & Sleep for Profit in Online Deadline Scheduling