  Two simple Markov processes are examined, one in discrete and one in
continuous time, arising from idealized versions of a transmission protocol for
mobile, delay-tolerant networks. We consider two independent walkers moving
with constant speed on either the discrete or continuous circle, and changing
directions at independent geometric (respectively, exponential) times. One of
the walkers carries a message that wishes to travel as far and as fast as
possible in the clockwise direction. The message stays with its current carrier
unless the two walkers meet, the carrier is moving counter-clockwise, and the
other walker is moving clockwise. In that case, the message jumps to the other
walker. The long-term average clockwise speed of the message is computed. An
explicit expression is derived via the solution of an associated boundary value
problem in terms of the generator of the underlying Markov process. The average
transmission cost is also similarly computed, measured as the long-term number
of jumps the message makes per unit time. The tradeoff between speed and cost
is examined, as a function of the underlying problem parameters.
