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**Scheduling networks of queues: Heavy traffic analysis of a two-station network with controllable inputs.**
*(English)*
Zbl 0724.90025

Summary: Motivated by a factory scheduling problem, we consider the problem of input control, subject to a specified product mix, and priority sequencing in a two-station multiclass queueing network with general service time distributions and a general routing structure. The objective is to minimize the long-run expected average number of customers in the system subject to a constraint on the long-run expected average output rate. Under balanced heavy loading conditions, this scheduling problem is approximated by a control problem involving Brownian motion. A reformulation of this Brownian control problem was solved exactly by the author [Math. Oper. Res. 15, No.2, 215-242 (1990; Zbl 0714.90042)]. In the present paper, this solution is interpreted in terms of the queueing network model in order to obtain an effective scheduling rule. The resulting sequencing policy dynamically prioritizes customers according to reduced costs calculated from a linear program. The input rule is a workload regulating input policy, where a customer is injected into the system whenever the expected total amount of work in the system for the two stations falls within a prescribed region. An example is presented 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 |

90C90 | Applications of mathematical programming |

60J65 | Brownian motion |

90B30 | Production models |