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Consider oriented graph nodes requiring periodic visits by a service agent.The agent moves among the nodes and receives a payoff for each completedservice 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 tomaximize its long-run average payoff per time unit. We show that the problem ofconstructing an $\varepsilon$-optimal schedule is PSPACE-hard for every fixed$\varepsilon \geq 0$, and that there exists an optimal periodic schedule ofexponential length. We propose randomized finite-memory (RFM) schedules as acompact description of the agent's strategies and design an efficient algorithmfor constructing RFM schedules. Furthermore, we construct deterministicperiodic schedules by sampling from RFM schedules.

Mean Payoff Optimization for Systems of Periodic Service and Maintenance