Sabine Schlegel

Ruin probabilities in perturbed risk models

Abstract We consider the asymptotical behaviour of the ruin function in perturbed and unperturbed non-standard risk models when the initial risk reserve tends to infinity. We give a characterization of this behaviour in terms of the unperturbed ruin function and the perturbation law provided that at least one of both is subexponential. By a number of examples for the claim arrival process as well as the perturbation process we show that our result is a generalization of previous work on this subject.

Asymptotics of stochastic networks with subexponential service times

We analyse the tail behaviour of stationary response times in the class of open stochastic networks with renewal input admitting a representation as (max,+)-linear systems. For a K-station tandem network of single server queues with infinite buffer capacity, which is one of the simplest models in this class, we first show that if the tail of the service time distribution of one server, say server i0 ∈ {1,...,K}, is subexponential and heavier than those of the other servers, then the stationary distribution of the response time until the completion of service at server j ⩾ i0 asymptotically behaves like the stationary response time distribution in an isolated single-server queue with server i0. Similar asymptotics are given in the case when several service time distributions are subexponential and asymptotically tail-equivalent. This result is then extended to the asymptotics of general (max,+)-linear systems associated with i.i.d. driving matrices having one (or more) dominant diagonal entry in the subexponential class. In the irreducible case, the asymptotics are surprisingly simple, in comparison with results of the same kind in the Cramér case: the asymptotics only involve the excess distribution of the dominant diagonal entry, the mean value of this entry, the intensity of the arrival process, and the Lyapunov exponent of the sequence of driving matrices. In the reducible case, asymptotics of the same kind, though somewhat more complex, are also obtained. As a direct application, we give the asymptotics of stationary response times in a class of stochastic Petri nets called event graphs. This is based on the assumption that the firing times are independent and that the tail of the firing times of one of the transitions is subexponential and heavier than those of the others. An extension of these results to nonrenewal input processes is discussed. Asymptotics of queue size processes are also considered.

A Tandem Queue with a Gate Mechanism

Inspired by a problem regarding cable access networks, we consider a two station tandem queue with Poisson arrivals. At station 1 we operate a gate mechanism, leading to batch arrivals at station 2. Upon arrival at station 1, customers join a queue in front of a gate. Whenever all customers present at the service area of station 1 have received service, the gate before as well as a gate behind the service facility open. Customers leave the service area and enter station 2 (as a batch), while all customers waiting at the gate in front of station 1 are admitted into the service area. For station 1 we analyse the batch size and the time between two successive gate openings, as well as waiting and sojourn times of individual customers for different service disciplines. For station 2, we investigate waiting times of batch customers, where we allow that service times may depend on the size of the batch and also on the interarrival time. In the analysis we use Wiener–Hopf factorization techniques for Markov modulated random walks.

Is intervertebral disc degeneration related to segmental instability?: An evaluation with two different grading systems based on clinical imaging

Background Several in vitro studies investigated how degeneration affects spinal motion. However, no consensus has emerged from these studies. Purpose To investigate how degeneration grading systems influence the kinematic output of spinal specimens. Material and Methods Flexibility testing was performed with ten human T12-S1 specimens. Degeneration was graded using two different classifications, one based on X-ray and the other one on magnetic resonance imaging (MRI). Intersegmental rotation (expressed by range of motion [ROM] and neutral zone [NZ]) was determined in all principal motion directions. Further, shear translation was measured during flexion/extension motion. Results The X-ray grading system yielded systematically lesser degeneration. In flexion/extension, only small differences in ROM and NZ were found between moderately degenerated motion segments, with only NZ for the MRI grading reaching statistical significance. In axial rotation, a significant increase in NZ for moderately degenerated segments was found for both grading systems, whereas the difference in ROM was significant only for the MRI scheme. Generally, the relative increases were more pronounced for the MRI classification compared to the X-ray grading scheme. In lateral bending, only relatively small differences between the degeneration groups were found. When evaluating shear translations, a non-significant increase was found for moderately degenerated segments. Motion segment segments tended to regain stability as degeneration progressed without reaching the level of statistical significance. Conclusion We found a fair agreement between the grading schemes which, nonetheless, yielded similar degeneration-related effects on intersegmental kinematics. However, as the trends were more pronounced using the Pfirrmann classification, this grading scheme appears superior for degeneration assessment.

ASYMPTOTICS FOR RANDOM WALKS WITH DEPENDENT HEAVY-TAILED INCREMENTS

We consider a random walk {Sn} with dependent heavy-tailed increments and negative drift. We study the asymptotics for the tail probability P{supnSn>x} as x→∞. If the increments of {Sn} are independent then the exact asymptotic behavior of P{supnSn>x} is well known. We investigate the case in which the increments are given as a one-sided asymptotically stationary linear process. The tail behavior of supnSn turns out to depend heavily on the coefficients of this linear process.

Expansion formulae for characteristics of cumulative cost in finite horizon production models

We consider the expected value and the tail probability of cumulative shortage and holding cost (i.e. the probability that cumulative cost is more than a certain value) in finite horizon production models. An exact expression is provided for the expected value of the cumulative cost for general production functions. This expression is then used to compute the optimal production rate when the production function is linear. An expansion formula (whose coefficients can be obtained recursively) is provided for the tail probability. Examples are given to illustrate the results.