Tandem Brownian queues

Lieshout, Pmd; Mandjes, Mrh Michel

doi:10.1007/s00186-007-0149-x

Cited by 36 publications

(29 citation statements)

References 14 publications

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“…A subject for future research concerns the identification of the corresponding exact asymptotics. The results for the special case of Brownian input [31] indicate that it can be expected that for α in some specific range these have a purely exponential shape, whereas for other α in addition a factor 1/ √ x will appear. In [5, Sect.…”

Section: Discussion and Concluding Remarksmentioning

confidence: 92%

“…We remark that, in the Brownian case, a pictorial illustration of the paths to overflow is given in [31,Fig. 3]; the paths in the non-Brownian case look similar.…”

Section: Proposition 32mentioning

confidence: 99%

“…With the transforms being available, one may attempt to use these in order to explicitly find the joint distribution of the stationary buffer contents. So far this turned out to be possible in just a few cases, see for instance [4,31] in the two-node tandem case.…”

Section: Introductionmentioning

confidence: 99%

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Asymptotic analysis of Lévy-driven tandem queues

Lieshout

Mandjes

2008

Queueing Syst

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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.

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Section: Discussion and Concluding Remarksmentioning

confidence: 92%

“…We remark that, in the Brownian case, a pictorial illustration of the paths to overflow is given in [31,Fig. 3]; the paths in the non-Brownian case look similar.…”

Section: Proposition 32mentioning

confidence: 99%

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Asymptotic analysis of Lévy-driven tandem queues

Lieshout

Mandjes

2008

Queueing Syst

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“…Some progress has been recently made in case n = 2; for a Brownian fluid system, Lieshout and Mandjes [30] calculate the distribution of W . Avram, Palmowski, and Pistorius [2] study a compound Poisson setting with exponential jumps.…”

Section: Generalitiesmentioning

confidence: 99%

Quasi-Product Forms for Lévy-Driven Fluid Networks

2007

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We study stochastic tree fluid networks driven by a multidimensional Lévy process. We are interested in (the joint distribution of) the steady-state content in each of the buffers, the busy periods, and the idle periods. To investigate these fluid networks, we relate the above three quantities to fluctuations of the input Lévy process by solving a multidimensional Skorokhod reflection problem. This leads to the analysis of the distribution of the componentwise maximums, the corresponding epochs at which they are attained, and the beginning of the first last-passage excursion. Using the notion of splitting times, we are able to find their Laplace transforms. It turns out that, if the components of the Lévy process are 'ordered', the Laplace transform has a so-called quasi-product form.The theory is illustrated by working out special cases, such as tandem networks and priority queues.

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“…In all of them, the model is such that the components are ordered: A s ≤ B s for all s, which implies that the epochs that the two components achieve their respective maximum values are almost surely ordered. These special cases cover tandem systems of M/D/1 queues [18] and a tandem Brownian queue [17]. In both models, let D s denote the amount of work that has arrived up to time s and let c 1 and c 2 denote the constant service rate of the upstream and downstream queue, respectively (assume that c 1 > c 2 ).…”

Section: Introductionmentioning

confidence: 99%

Estimating Large Delay Probabilities in Two Correlated Queues

Cahen

Mandjes²,

Zwart

2018

ACM Trans. Model. Comput. Simul.

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This article focuses on evaluating the probability that both components of a two-dimensional stochastic process will ever, but not necessarily at the same time, exceed some large level u. An important application is in determining the probability of large delays occurring in two correlated queues. Since exact analysis of this probability seems prohibitive, we focus on deriving asymptotics and on developing efficient simulations techniques. Large deviations theory is used to characterise logarithmic asymptotics. The second part of this article focuses on efficient simulation techniques. Using "nearest-neighbour random walk" as an example, we first show that a "naive" implementation of importance sampling, based on the decay rate, is not asymptotically efficient. A different approach, which we call partitioned importance sampling, is developed and shown to be asymptotically efficient. The results are illustrated through various simulation experiments.

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Tandem Brownian queues

Cited by 36 publications

References 14 publications

Asymptotic analysis of Lévy-driven tandem queues

Asymptotic analysis of Lévy-driven tandem queues

Quasi-Product Forms for Lévy-Driven Fluid Networks

Estimating Large Delay Probabilities in Two Correlated Queues

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