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**Qualitative properties of \(\alpha\)-fair policies in bandwidth-sharing networks.**
*(English)*
Zbl 1304.60102

Flow-level network models are simplified versions of complex models which are useful, e.g., for Internet traffic description and evaluation for admission control. The most important question to be attacked with such models is to find conditions for stabilization of the network, i.e., for ergodicity of the describing Markov processes. Given this, the natural question is to decide about finiteness of stationary flows and to provide bounds for the mean number of ongoing flows in the stationary state. Under the so-called \(\alpha\)-fair bandwidth sharing policy of network capacities, the authors derive such bounds for the transient finite time phase of the network and for the stationary state as well. For underloaded networks, exponentially decaying bounds for the tails of the steady state distribution are derived and under heavy-traffic assumptions, the diffusion scaling provides insight into the transient as well as the limiting behaviour.

It turns out that the (positive) value of \(\alpha\) is decisive for obtaining some of the results. Another interesting aspect is that introducing new Lyapunov functions can lead to better bounds than those so far available in the literature.

It turns out that the (positive) value of \(\alpha\) is decisive for obtaining some of the results. Another interesting aspect is that introducing new Lyapunov functions can lead to better bounds than those so far available in the literature.

Reviewer: Hans Daduna (Hamburg)

### MSC:

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

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

68M12 | Network protocols |

68M20 | Performance evaluation, queueing, and scheduling in the context of computer systems |

60J70 | Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.) |

### Keywords:

\(\alpha\)-fair bandwidth-sharing policy; Lyapunov function; steady state distributions; state-space collapse; heavy traffic; flow-level description
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\textit{D. Shah} et al., Ann. Appl. Probab. 24, No. 1, 76--113 (2014; Zbl 1304.60102)

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