Dieter König

Queueing Networks: A Survey of Their Random Processes

In this paper we review three topics in queueing network theory: queue length processes, sojourn times, and flow processes. In the discussion of the queue length processes we present results for the continuous-time process and several embedded processes. Then we compare continuous-time processes with embedded processes. In considerable generality we present results for mean sojourn times and discuss the distributions of sojourn times. In the discussion of flow processes we present results for various queueing systems. Our bibliography of over 300 references, while not exhaustive, does cover the major papers for the topics considered.

Stationary queue-length and waiting-time distributions in single-server feedback queues

For M/GI/1/oo queues with instantaneous Bernoulli feedback timeand customer-stationary characteristics of the number of customers in the system and of the waiting time are investigated. Customer-stationary characteristics are thereby obtained describing the behaviour of the queueing processes, for example, at arrival epochs, at feedback epochs, and at times at which an arbitrary (arriving or fed-back) customer enters the waiting room. The method used to obtain these characteristics consists of simple relationships between them and the time-stationary distribution of the number of customers in the system at an arbitrary point in time. The latter is obtained from the wellknown Pollaczek-Khinchine formula for M/GI/1/oo queues without feedback. POISSON ARRIVALS; TIMEAND CUSTOMER-STATIONARY CHARACTERISTICS

EXTENDED AND CONDITIONAL VERSIONS OF THE PASTA PROPERTY

Two types of conditions are discussed ensuring the equality between long-run time fractions and long-run event fractions of stochastic processes with embedded point processes. Modifications of this equality statement are considered. In the current literature (see the references below) several lines of investigation can be observed dealing with generalizations of the fact that Poisson Arrivals see time averages (PASTA). Roughly speaking, the methods used for proving such results can be divided into two classes: (i) Martingale techniques are exploited, where the embedded point process is assumed to admit a stochastic intensity. (ii) Conditional intensities are considered, including their local characterization in the sense of Korolyuk, where the existence of these conditional intensities is either assumed a priori or ensured by the existence of a sufficiently rich family of regular subsets of the state space. We discuss these two approaches within the stationary framework. In particular, a general form of the conditional PASTA property is stated.

On the identification of Poisson arrivals in queues with coinciding time-stationary and customer-stationary state distributions

For several queueing systems, sufficient conditions are given ensuring that from the coincidence of some time-stationary and customer-stationary characteristics of the number of customers in the system such as idle or loss probabilities it follows that the arrival process is Poisson. STATIONARY CHARACTERISTICS; OUEUE LENGTH; INPUT IDENTIFICATION

EPSTA: the coincidence of time-stationary and customer-stationary distributions

In the first part of this paper we present an overview of relationships between time- and customer-stationary distributions of queueing processes. These have been proved by using the properties of random marked point processes, stochastic processes with embedded point processes, Palm distributions and an intensity conservation principle. In the second part a necessary and sufficient condition is established for the coincidence of the two types of stationary distributions, using conditional intensities. We also formulate the property of EPSTA that includes PASTA and ASTA as particular cases. A further result concerns the conditional EPSTA property. Applications to particular queueing systems are considered.

STATIONARY INCREMENTS OF ACCUMULATION PROCESSES IN QUEUES AND GENERALIZED SEMI-MARKOV SCHEMES

Let Tx be the length of time to accumulate x units of a resource. In queueing, the resource could be service. We derive a sufficient condition for the process (Tx, x -: 0) to have stationary increments where Tx is an additive functional of a Markov process. This condition is satisfied in symmetric queues and generalized semiMarkov schemes with insensitive components. As a corollary, we show that the conditional expected response time in a symmetric queue is linear in the service requirement. A similar result holds for the conditional average residence time of an insensitive component in a GSMS.

Directional distributions for multi-dimensional random point processes

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...

THE PALM-TYPE GRAIN SIZE DISTRIBUTION IN STATIONARY GRAIN MODELS

In this paper a point-process approach is given for determining the Palm-type (number-weighted) distribution of the size factor of the grains of a stationary grain model in the plane with non-overlapping, identically shaped and identically orientated convex grains starting from a suitably chosen characteristic of the grain model observed in fixed points of the plane.

Conditional intensities and coincidence properties of stochastic processes with embedded point processes

The aim of the present paper is to discuss three types of coincidence properties (EPSTA, CEPSTA, MUSTA) of stationary continuous-time stochastic processes with embedded point processes. It turns out that not only EPSTA and CEPSTA, but also MUSTA can be characterized by certain invariance properties of conditional intensities of the embedded point processes.

Markierte Punktprozesse im R d . Anwendungen in der stochastischen Geometrie und Stereologie

Zufallige markierte Punktprozesse im R d spielen eine wichtige Rolle in der stochastischen Geometrie (vgl. Abschnitt 1.3). Von besonderem Interesse ist dabei der Fall, das durch die Marken zufallige geometrische Figuren beschrieben werden und das durch die Punkte selbst die Lage dieser Figuren im R d erfast wird. Solche markierten Punktprozesse, genannt Keim-Korn-Prozesse, bilden den Gegenstand der Betrachtungen des vorliegenden Kapitels. Werden spezielle Kornformen vorausgesetzt, dann konnen auf diese Weise zum Beispiel zufallige Faserbzw. Kugelsysteme modelliert werden. Im Zusammenhang mit stationaren Keim-Korn-Prozessen wird in Abschnitt 13.3 insbesondere der Fall betrachtet, das der zugrundeliegende Punktprozes poissonsch ist, was dann zum Begriff des Booleschen Modells fuhrt. Fur diesen Spezialfall werden leicht handhabbare Formeln fur das Kapazitatsfunktional, die Kovarianzfunktion und die spharische Kontaktverteilungsfunktion angegeben. In Abschnitt 13.4 wird anhand ausgewahlter Beispiele gezeigt, wie sich Charakteristiken stationarer Keim-Korn-Prozesse mittels stereologischer Formeln aus linearen bzw. ebenen Schnitten dieser Prozesse bestimmen lassen.