# Optimal control of a single server in a finite-population queueing network

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## Abstract

We study the optimal dynamic assignment of a single server to multiple stations in a finite-population queueing network. The objective is to maximize the long-run average reward/throughput. We use sample-path comparisons to identify conditions on the network structure and service time distributions under which the optimal policy is an index policy. This index policy assigns the server to the non-empty station where it takes the shortest amount of time (in some stochastic sense) to complete a job. For example, in a network of multiple parallel stations, the optimal policy assigns the highest priority to the fastest station if service times can be ordered in likelihood ratios. Finally, by means of a numerical study, we test the shortest-expected-remaining-service-time policy on parallel-series networks with three stations and find that this index policy either coincides with the optimal policy or provides a near-optimal performance.

## Keywords

Scheduling Queueing networks Likelihood ratio ordering SEPT## Mathematics Subject Classification

68M20 60K25 90B22 90B36## Notes

### Acknowledgements

The authors thank the associate editor and an anonymous referee for their comments on an earlier version of this article, which significantly improved it. The work of the first author was partially supported by the National Science Foundation under grants CMMI–1234212 and CMMI–1635574.

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