Exit problem of a two-dimensional risk process from the quadrant: Exact and asymptotic results

Avram, Florin; Palmowski, Zbigniew; Pistorius, Martijn

doi:10.1214/08-aap529

Cited by 68 publications

(128 citation statements)

References 27 publications

(35 reference statements)

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“…Those asymptotic results exhibit some analogy to the discrete time counterpart, that is, the reflected random walk. Similar exact asymptotics are also reported for hitting probabilities of two dimensional risk processes in [3]. As a by product, we also get some asymptotics for the tail distribution of a convex combination of the two buffer contents in the Brownian input case.…”

supporting

confidence: 78%

Tail asymptotics for a Lévy-driven tandem queue with an intermediate input

Miyazawa

Rolski

2009

Queueing Syst

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We consider a Lévy-driven tandem queue with an intermediate input assuming that its buffer content process obtained by a reflection mapping has the stationary distribution. For this queue, no closed form formula is known, not only for its distribution but also for the corresponding transform. In this paper, we consider only light-tailed inputs. For the Brownian input case, we derive exact tail asymptotics for the marginal stationary distribution of the second buffer content, while weaker asymptotic results are obtained for the general Lévy input case. The results generalize those of Lieshout and Mandjes from the recent papers (Lieshout and Mandjes in Math. Methods Oper. Res. 66:275-298, 2007 and Queueing Syst. 60:203-226, 2008) for the corresponding tandem queue without an intermediate input.Keywords Brownian tandem queue · Levy driven tandem queue · Intermediate input · Two dimensional semi-martingale reflected process · Stationary buffer content · Exact tail asymptotics · Tail decay rate · Large deviations Mathematics Subject Classification (2000) 60K25 · 68M20 · 90B15

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confidence: 78%

Tail asymptotics for a Lévy-driven tandem queue with an intermediate input

Miyazawa

Rolski

2009

Queueing Syst

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“…5], which has been written simultaneously with the present paper, this has already been done for the compound Poisson case as well as the purely Brownian case. We remark that the techniques used in [5] substantially differ from ours; in addition, our approach also yields the 'most likely paths to overflow,' as determined in the proof of Proposition 3.5. Also, [5] does not consider the heavy-tailed case.…”

Section: Discussion and Concluding Remarksmentioning

confidence: 93%

“…We remark that the techniques used in [5] substantially differ from ours; in addition, our approach also yields the 'most likely paths to overflow,' as determined in the proof of Proposition 3.5. Also, [5] does not consider the heavy-tailed case.…”

Section: Discussion and Concluding Remarksmentioning

confidence: 93%

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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“…Recent results pertaining to multidimensional risk models can be found in, e.g. Chan et al (2003), Cai and Li (2005), (2007), Yuen et al (2006), Li et al (2007), Avram et al (2008aAvram et al ( ), (2008b, Dang et al (2009), Rabehasaina (2009), andGong et al (2010). Among these, the work that motivates the present work is that of Avram et al (2008a).…”

Section: Introductionmentioning

confidence: 99%

A Two-Dimensional Risk Model with Proportional Reinsurance

Badescu

Cheung

2011

J. Appl. Probab.

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In this paper we consider an extension of the two-dimensional risk model introduced in Avram, Palmowski and Pistorius (2008a). To this end, we assume that there are two insurers. The first insurer is subject to claims arising from two independent compound Poisson processes. The second insurer, which can be viewed as a different line of business of the same insurer or as a reinsurer, covers a proportion of the claims arising from one of these two compound Poisson processes. We derive the Laplace transform of the time until ruin of at least one insurer when the claim sizes follow a general distribution. The surplus level of the first insurer when the second insurer is ruined first is discussed at the end in connection with some open problems.

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Exit problem of a two-dimensional risk process from the quadrant: Exact and asymptotic results

Cited by 68 publications

References 27 publications

Tail asymptotics for a Lévy-driven tandem queue with an intermediate input

Tail asymptotics for a Lévy-driven tandem queue with an intermediate input

Asymptotic analysis of Lévy-driven tandem queues

A Two-Dimensional Risk Model with Proportional Reinsurance

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