### Abstract

We consider networks where at each node there is a single exponential server with a service rate which is a non-decreasing function of the queue length. The asymptotic profile of a sequence of networks consists of the set of persistent service rates, the limiting customer-to-node ratio, and the limiting service-rate measure. For a sequence of cyclic networks whose asymptotic profile exists, we compute upper and lower bounds for the limit points of the sequence of throughputs as functions of the limiting customer-to-node ratio. We then find conditions under which the limiting throughput exists and is expressible in terms of the asymptotic profile. Under these conditions, we determine the limiting queue-length distributions for persistent service rates. In the absence of these conditions, the limiting throughput need not exist, even for increasing sequences of cyclic networks.

Original language | English (US) |
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Pages (from-to) | 191-219 |

Number of pages | 29 |

Journal | Queueing Systems |

Volume | 58 |

Issue number | 3 |

DOIs | |

State | Published - Mar 1 2008 |

### Keywords

- Asymptotic queue length
- Convergence of throughput
- Cyclic networks
- Product form
- State-dependent service

### ASJC Scopus subject areas

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