Interweaving Real-Time Jobs with Energy Harvesting to Maximize Throughput

Motivated by baterryless IoT devices, we consider the following scheduling problem. The input includes $n$ unit time jobs $\mathcal{J} = \{J_1, \ldots, J_n\}$, where each job $J_i$ has a release time $r_i$, due date $d_i$, energy requirement $e_i$, and weight $w_i$. We consider time to be slotted; hence, all time related job values refer to slots. Let $T=\max_i\{d_i\}$. The input also includes an $h_t$ value for every time slot $t$ ($1 \leq t \leq T$), which is the energy harvestable on that slot. Energy is harvested at time slots when no job is executed. The objective is to find a feasible schedule that maximizes the weight of the scheduled jobs. A schedule is feasible if for every job $J_j$ in the schedule and its corresponding slot $t_j$, $t_{j} \neq t_{j'}$ if ${j} \neq {j'}$, $r_j \leq t_j \leq d_j$, and the available energy before $t_j$ is at least $e_j$. To the best of our knowledge, we are the first to consider the theoretical aspects of this problem. In this work we show the following. (1) A polynomial time algorithm when all jobs have identical $r_i, d_i$ and $w_i$. (2) A $\frac{1}{2}$-approximation algorithm when all jobs have identical $w_i$ but arbitrary $r_i$ and $d_i$. (3) An FPTAS when all jobs have identical $r_i$ and $d_i$ but arbitrary $w_i$. (4) Reductions showing that all the variants of the problem in which at least one of the attributes $r_i$, $d_i$, or $w_i$ are not identical for all jobs are NP-Hard.