Showing papers by "Volker Schmidt published in 1992"

A continuous polling system with general service times

Abstract: Consider a ring on which a server travels at constant speed. Customers arrive on the ring according to a Poisson process, at locations independently and uniformly distributed over the circle. Whenever the server encounters a customer, he stops and serves the client according to a general service time distribution. After the service is completed, the server removes the customer from the ring and resumes his round. The model is analyzed by means of point processes and regenerative processes in combination with some stochastic integration theory. This approach clarifies the analysis of the continuous polling model and provides the means for further generalizations. For every time $t$, the locations of customers that are waiting for service and the positions of clients that have been served during the last tour of the server are represented by random counting measures. These measures converge in distribution as $t \rightarrow \infty$. A recursive expression for the Laplace functionals of the limiting random measures is found, from which the corresponding $k$th moment measures can be derived.

Queueing systems on a Circle

Abstract: Consider a ring on which customers arrive according to a Poisson process. Arriving customers drop somewhere on the circle and wait there for a server who travels on the ring. Whenever this server encounters a customer, he stops and serves the customer according to an arbitrary service time distribution. After the service is completed, the server removes the client from the circle and resumes his journey. We are interested in the number and the locations of customers that are waiting for service. These locations are modeled as random counting measures on the circle. Two different types of servers are considered: The polling server and the Brownian (or drunken) server. It is shown that under both server motions the system is stable if the traffic intensity is less than 1. Furthermore, several earlier results on the configuration of waiting customers are extended, by combining results from random measure theory, stochastic integration and renewal theory.

An insensitivity property of ladder height distributions

Abstract: This paper considers the undershoot of a general continuous-time risk process with dependent increments under a certain initial level. The increments are given by the locations and amounts of claims which are described by a stationary marked point process. Under a certain balance condition, it is shown that the distribution of the undershoot depends only on the mark distribution and on the intensity of the underlying point process, but not on the form of its distribution. In this way an insensitivity property is extended which has been proved in Bjork and Grandell [3] for the ruin probability, i.e. for the probability that after a finite time interval the initial level will be crossed from above.

Directional distributions for multi-dimensional random point processes

Abstract: In biology (histology), material science, ecology and further areas there are oriented aggregates of particles which can be stochastically modelled with the help of anisotropic point processes. Furthermore, distance-dependent orientations do appear in practice. The anisotropies of a random point process Φ are described by the directions of vectors connecting pairs of points of Φ. Vectors, the lengths of which are of different size, can have different favoured directions. For investigating this effect we consider the direction of the vector between a given point x of Φ and its nearest neighbour chosen within those points of Φ the distance to which from a; exceeds some minimum distance. In this way we define a distance-dependent directional distribution. Moreover, a second type of such a directional distribution is considered which is based on counting the numbers of points in sectors of some given spherical shell. Assuming that Φ is stationary we discuss asymptotic properties of estimators of these directi...

Zufällige Punktprozesse : eine Einführung mit Anwendungsbeispielen

Abstract: Die Fachliteratur liber zufallige Punktprozesse und deren Anwendungen hat in den letzten Jahrzehnten stark zugenommen. Die Anzahl der Lehrbiicher dagegen ist minimal, und zum Teil von speziellem Charakter, z. B. durch Beschrankung auf die reelle Achse oder auf den Martingalzugang fUr Punktprozesse. (So wie hier werden wir, wenn keine Mifiverstandnisse auftreten konnen, oft nur kurz von "Punktprozessen" sprechen und damit "zufallige Punktprozesse" meinen. ) Wir mochten hiermit eine EinfUhrung in wesentliche Teile der Theorie mar kierter Punktprozesse im mehrdimensionalen Raum fUr mathematisch sowie an Anwendungen interessierte Leser anbieten, die Kenntnisse in der Wahrscheinlich keitstheorie mit ihrem mafi- und mengentheoretischen Aufbau besitzen. Einige benotigte Grundbegriffe der Mafitheorie werden wir jedoch erklaren; denn unser Hauptzugang zu Punktprozessen ist derjenige liber Zahlmafie, der auf der Proze dur des Zahlens zufalliger Anzahlen von Punkten in fest vorgegebenen Intervallen oder Mengen basiert. Die Darstellung von Punktprozessen als Folgen von Punkten ergibt sich aber von selbst. U nd fUr Punktprozesse auf der reellen Achse werden wir noch weitere Darstellungsformen, z. B. als Folgen von Intervallen, darlegen. Leser, denen Poisson-, Cox-, Erneuerungs-, Cluster- und semi-markowsche Prozesse auf der reellen Achse vertraut sind, finden in unserem Buch u. a. die De finition und Darstellung dieser und weiterer Prozesse aus der einheitlichen Sicht des Punktprozefizuganges. Vorkenntnisse liber die genannten Prozefiklassen wer den jedoch nicht vorausgesetzt.

Definition, Existenz und Eindeutigkeit zufälliger Punktprozesse

Abstract: Wir definieren, ausgehend von den inhaltlichen Ausfuhrungen in Kapitel 1, einen zufalligen Punktprozes in R zunachst als ein zufalliges Zahlmas auf der σ-Algebra ℛ der Borel-Mengen von R.