### Abstract

For a discrete-time, closed, cyclic queueing network, where the nodes have independent, geometric service times, the equilibrium rate of local progress is determined. Faster nodes are shown to have a capacity depending only on the service probabilities. A family of such networks, each with the same number of types of nodes, is analyzed. If the number of nodes approaches infinity, and if the ratio of jobs to nodes has a positive limit and each node type has an asymptotic density, then for a given node type, the limits of the proportion of occupied nodes and the expected queue length are calculated. These values depend on the service parameter and on the asymptotic rate of local progress. The faster nodes can attain their capacity only when the limiting density of nodes of slowest type is zero.

Original language | English (US) |
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Pages (from-to) | 327-357 |

Number of pages | 31 |

Journal | Queueing Systems |

Volume | 31 |

Issue number | 3-4 |

State | Published - Jul 1 1999 |

### Keywords

- Asymptotic analysis
- Cyclic network
- Discrete-time queue
- Equilibrium distribution
- Queue length

### ASJC Scopus subject areas

- Statistics and Probability
- Computer Science Applications
- Management Science and Operations Research
- Computational Theory and Mathematics