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An efficient task scheduling in a heterogeneous computing (HC) system is an important issue to meet the computational requirements of a large number of parallel applications. In this paper, we have proposed a non-preemptive static scheduling algorithm for the HC systems, called the multiple strict bound constraints (MSBC) algorithm. The MSBC algorithm employs an alternative task-prioritized scheme in stead of that used by the heterogeneous earliest finish time (HEFT) algorithm. Additionally, we also exploit available idle time slots in every scheduling round of the MSBC algorithm. In order to provide an unbiased comparison with the existing algorithms, the randomly generated graphs are considered as experimental test-beds. The experimental results show the superiority of our proposed algorithm to those presented previously. Therefore, our proposed algorithm may be used to raise the performance of the parallel applications executed in the HC systems.