Dynamic visit-order rules for batch-service polling

We explore visit-order policies in nonsymmetric polling systems with switch-in and switch-out times, where service is in batches of unlimited size. We concentrate on so-called "Hamiltonian tour" policies in which, in order to give a fair treatment to the various users, the server attends every nonempty queue exactly once during each round of visits (cycle). The server dynamically generates a new visit schedule at the start of each round, depending on the current state of the system (number of jobs in each queue) and on the various nonhomogeneous system parameters. We consider three service regimes, globally gated, (locally) gated, and exhaustive, and study three different performance measures: (1) minimizing the expected weighted sum of all sojourn times of jobs within a cycle; (2) minimizing the expected length of the next cycle, and (3) maximizing the expected weighted throughput in a cycle. For each combination of performance measure and service regime, we derive characteristics of the optimal Hamiltonian tour. Some of the resulting optimal policies are shown to be elegant index-type rules. Others are the solutions of deterministic NP-hard problems. Special cases are reduced to assignment problems with specific cost matrices. The index-type rules can further be used to construct fixed-order, cyclic-type polling tables in cases where dynamic control is not applicable.