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**Scheduling networks of queues: Heavy traffic analysis of a two-station closed network.**
*(English)*
Zbl 0723.90026

Summary: We consider a multiclass closed queueing network with two single-server stations. Each class requires service at a particular station, and customers change class after service according to specified probabilities. There is a general service time distribution for each class. The problem is to schedule the two servers to maximize the long- run average throughput of the network. By assuming a large customer population and nearly balanced loading of the two stations, the scheduling problem can be approximated by a dynamic control problem involving Brownian motion. A reformulation of this control problem is solved exactly and the solution is interpreted in terms of the queueing network to obtain a scheduling rule. (We conjecture, quite naturally, that the resulting scheduling rule is asymptotically optimal under heavy traffic conditions, but no attempt is made to prove that.) The scheduling rule is a static priority policy that computes an index for each class and awards higher priority at station 1 (respectively, station 2) to classes with the smaller (respectively, larger) values of this index. An analytic comparison of this rule to any other static policy is also obtained. An example is given that illustrates the procedure and demonstrates its effectiveness.

### MSC:

90B22 | Queues and service in operations research |

60K20 | Applications of Markov renewal processes (reliability, queueing networks, etc.) |

93C95 | Application models in control theory |

90B35 | Deterministic scheduling theory in operations research |

90B15 | Stochastic network models in operations research |

60J65 | Brownian motion |

49J55 | Existence of optimal solutions to problems involving randomness |