This paper analyzes transient characteristics of Gaussian queues. More specifically, we determine the logarithmic asymptotics of P(Q_0 > pB,Q_TB > qB), where Q_t denotes the workload at time t. For any pair (p, q) three regimes can be distinguished: (A) For small values of T, one of the events {Q_0 > pB} and {Q_TB > qB} will essentially imply the other. (B) Then there is an intermediate range of values of T for which it is to be expected that both {Q_0 > pB} and {Q_TB > qB} are tight (in that none of them essentially implies the other), but that the time epochs 0 and T lie in the same busy period with overwhelming probability. (C) Finally, for large T still both events are tight, but now they occur in different busy periods with overwhelming probability. For the short-range dependent case explicit calculations are presented, whereas for the long-range dependent case structural results are proven.
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CWI. Probability, Networks and Algorithms [PNA]

Dȩbicki, K.G, Es-Saghouani, A, & Mandjes, M.R.H. (2008). Transient characteristics of Gaussian queues. CWI. Probability, Networks and Algorithms [PNA]. CWI.