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We have identified the limitations of the existing §work in the area of processor allocation in mesh-§connected multicomputers. These limitations form the §motives behind undertaking such a research. We can §summarise the main contributions of this research §into three main points. The first part of this §research presents an efficient contiguous processor §allocation algorithm, referred to as Turning Busy §List (or TBL for short), for 3D mesh-connected §multicomputers that overcome the limitations of the §existing strategies suggested previously for the 3D §mesh networks. The second part of this research §proposes a new non-contiguous processor allocation §algorithm, referred to as Greedy Available Busy List §(or GABL for short), for the 2D mesh-connected §multicomputer. And finally, we study the impact of §heavy-tailed job execution times on the performance §of the existing contiguous allocation strategies for §3D mesh-connected multicomputers, including the ones §proposed in this research.