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In this paper, we consider the problem of processor allocation on mesh-based multiprocessor systems. We employ the idea of using migration to minimize fragmentation and the overall processing time of the tasks. In our schemes, we consider the use of task migration whenever required to improve the problem of fragmentation. To this end, we propose three efficient schemes to improve the performance of first-fit allocation strategies commonly used in practice. The first scheme, called the first-fit mesh-bifurcation (FFMB) scheme, attempts to start the search for a free submesh from either the bottom-left corner or the top-left corner of the mesh so as to reduce the amount of fragmentation in the mesh. The next two schemes, called the online dynamic compaction-single corner (ODC-SC) and online dynamic compaction-four corners (ODC-FC) schemes, use task migration to improve the performance of existing submesh allocation strategies. We perform rigorous simulation experiments based on practical workloads as reported in the literature to quantify all our proposed schemes and compare them against standard schemes existing in the literature. Based on the results, we make clear recommendations on the choice of the strategies.