A Heuristic Scheduling of Independent Tasks with Bottleneck Resource Constraints

Abstract

In this paper, the problem of scheduling independent tasks with bottleneck resource constraints is investigated. There is a set of independent tasks $\mathcal{T} = \{T_{1}, \dotsc , T_{n}\}$ and a set of resources $\mathcal{R} = \{R_{1}, \dotsc , R_{m}\}$. The available amount of resource $R_{j}$, for $j = 1, \dotsc , m$, is (normalized to) 1. Each task $T_{i}$’s execution requires at least $\lambda_{(i,j)} \leq 1$ units of resource $R_{j} \in \mathcal{R}$, and with these minimal resources, the execution time of $T_{i}$, is $\tau_{i}$. If there is a resource $R_{\beta (i)}$ so that $T_{i}$ can use $\xi$ units $(\xi \geq \lambda_{(i,\beta (j))})$ and reduce the execution time to $\frac{{\lambda _{(i,\beta (i))} }}{\xi }\tau _i $, we say that $T_{i}$ can achieve linear speed-up with respect to $R_{\beta (i)}$. It is assumed that for each $T_{i}$, there is one, and only one, resource $R_{\beta (i)}$ that is $T_{i}$’s bottleneck resource: $T_{i}$ can use $\xi$ units of ${R_{(i,\beta (i))} }$ and achieve linear speed-up, where $\xi$ is between $\lambda _{(i,\beta (i))}$ and $\Lambda _{(i,\beta (i))} \leq 1$. The problem is to find a feasible schedule for all the tasks in $\mathcal{T}$ that has a shortest overall makespan. This problem is a combination of two previously studied problems—scheduling tasks with resource constraints [SIAM J. Comput., 4 (1975), pp. 187–200.] scheduling parallel tasks [SIAM J. Comput., 21 (1992), pp. 281–294]. A variant of this problem was studied in [J. Combin. Theory, 21 (1976), pp. 257–298]. Because the problem is NP-hard, we propose the ECT (earliest completion time) algorithm as a heuristic solution and show that the performance ratio of the ECT makespan $M_{\text{ECT}}$ to the optimal makespan $M_{\text{OPT}}$ is bounded by $2l + m + 1$, where l is the number of the bottleneck resources in $\mathcal{R}$. When $l = 0$, this is exactly the performance bound shown by Garey and Graham in [SIAM J. Comput., 4 (1975), pp. 187–200].