  Unlimited asynchronism is intolerable in real physically distributed computer
systems. Such systems, synchronous or not, use clocks and timeouts. Therefore
the magnitudes of elapsed absolute time in the system need to satisfy the axiom
of Archimedes. Under this restriction of asynchronicity logically
time-independent solutions can be derived which are nonetheless better (in
number of message passes) than is possible otherwise. The use of clocks by the
individual processors, in elections in a ring of asynchronous processors
without central control, allows a deterministic solution which requires but a
linear number of message passes. To obtain the result it has to be assumed that
the clocks measure finitely proportional absolute time-spans for their time
units, that is, the magnitudes of elapsed time in the ring network satisfy the
axiom of Archimedes. As a result, some basic subtilities associated with
distributed computations are highlighted. For instance, the known nonlinear
lower bound on the required number of message passes is cracked. For the
synchronous case, in which the necessary assumptions hold a fortiori, the
method is -asymptotically- the most efficient one yet, and of optimal order of
magnitude. The deterministic algorithm is of -asymptotically- optimal bit
complexity, and, in the synchronous case, also yields an optimal method to
determine the ring size. All of these results improve the known ones.
