jats:pTwo simple Markov processes are examined, one in discrete and one in continuous time, arising from idealized versions of a transmission protocol for mobile networks. We consider two independent walkers moving with constant speed on 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. Then the message jumps to the other walker. Explicit expressions are derived for the long‐term average clockwise speed and number of jumps made of the message, via the solution of associated boundary value problems. The tradeoff between speed and cost (measured as the rate of jumps) is also examined.</jats:p>