  Consider oriented graph nodes requiring periodic visits by a service agent.
The agent moves among the nodes and receives a payoff for each completed
service task, depending on the time elapsed since the previous visit to a node.
We consider the problem of finding a suitable schedule for the agent to
maximize its long-run average payoff per time unit. We show that the problem of
constructing an $\varepsilon$-optimal schedule is PSPACE-hard for every fixed
$\varepsilon \geq 0$, and that there exists an optimal periodic schedule of
exponential length. We propose randomized finite-memory (RFM) schedules as a
compact description of the agent's strategies and design an efficient algorithm
for constructing RFM schedules. Furthermore, we construct deterministic
periodic schedules by sampling from RFM schedules.
