### Abstract

We consider a two-node fluid network with batch arrivals of random size having a heavy-tailed distribution. We are interested in the tail asymptotics for the stationary distribution of a two-dimensional workload process. Tail asymptotics have been well studied for twodimensional reflecting processes where jumps have either a bounded or an unbounded light-tailed distribution. However, the presence of heavy tails totally changes these asymptotics. Here we focus on the case of strong stability where both nodes release fluid at sufficiently high speeds to minimise their mutual influence. We show that, as in the one-dimensional case, big jumps provide the main cause for workloads to become large, but now they can have multidimensional features. We first find the weak tail asymptotics of an arbitrary directional marginal of the stationary distribution at Poisson arrival epochs. In this analysis, decomposition formulae for the stationary distribution play a key role. Then we employ sample-path arguments to find the exact tail asymptotics of a directional marginal at renewal arrival epochs assuming one-dimensional batch arrivals.

Original language | English |
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Pages (from-to) | 249-265 |

Number of pages | 17 |

Journal | Journal of Applied Probability |

Volume | 51A |

Publication status | Published - 1 Jan 2014 |

### Keywords

- Fluid network
- Heavy-tailed distribution of batch size
- Poisson and renewal arrivals
- Stability
- Strong stability
- Workload process

### ASJC Scopus subject areas

- Mathematics(all)
- Statistics and Probability
- Statistics, Probability and Uncertainty

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## Cite this

*Journal of Applied Probability*,

*51A*, 249-265.