### Abstract

In queueing theory, an important class of events can be written as ‘infinite intersections’. For instance, in a queue with constant service rate c, busy periods starting at 0 and exceeding L > 0 are determined by the intersection of the events ⋂

_{t∈[0,L]}{Q0=0,At>ct} , i.e., queue Q_{ t}is empty at 0 and for all t∊ [0, L] the amount of traffic A_{t}arriving in [0,t) exceeds the server capacity. Also the event of exceeding some predefined threshold in a tandem queue, or a priority queue, can be written in terms of this kind of infinite intersections. This paper studies the probability of such infinite intersections in queueing systems fed by a large number n of i.i.d. traffic sources (the so-called ‘many-sources regime’). If the sources are of the exponential on-off type, and the queueing resources are scaled proportional to n, the probabilities under consideration decay exponentially; we explicitly characterize the corresponding decay rate. The techniques used stem from large deviations theory (particularly sample-path large deviations).Original language | English |
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Pages (from-to) | 5-20 |

Journal | Queueing Systems |

Volume | 54 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2006 |

MoE publication type | A1 Journal article-refereed |

### Keywords

- sample-path large deviations
- on-off processes
- busy period
- tandem queue
- priority queue
- queueing theory

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## Cite this

Mandjes, M., & Mannersalo, P. (2006). Queueing systems fed by many exponential on-off sources: An infinite-intersection approach.

*Queueing Systems*,*54*(1), 5-20. https://doi.org/10.1007/s11134-006-7781-7