The majority of current distributed mutual exclusion algorithms are not suited for parallel or distributed applications on a Grid as they do not consider the heterogeneity of latency on Grids. We propose two distributed mutual exclusion algorithms, based on Naimi-Trehel's token-based algorithm, which take into account latency gaps, especially those between local and remote clusters of machines. Our first algorithm exploits cluster locality by giving higher priority to critical section requests issued from nodes of the same cluster when compared to those from remote nodes. Our second algorithm adds a router layer to the first algorithm, bringing it closer to Grid network topology. Viewing each cluster as a single node, the Naimi-Trehel algorithm is applied to this router layer. Redirection of inter-cluster messages to cluster's nodes is then minimized.