  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.
