E.A. van Doorn

A fluid queue driven by an infinite-state birth-death process

We describe an approach to analysing the stochastic behaviour of the content of a buffer which receives and releases fluid flows at rates which are determined by the state of an infinite-state birth-death process evolving in the background. The approach is illustrated by a concrete example.

Limiting conditional distributions for birth-death processes

In a recent paper one of us identified all of the quasi-stationary distributions for a non-explosive, evanescent birth-death process for which absorption is certain, and established conditions for the existence of the corresponding limiting conditional distributions. Our purpose is to extend these results in a number of directions. We shall consider separately two cases depending on whether or not the process is evanescent. In the former case we shall relax the condition that absorption is certain. Furthermore, we shall allow for the possibility that the minimal process might be explosive, so that the transition rates alone will not necessarily determine the birth-death process uniquely. Although we shall be concerned mainly with the minimal process, our most general results hold for any birth-death process whose transition probabilities satisfy both the backward and the forward Kolmogorov differential equations.

Analysis of a single-server queue interacting with a fluid reservoir

We consider a single-server queueing system with Poisson arrivals in which the speed of the server depends on whether an associated fluid reservoir is empty or not. Conversely, the rate of change of the content of the reservoir is determined by the state of the queueing system, since the reservoir fills during idle periods and depletes during busy periods of the server. Our interest focuses on the stationary joint distribution of the number of customers in the system and the content of the fluid reservoir, from which various performance measures such as the steady-state sojourn time distribution of a customer may be obtained. We study two variants of the system. For the first, in which the fluid reservoir is infinitely large, we present an exact analysis. The variant in which the fluid reservoir is finite is analysed approximatively through a discretization technique. The system may serve as a mathematical model for a traffic regulation mechanism - a two-level traffic shaper - at the edge of an ATM network, regulating a very bursty source. We present some numerical results showing the effect of the mechanism.

On the “pasta” property and a further relationship between customer and time averages in stationary queueing systems

We consider queueing systems in statistical equilibrium and discuss a property which is known as PASTA (Poisson Arrivals See Time Averages). We give examples showing that for this property as well as for a related property called Conditional PASTA to be valid the independence assumption usually made in this connection is not necessary. Furthermore, we extend to queueing systems with preemptive service discipline a formula of Brumelles relating the time-average workload to the delay in queue and the service time of a customer.

Orthogonal polynomials on

The purpose of this paper is to extend some results of Karlin and McGregors and Chiharas concerning the three-terms recurrence relation for polynomials orthogonal with respect to a measure on the nonnegative real axis. Our findings are relevant for the analysis of a type of Markov chains known as birth-death processes with killing.

R^+

We consider a discrete closed-loop conveyor system consisting of a loading station, an unloading station, and a number of carriers which move with constant speed along a closed track. At the loading station units arrive in batches while the arrival of batches is governed by a Poisson process. The units queue at the loading station and await there the arrival of an empty carrier. Each unit requires some amount of service which is provided while the unit is on a carrier. After completion of service a unit leaves the system as soon as it reaches the unloading station. Our interest focuses on the steady-state queue length at the loading station. We obtain explicit results for the case where units leave their carriers on passing the unloading station for the second time. We were motivated to study this case by a slotted ring protocol for local area networks.

and birth-death processes with killing

We study the stationary distribution of the number of busy servers in a GI/GI/∞ system in which the service-time distribution is identical to the interarrival-time distribution, and obtain several representations for the variance. As a result we can verify an expression for the variance, conjectured by Rajaratnam and Takawira (IEEE Trans. Vehicular Technol. 50 (2001) 954–970), when the common distribution of interarrival and service times is a gamma distribution.

Queueing analysis of a discrete closed-loop conveyor with service facilities

Reconfiguring a network to counter variations in traffic is expected to greatly enhance optimal usage of network resources. But an important input to this method is the traffic fluctuations themselves. We have developed two models for this purpose to describe the time-dependent variations in traffic at a base station in a nomadic computing, wireless environment. The first model is rather simple and does not take into account details of human behavior. It takes into account the probabilities of choosing different applications. The model is also analyzed and experimented with to identify the important input parameters. The second model, a refined version of the first model, takes into account details of relevant human behavior (in the context of a wireless nomadic computing environment). Finally, we have compared the two models on the basis of their complexity and validity in different situations.