The BCD queue and its ramifications

We introduce a process similar to the birth-death process and use it as a starting point for defining queueing models much in the same way as the birth-death process can be used for this purpose. Steady-state distributions for this process and the corresponding queues are derived. Generalizations allowing nonexponential service times are also studied.     

The doubly stochastic server: A time-sharing model

The doubly stochastic server is a time-sharing model of slightly greater versatility than any comparable model that has been analysed previously.The stationary distribution of its state is presented along with a necessary and sufficient condition of existence. Moreover, the conditional average response time of a job, given its requested processing time, is derived, and the stochastic law of the output stream is determined. The results are of a simple and useful form.

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Zusammenfassung Es wird ein Modell eines einzelnen Bedieners vorgestellt und unter der Annahme einer diskreten Zeitskala untersucht. Die betrachteten Bedienungsdisziplinen enthalten solche, die für den time-sharing Betrieb von Rechenanlagen interessant sind, wie etwa eine round-robin Disziplin.

     

Ergodicity of a polling network

The polling network considered here consists of a finite collection of stations visited successively by a single server who is following a Markovian routing scheme. At every visit of a station a positive random number of the customers present at the start of the visit are served, whereupon the server takes a positive random time to walk to the station to be visited next. The network receives arrivals of customer groups at Poisson instants, and all customers wait until served, whereupon they depart from the network. Necessary and sufficient conditions are derived for the server to be able to cope with the traffic. For the proof a multidimensional imbedded Markov chain is studied.     

Weak convergence for generalized semi-Markov processes

Generalized semi-Markov schemes were introduced by Matthes in 1962 under the designation ‘Bedienungsschemata’ (service schemes). They include a large variety of familiar stochastic models. It is shown in this paper that under appropriate regularity conditions the associated stochastic process describing the state at timet,t≥0, and the stationary distribution are continuous functions of the life-times of the active components. The supplementary-variable Markov process is shown to be the limit process of a sequence of discrete-state-process obtained through approximating the life-time distributions by mixtures of Erlang distributions and measuring ages and residual life-times in phases. This approach supplements the phase method.     

Ergodicity of a polling network with an infinite number of stations

A Markov polling system with infinitely many stations is studied. The topic is the ergodicity of the infinite-dimensional process of queue lengths. For the infinite-dimensional process, the usual type of ergodicity cannot prevail in general and we introduce a modified concept of ergodicity, namely, weak ergodicity. It means the convergence of finite-dimensional distributions of the process. We give necessary and sufficient conditions for weak ergodicity. Also, the “usual” ergodicity of the system is studied, as well as convergence of functionals which are continuous in some norm. This revised version was published online in June 2006 with corrections to the Cover Date.     

On the Existence of Invariant Measures for Networks with String Transitions

Recently a new class of Markov network processes was introduced, characterized by so-called string transitions. These are continuous-time Markov processes on a discrete state space. It is known that they possess an invariant measure of a special form, called a product-form, provided that a certain system of so-called traffic equations possesses a solution. Little is known about the existence of solutions of the traffic equations. The present paper deals with this question, focussing on the most important special case of unit vector string transitions. It is shown for open networks with unit vector string transitions of bounded lengths that the traffic equations possess a solution. Furthermore, it is shown for a prominent example of a network featuring signals and batch services that the traffic equations possess a solution.     

On the M/G/1 foreground-background processor-sharing queue

We consider the M/G/1 queue under the foreground-background processor-sharing discipline. Using a result on the stationary distribution of the total number of customers we give a direct derivation of the distribution of the random counting measure representing the steady state of the queue in all detail.This work was done during a sabbatical at INRIA, France.     

On the second derivative of the spherical contact distribution function of smooth grain models

We consider a stationary grain model Ξ in ℝ ^d with convex, compact and smoothly bounded grains. We study the spherical contact distribution function F of Ξ and derive (under suitable assumptions) an explicit formula for its second derivative F″. The value F″(0) is of a simple form and admits a clear geometric interpretation.For the Boolean model we obtain an interesting new formula for the(d− 1)-st quermass density. Received: 22 November 1999 / Revised version: 2 November 2000 /?Published online: 14 June 2001     

A definition of discrete product form distributions

Summary A number of important applied probability models, many of which can be interpreted as networks of queues, lead to a discrete probability distribution of states exhibiting a so-called product form. This phenomenon is explained by a certain form of decomposability, and a definition of the notion of product form emerges.

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Zusammenfassung Bei einer Reihe von wichtigen Modellen aus dem Anwendungsbereich der Wahrscheinlichkeitstheorie, viele davon als Netzwerke von Warteschlangen interpretierbar, treten diskrete Wahrscheinlichkeitsverteilungen auf, deren Gestalt von einer Produktform ist. Dieses Phänomen wird durch eine gleichzeitig auftretende Art der Zerlegbarkeit erklärt, woraus eine Definition des Begriffs der Produktform resultiert.