Efficient simulation of tail probabilities in a queueing model with heterogeneous servers

Kühn, Julia; Mandjes, Michel

doi:10.1080/15326349.2018.1458629

Cited by 5 publications

(3 citation statements)

References 24 publications

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“…Though notationally rather involved, conceptually such extensions are relatively straightforward; see e.g. [26] for such computations in a related model. In addition, one could consider the specific case of Gamma claims; cf.…”

Section: Discussion and Concluding Remarksmentioning

confidence: 99%

A ruin model with a resampled environment

Constantinescu

Delsing

Mandjes

et al. 2019

Scandinavian Actuarial Journal

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This paper considers a Cramér-Lundberg risk setting, where the components of the underlying model change over time. These components could be thought of as the claim arrival rate, the claim-size distribution, and the premium rate, but we allow the more general setting of the cumulative claim process being modelled as a spectrally positive Lévy process. We provide an intuitively appealing mechanism to create such parameter uncertainty: at Poisson epochs we resample the model components from a finite number of d settings. It results in a setup that is particularly suited to describe situations in which the risk reserve dynamics are affected by external processes (such as the state of the economy, political developments, weather or climate conditions, and policy regulations). We extend the classical Cramér-Lundberg approximation (asymptotically characterizing the all-time ruin probability in a light-tailed setting) to this more general setup. In addition, for the situation that the driving Lévy processes are sums of Brownian motions and compound Poisson processes, we find an explicit uniform bound on the ruin probability, which can be viewed as an extension of Lundberg's inequality; importantly, here it is not required that the Lévy processes be spectrally one-sided. In passing we propose an importance-sampling algorithm facilitating efficient estimation, and prove it has bounded relative error. In a series of numerical experiments we assess the accuracy of the asymptotics and bounds, and illustrate that neglecting the resampling can lead to substantial underestimation of the risk. Keywords. Lévy risk processes • parameter uncertainty • ruin probabilities • Cramér-Lundberg asymptotics • Lundberg's inequality Affiliations. Corina Constantinescu and Leonardo Rojas Nandayapa are with the Institute

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Section: Discussion and Concluding Remarksmentioning

confidence: 99%

A ruin model with a resampled environment

Constantinescu

Delsing

Mandjes

et al. 2019

Scandinavian Actuarial Journal

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“…The asymptotic (co-)variances v A , v S , and c A, S can be computed in an alternative way, using results from large deviations theory [17] ; a similar approach has been followed in e.g., [28,39] . With this approach, also higher (centered) moments of AðtÞ=t and SðtÞ=t can be calculated in closed form in the regime that t !…”

Section: A Appendix: Alternative Computation Of the Asymptotic Variancementioning

confidence: 99%

Single-server queues under overdispersion in the heavy-traffic regime

2020

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This paper addresses the analysis of the queue-length process of single-server queues under overdispersion, i.e., queues fed by an arrival process for which the variance of the number of arrivals in a given time window exceeds the corresponding mean. Several variants are considered, using concepts as mixing and Markov modulation, resulting in different models with either endogenously triggered or exogenously triggered random environments. Only in special cases explicit expressions can be obtained, e.g., when the random arrival and/or service rate can attain just finitely many values. While for more general model variants exact analysis is challenging, one can derive limit theorems in the heavy-traffic regime. In some of our derivations we rely on evaluating the relevant Laplace transform in the heavy-traffic scaling using Taylor expansions, whereas other results are obtained by applying the continuous mapping theorem.

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“…The asymptotic (co-)variances v A , v S , and c A,S can be computed in an alternative way, using results from large deviations theory [16]; a similar approach has been followed in e.g. [25,35]. With this approach, also higher (centered) moments of A(t)/t and S(t)/t can be calculated in closed form in the regime that t → ∞, as we point out below.…”

Section: A Appendix: Alternative Computation Of the Asymptotic Variancementioning

confidence: 99%

Single-server queues under overdispersion in the heavy-traffic regime

Boxma¹,

Heemskerk²,

Mandjes³

2019

Preprint

Self Cite

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This paper addresses the analysis of the queue-length process of single-server queues under overdispersion, i.e., having arrival and/or service rates that are not constant but instead randomly evolve over time. Several variants are considered, using concepts as mixing and Markov modulation, resulting in different models with either endogenously triggered or exogenously triggered random environments. Only in special cases explicit expressions can be obtained, e.g. when the random arrival and/or service rate can attain just finitely many values. While for more general model variants exact analysis is challenging, one can derive limit theorems in the heavy-traffic regime. In some of our derivations we rely on evaluating the relevant Laplace transform in the heavy-traffic scaling using Taylor expansions, whereas other results are obtained by applying the continuous mapping theorem.

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Efficient simulation of tail probabilities in a queueing model with heterogeneous servers

Cited by 5 publications

References 24 publications

A ruin model with a resampled environment

A ruin model with a resampled environment

Single-server queues under overdispersion in the heavy-traffic regime

Single-server queues under overdispersion in the heavy-traffic regime

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