The First Passage Time of a Level and the Behavior at Infinity for a Class of Processes with Independent Increments

Золотарев, В. М.

doi:10.1137/1109090

Cited by 117 publications

(78 citation statements)

References 2 publications

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“…Then the steady-state distribution Q := lim t→∞ Q t , which exists due to EX 1 < 0, is known (in terms of its Laplace transform) for both the spectrally positive and spectrally negative case. For spectrally positive input, we have the generalized Pollaczek-Khinchine formula, usually attributed to (Zolotarev 1964):…”

Section: Reflected Lévy Processes; Queuesmentioning

confidence: 99%

Simulation-Based Computation of the Workload Correlation Function in a Lévy-Driven Queue

Mandjes

2011

J. Appl. Probab.

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In this paper we consider a single-server queue with Lévy input, and, in particular, its workload process (Q t ) t≥0, focusing on its correlation structure. With the correlation function defined as r(t):= cov(Q 0, Q t ) / varQ 0 (assuming that the workload process is in stationarity at time 0), we first study its transform ∫0 ∞ r(t)e-ϑt dt, both for when the Lévy process has positive jumps and when it has negative jumps. These expressions allow us to prove that r(·) is positive, decreasing, and convex, relying on the machinery of completely monotone functions. For the light-tailed case, we estimate the behavior of r(t) for large t. We then focus on techniques to estimate r(t) by simulation. Naive simulation techniques require roughly (r(t))-2 runs to obtain an estimate of a given precision, but we develop a coupling technique that leads to substantial variance reduction (the required number of runs being roughly (r(t))-1). If this is augmented with importance sampling, it even leads to a logarithmically efficient algorithm.

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Section: Reflected Lévy Processes; Queuesmentioning

confidence: 99%

Simulation-Based Computation of the Workload Correlation Function in a Lévy-Driven Queue

Mandjes

2011

J. Appl. Probab.

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“…The Laplace transform of Q 1 dates back to, at least, Zolotarev [42]: with ϑ(s) := κ(s) + c 1 s, the so-called generalized Pollaczek-Khinchine formula states that, for s ≥ 0,…”

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

confidence: 99%

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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“…The first observation is that Proposition 3 jointly with Theorem 1 give another proof of the famous Zolotarev's theorem from [22]. The second observation is that immediately from Theorem 1 it follows that the class of Instead of reversing this LST we find distribution of W using convergence ω n D → W.…”

Section: Relation Between Lévy Processes and M/g/1 Queuesmentioning

confidence: 91%

Queueing approximation of suprema of spectrally positive Lévy process

Czystołowski

Szczotka

2010

Queueing Syst

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Summary: Let W = sup 0≤t<∞ (X(t) − βt), where X is a spectrally positive Lévy process with expectation zero and 0 < β < ∞. One of the main results of the paper says that for such a process X there exists a sequence of M/GI/1 queues for which stationary waiting times converge in distribution to W. The second result shows that condition (III) of Proposition 2 in the paper is not implied by all other conditions.

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The First Passage Time of a Level and the Behavior at Infinity for a Class of Processes with Independent Increments

Cited by 117 publications

References 2 publications

Simulation-Based Computation of the Workload Correlation Function in a Lévy-Driven Queue

Simulation-Based Computation of the Workload Correlation Function in a Lévy-Driven Queue

Asymptotic analysis of Lévy-driven tandem queues

Queueing approximation of suprema of spectrally positive Lévy process

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