### Abstract

We consider the asymptotic variance of the departure counting process D(t) of the GI/G/1 queue; D(t) denotes the number of departures up to time t. We focus on the case where the system load ϱ equals 1, and prove that the asymptotic variance rate satisfies limt→∞varD(t) / t = λ(1 - 2 / π)(ca2 + cs2), where λ is the arrival rate, and ca2 and cs2 are squared coefficients of variation of the interarrival and service times, respectively. As a consequence, the departures variability has a remarkable singularity in the case in which ϱ equals 1, in line with the BRAVO (balancing reduces asymptotic variance of outputs) effect which was previously encountered in finite-capacity birth-death queues. Under certain technical conditions, our result generalizes to multiserver queues, as well as to queues with more general arrival and service patterns. For the M/M/1 queue, we present an explicit expression of the variance of D(t) for any t.

Original language | English |
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Pages (from-to) | 243-263 |

Journal | Advances in applied probability |

Volume | 43 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2011 |

### Keywords

- departure process
- critically loaded system
- uniform integrability
- renewal theory
- IR-76966
- GI/G/1 queue
- Multi-server queue
- METIS-275143
- Brownian bridge

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## Cite this

Al Hanbali, A., Mandjes, M. R. H., Nazarathy, Y., & Whitt, W. (2011). The asymptotic variance of departures in critically loaded queues.

*Advances in applied probability*,*43*(1), 243-263. https://doi.org/10.1239/aap/1300198521