On the Dependence Structure of Gaussian Queues

Es-Saghouani, Abdelghafour; Mandjes, Michel

doi:10.1080/15326340902869887

Cited by 4 publications

(4 citation statements)

References 28 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The above conjectures are to some extent supported by findings in [9,11], where the asymptotics of P(Q(0) > u, Q(t u ) > u), as u → ∞ were derived for various classes of functions t u .…”

Section: Correlation Structure Of Gaussian Queuesupporting

confidence: 56%

Open problems in Gaussian fluid queueing theory

Dȩbicki

Mandjes

2011

Queueing Syst

View full text Add to dashboard Cite

We present three challenging open problems that originate from the analysis of the asymptotic behavior of Gaussian fluid queueing models. In particular, we address the problem of characterizing the correlation structure of the stationary buffer content process, the speed of convergence to stationarity, and analysis of an asymptotic constant associated with the stationary buffer content distribution (the so-called Pickands constant).

show abstract

Section: Correlation Structure Of Gaussian Queuesupporting

confidence: 56%

Open problems in Gaussian fluid queueing theory

Dȩbicki

Mandjes

2011

Queueing Syst

View full text Add to dashboard Cite

show abstract

“…Importantly, however, so far hardly any attention has been paid to transient properties. A notable exception is the recent paper [10], where asymptotics of transient probabilities under a so-called many-sources scaling were found (for specific Gaussian inputs).…”

Section: Introductionmentioning

confidence: 99%

“…A crucial element in the reasoning is that for T large enough, the time epochs 0 and T lie in separate busy periods, thus simplifying the analysis substantially. A conclusion drawn in [10] is that the correlation structure of the input process essentially carries over to the workload process.…”

Section: Introductionmentioning

confidence: 99%

Transient characteristics of Gaussian queues

Dȩbicki

Es-Saghouani²,

Mandjes

2009

Queueing Syst

Self Cite

View full text Add to dashboard Cite

This paper analyzes transient characteristics of Gaussian queues. More specifically, we determine the logarithmic asymptotics of P(Q 0 > pB, Q T B > 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 T B > 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 T B > 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 calcula- Queueing Syst (2009) 62: 383-409 tions are presented, whereas for the long-range dependent case, structural results are proven.

show abstract

“…Based on the recent results in [13], however, we expect that this is not true. Instead, we anticipate that the asymptotics of Cov(Q(0), Q(t)) are roughly polynomially, or, more precisely, decaying as t 2H−2 , which is equally fast as the asymptotics of Cov (A(0, 1), A(t, t + 1)).…”

Section: Convergence To Stationarity Of Fractional Brownian Storage 17mentioning

confidence: 89%

On convergence to stationarity of fractional Brownian storage

Mandjes¹,

Norros²

2009

Ann. Appl. Probab.

Self Cite

View full text Add to dashboard Cite

With $M(t):=\sup_{s\in[0,t]}A(s)-s$ denoting the running maximum of a fractional Brownian motion $A(\cdot)$ with negative drift, this paper studies the rate of convergence of $\mathbb {P}(M(t)>x)$ to $\mathbb{P}(M>x)$. We define two metrics that measure the distance between the (complementary) distribution functions $\mathbb{P}(M(t)>\cdot)$ and $\mathbb{P}(M>\cdot)$. Our main result states that both metrics roughly decay as $\exp(-\vartheta t^{2-2H})$, where $\vartheta$ is the decay rate corresponding to the tail distribution of the busy period in an fBm-driven queue, which was computed recently [Stochastic Process. Appl. (2006) 116 1269--1293]. The proofs extensively rely on application of the well-known large deviations theorem for Gaussian processes. We also show that the identified relation between the decay of the convergence metrics and busy-period asymptotics holds in other settings as well, most notably when G\"artner--Ellis-type conditions are fulfilled.Comment: Published in at http://dx.doi.org/10.1214/08-AAP578 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

show abstract

On the Dependence Structure of Gaussian Queues

Cited by 4 publications

References 28 publications

Open problems in Gaussian fluid queueing theory

Open problems in Gaussian fluid queueing theory

Transient characteristics of Gaussian queues

On convergence to stationarity of fractional Brownian storage

Contact Info

Product

Resources

About