Concurrent systems in which there is a known upper bound $\Delta$ on memory access time are considered. Two prototypical synchronization problems, mutual exclusion and consensus, are studied and solutions that have constant (i.e. independent of $\Delta$ and the total number of processes) time complexity in the absence of contention are presented. For mutual exclusion, in the absence of contention, a process needs only five accesses to the shared memory to enter its critical section, and in the presence of contention, the winning process may need to delay itself for $4\Delta$ time units. For consensus, in absence of contention, a process decides after four accesses to the shared memory, and in the presence of contention, it may need to delay itself for $\Delta$ time units.
Distributed Computing 10(1): 1-10, 1996. A preliminary version appeared in Proceedings of the 13th IEEE Symposium on Real-Time Systems (RTSS 1992), pp. 12-21.