Transient characteristics of Gaussian queues

This paper analyzes transient characteristics of Gaussian queues. More specifically, we determine the logarithmic asymptotics of P(Q0 > pB,QTB > qB), where Qt 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 {Q0 > pB} and {QTB > 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 {Q0 > pB} and {QTB > 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 calculaK. Dȩbicki was partially supported by MNiSW Grant No. N N2014079 33 (2007–2009) and by a Marie Curie Transfer of Knowledge Fellowship of the European Community’s Sixth Framework Programme under Contract MTKD-CT-2004-013389. Part of this work was done while M. Mandjes was at Stanford University, Stanford, CA 94305, US. K. Dȩbicki Instytut Matematyczny, University of Wrocław, pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland e-mail: [email protected] A. Es-Saghouani · M. Mandjes ( ) Korteweg-de Vries Institute for Mathematics, Plantage Muidergracht 24, 1018 TV Amsterdam, The Netherlands e-mail: [email protected] A. Es-Saghouani e-mail: [email protected] M. Mandjes CWI, Amsterdam, The Netherlands M. Mandjes EURANDOM, Eindhoven, The Netherlands 384 Queueing Syst (2009) 62: 383–409 tions are presented, whereas for the long-range dependent case, structural results are proven.