A Tandem Queue with Coupled Processors: Computational Issues

In Resing and Örmeci [16] it is shown that the two-stage tandem queue with coupled processors can be solved using the theory of boundary value problems. In this paper we consider the issues that arise when calculating performance measures like the mean queue length and the fraction of time a station is empty. It is assumed that jobs arrive at the first station according to a Poisson process and require service at both stations before leaving the system. The amount of work that a job requires at each of the stations is an independent, exponentially distributed random variable. When both stations are nonempty, the total service capacity is shared among the stations according to fixed proportions. When one of the stations becomes empty, the total service capacity is given to the nonempty station. We study the two-dimensional Markov process representing the numbers of jobs at the two stations. The problem of finding the generating function of the stationary distribution can be reduced to two different Riemann-Hilbert boundary value problems, where both problems yield a complete analytical solution. We discuss the similarities and differences between the two problems, and relate them to the computational aspects of obtaining performance measures.     

An M/G/1 Queueing Model with Gated Random Order of Service

We analyse an M/G/1 queueing model with gated random order of service. In this service discipline there are a waiting room, in which arriving customers are collected, and a service queue. Each time the service queue becomes empty, all customers in the waiting room are put instantaneously and in random order into the service queue. The service times of customers are generally distributed with finite mean. We derive various bivariate steady-state probabilities and the bivariate Laplace–Stieltjes transform (LST) of the joint distribution of the sojourn times in the waiting room and the service queue. The derivation follows the line of reasoning of Avi-Itzhak and Halfin [4]. As a by-product, we obtain the joint sojourn times LST for several other gated service disciplines.     

Tandem queues with deterministic service times

In this paper we consider a tandem queueing model for a sequence of multiplexers at the edge of an ATM network. All queues of the tandem queueing model have unit service times. Each successive queue receives the output of the previous queue plus some external arrivals. For the case of two queues in series, we study the end-to-end delay of a cell (customer) arriving at the first queue, and the covariance of its delays at both queues. The joint queue length process at all queues is studied in detail for the 2-queue and 3-queue cases, and we outline an approach to the case of an arbitrary number of queues in series.Part of the research of this author has been supported by the European Grant BRA-QMIPS of CEC DG XIII.The research of this author was done during the time that he was affiliated with CWI, in a joint project with PTT Research.     

A tandem queueing model with coupled processors

We consider a tandem queue with coupled processors and analyze the two-dimensional Markov process representing the numbers of jobs in the two stations. A functional equation for the generating function of the stationary distribution of this two-dimensional process is derived and solved through the theory of Riemann-Hilbert boundary value problems.     

Polling systems and multitype branching processes

The joint queue length process in polling systems with and without switchover times is studied. If the service discipline in each queue satisfies a certain property it is shown that the joint queue length process at polling instants of a fixed queue is a multitype branching process (MTBP) with immigration. In the case of polling models with switchover times, it turns out that we are dealing with an MTBP with immigration in each state, whereas in the case of polling models without switchover times we are dealing with an MTBP with immigration in state zero. The theory of MTBPs leads to expressions for the generating function of the joint queue length process at polling instants. Sufficient conditions for ergodicity and moment calculations are also given.This work was done while the author was at the Centre for Mathematics and Computer Science (CWI) in Amsterdam, The Netherlands.     

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. This revised version was published online in June 2006 with corrections to the Cover Date.     

NMR observations of fluorine ion-implanted into polyacetylene and graphite

¹⁹FNMR spectroscopy has been used to characterize fluorine implanted into polyacetylene and highly orientated pyrolytic graphite. The intensities of the ¹⁹F NMR spectra at 4.5 K show that essentially all the fluorine is retained in these materials after the implantation process is over. The Une width of the spectrum of the graphite sample indicates that the fluorine remains where it was deposited.For the polyacetylene sample the ratio of the NMR solid echo height to the spin echo height, as calibrated herein, shows that the predominant nuclear diploar interaction of the implanted fluorine is heteronuclear.     

Queueing Models with Multiple Waiting Lines

This paper discusses analytic solution methods for queueing models with multiple waiting lines. The methods are briefly illustrated, using key models like the 2×2 switch, the shortest queue and the cyclic polling system.