Robust Online Speed Scaling With Deadline Uncertainty.

A speed scaling problem is considered, where time is divided into slots, and jobs with payoff $v$ arrive at the beginning of the slot with associated deadlines $d$. Each job takes one slot to be processed, and multiple jobs can be processed by the server in each slot with energy cost $g(k)$ for processing $k$ jobs in one slot. The payoff is accrued by the algorithm only if the job is processed by its deadline. We consider a robust version of this speed scaling problem, where a job on its arrival reveals its payoff $v$, however, the deadline is hidden to the online algorithm, which could potentially be chosen adversarially and known to the optimal offline algorithm. The objective is to derive a robust (to deadlines) and optimal online algorithm that achieves the best competitive ratio. We propose an algorithm (called min-LCR) and show that it is an optimal online algorithm for any convex energy cost function $g(.)$. We do so without actually evaluating the optimal competitive ratio, and give a general proof that works for any convex $g$, which is rather novel. For the popular choice of energy cost function $g(k) = k^\alpha, \alpha \ge 2$, we give concrete bounds on the competitive ratio of the algorithm, which ranges between $2.618$ and $3$ depending on the value of $\alpha$. The best known online algorithm for the same problem, but where deadlines are revealed to the online algorithm has competitive ratio of $2$ and a lower bound of $\sqrt{2}$. Thus, importantly, lack of deadline knowledge does not make the problem degenerate, and the effect of deadline information on the optimal competitive ratio is limited.