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Today’s multi-computer systems are heterogeneous in nature, ie, the machines they are composed of, have varying processing capabilities and are interconnected through high speed networks, thus, making them suitable for performing diverse set of computing-intensive applications. In order to exploit the high performance of such a distributed system, efficient mapping of the tasks on available machines is necessary. This is an active research topic and different strategies have been adopted in literature for the mapping problem. A novel approach has been introduced in the paper for the efficient mapping of the DAG-based applications. The approach that takes into account the lower and upper bounds for the start time of the tasks. The algorithm is based on list scheduling approach and has been compared with the well known list scheduling algorithms existing in the literature. The comparison results for the randomly synthesized graphs as well as the graphs from the real world elucidate that the proposed algorithm significantly outperforms the existing ones on the basis of different cost and performance metrics.