Among several multiprocessor topologies, two-dimensional (2D) mesh topology has become popular due to its simplicity and efficiency. Even though a number of scheduling and processor allocation schemes for 2D meshes have been proposed in the literature, little study has been done aimed for real-time environment. In this paper, we propose an on-line scheduling and allocation scheme for real-time tasks that require the exclusive use of submeshes in 2D mesh system. By effectively manipulating the information on allocated or reserved submeshes, the proposed scheme can quickly identify the earliest available time of a free submesh for a newly arrived task. We employ a limited preemption approach to reduce the complexity of the search for a feasible schedule. Computer simulation reveals that the proposed scheme allows high throughput by decreasing the number of tasks rejected.