Quasi-Product Forms for Lévy-Driven Fluid Networks

Dȩbicki, Krzysztof; Dieker, A. B.; Rolski, Tomasz

doi:10.1287/moor.1070.0259

Cited by 30 publications

(29 citation statements)

References 36 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…This result for a single queue has been extended more recently to the network setting [18,29]; it was found that for (s, t) ∈ R 2 + ,…”

Section: A(s T) = A(t) − A(s) Denote the Amount Of Traffic Generatedmentioning

confidence: 85%

“…For the important case of spectrally-positive Lévy input (that is, the driving Lévy process does not have negative jumps), the joint Laplace transform of the stationary buffer contents has been found for a broad class of network structures, including tandem queues [18,19,29]. With the transforms being available, one may attempt to use these in order to explicitly find the joint distribution of the stationary buffer contents.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Asymptotic analysis of Lévy-driven tandem queues

Lieshout

Mandjes

2008

Queueing Syst

View full text Add to dashboard Cite

We analyze tail asymptotics of a two-node tandem queue with spectrallypositive Lévy input. A first focus lies in the tail probabilities of the type, for α ∈ (0, 1) and x large, and Q i denoting the steadystate workload in the ith queue. In case of light-tailed input, our analysis heavily uses the joint Laplace transform of the stationary buffer contents of the first and second queue; the logarithmic asymptotics can be expressed as the solution to a convex programming problem. In case of heavy-tailed input we rely on sample-path methods to derive the exact asymptotics. Then we specialize in the tail asymptotics of the downstream queue, again in case of both light-tailed and heavy-tailed Lévy inputs. It is also indicated how the results can be extended to tandem queues with more than two nodes.

show abstract

“…This result for a single queue has been extended more recently to the network setting [18,29]; it was found that for (s, t) ∈ R 2 + ,…”

Section: A(s T) = A(t) − A(s) Denote the Amount Of Traffic Generatedmentioning

confidence: 85%

Section: Introductionmentioning

confidence: 99%

Asymptotic analysis of Lévy-driven tandem queues

Lieshout

Mandjes

2008

Queueing Syst

View full text Add to dashboard Cite

show abstract

“…The results concerning the joint distribution of the steady state of the workloads were studied in a more general network setting in, for example, [9]. These results play an important role for our analysis and are therefore summarized in Sect.…”

Section: Introductionmentioning

confidence: 99%

A tandem fluid network with Lévy input in heavy traffic

2016

View full text Add to dashboard Cite

In this paper we study the stationary workload distribution of a fluid tandem queue in heavy traffic. We consider different types of Lévy input, covering compound Poisson, α-stable Lévy motion (with 1 < α < 2), and Brownian motion. In our analysis, we separately deal with Lévy input processes with increments that have finite and infinite variance. A distinguishing feature of this paper is that we do not only consider the usual heavy traffic regime, in which the load at one of the nodes goes to unity, but also a regime in which we simultaneously let the load of both servers tend to one, which, as it turns out, leads to entirely different heavy traffic asymptotics. Numerical experiments indicate that under specific conditions the resulting simultaneous heavy traffic approximation significantly outperforms the usual heavy traffic approximation.

show abstract

“…The transform can be obtained for tandem and feed-forward networks with decreasing service rates (in the direction of flow), see e.g. Kella and Whitt (1992a) and Debicki, Dieker and Rolski (2007). In such networks, if one queue is empty then all the queues preceding it are empty as well.…”

mentioning

confidence: 99%

Two Coupled Lévy Queues with Independent Input

Boxma

Ivanovs

2013

Stochastic Systems

View full text Add to dashboard Cite

We consider a pair of coupled queues driven by independent spectrally-positive Lévy processes. With respect to the bi-variate workload process this framework includes both the coupled processor model and the two-server fluid network with independent Lévy inputs. We identify the joint transform of the stationary workload distribution in terms of Wiener-Hopf factors corresponding to two auxiliary Lévy processes with explicit Laplace exponents. We reinterpret and extend the ideas of Cohen and Boxma (1983) to provide a general and uniform result with a neat transform expression.1. Introduction. In the queueing literature, several studies have been devoted to a queueing model of two servers, each with their own customer arrival process, with the special feature that the speed of one server changes when the other server becomes idle. This has become known as the coupled processor model. A possibly even more popular model of two servers is a fluid network with independent arrival processes, where fixed fractions of fluid exiting one queue are routed into the same and the other queue, as well as out of the system. These models are intimately related and in the case of Lévy input both can be put in our framework below; see Section 1.1 and Section 1.3 for possible interpretations of our model and for related literature respectively.More specifically, we assume that our queues are driven by two independent Lévy processes X 1 (t) and X 2 (t) without negative jumps. We model a pair of workload processes (W 1 (t), W 2 (t)) as a 2-dimensional reflected process, see e.g. Harrison and Reiman (1981); Kella (2006),

show abstract

Quasi-Product Forms for Lévy-Driven Fluid Networks

Cited by 30 publications

References 36 publications

Asymptotic analysis of Lévy-driven tandem queues

Asymptotic analysis of Lévy-driven tandem queues

A tandem fluid network with Lévy input in heavy traffic

Two Coupled Lévy Queues with Independent Input

Contact Info

Product

Resources

About