We propose two modifications of the n-process mutual exclusion algorithm by Peterson for the asynchronous multi-writer/reader shared memory model. By any of the modifications we can speed up the original n-process algorithm. The running times for the trying regions of the first modified algorithm and the second modified algorithm are (2n - 3)c + O(n3 l) and (n - 1)c + O(n3 l), respectively, where n is the number of processes, l is an upper bound on the time between two steps, and c is an upper bound on the time that any user spends in the critical region. These running times are improvements on the running time, O(n2c + n4 l) of the original n-process algorithm for the same asynchronous shared memory model.