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**On the improvement from scheduling a two-station queueing network in heavy traffic.**
*(English)*
Zbl 0755.60075

Summary: For a two-station multiclass queueing network in heavy traffic, we assess the improvement from scheduling (job release and priority sequencing) that can occur relative to Poisson input and first-come first-served (FCFS) sequencing. In particular, simple upper bounds are derived on the optimal objective function value [found in the paper of the second author, Math. Oper. Res. 15, No. 2, 215-242 (1990; Zbl 0714.90042)] of a Brownian control problem that approximates (via J. M. Harrison’s model [Stochastic differential systems, stochastic control theory and applications, Proc. Workshop Minneapolis/Minn. 1986, IMA Vol. Math. Appl. 10, 147-186 (1988; Zbl 0658.60123)]) a two-station queueing network scheduling problem in heavy traffic. When the system is perfectly balanced, the Brownian analysis predicts that optimal scheduling will reduce the long run expected average number of customers in the network by at least a factor of four relative to the Poisson input, FCFS sequencing policy that achieves the same throughput rate. When the system is not perfectly balanced, the corresponding factor is slightly smaller than two.

### MSC:

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

60K25 | Queueing theory (aspects of probability theory) |

90B22 | Queues and service in operations research |

### Keywords:

Brownian approximations; queueing network in heavy traffic; two-station queueing network scheduling problem in heavy traffic
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\textit{J. Ou} and \textit{L. M. Wein}, Oper. Res. Lett. 11, No. 4, 225--232 (1992; Zbl 0755.60075)

### References:

[1] | Baskett, F.; Chandy, K. M.; Muntz, R. R.; Palacios, F. G., Open, closed and mixed networks of queues with different classes of customers, J. Assoc. Comput. Mach., 22, 248-260 (1975) · Zbl 0313.68055 |

[2] | Harrison, J. M., Brownian models of queueing networks with heterogeneous customer populations, (Fleming, W.; Lions, P. L., Stochastic Differential Systems, Stochastic Control Theory and Applications, IMA Volume 10 (1988), Springer-Verlag: Springer-Verlag New York), 147-186 · Zbl 0658.60123 |

[3] | Kelly, F. P., Reversibility and Stochastic Networks (1979), John Wiley: John Wiley New York · Zbl 0422.60001 |

[4] | Kushner, H. J.; Martins, L. F., Limit theorems for pathwise average cost per unit time problems for queues in heavy traffic, (Lefschetz Center for Dynamical Systems Report #91-2 (1991), Brown University) · Zbl 0806.60019 |

[5] | Kushner, H. J.; Ramachandran, K. M., Optimal and approximately optimal control policies for queues in heavy traffic, SIAM J. Control Optim., 27, 1293-1318 (1989) · Zbl 0691.93068 |

[6] | Martins, L. F.; Kushner, H. J., Routing and singular control for queueing networks in heavy traffic, SIAM J. Control Optim., 28, 1209-1233 (1990) · Zbl 0718.90029 |

[7] | Wein, L. M., Optimal control of a two-station Brownian network, Math. Oper. Res., 15, 215-242 (1990) · Zbl 0714.90042 |

[8] | Wein, L. M., Scheduling networks of queues: Heavy traffic analysis of a two-station network with controllable inputs, Oper. Res., 38, 1065-1078 (1991) · Zbl 0724.90025 |

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