Jac Jacques Resing

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.

Fitting discrete distributions on the first two moments

In this paper we present a simple method to fit a discrete distribution on the first two moments of a given random variable. With the Fitted distribution we solve approximately Lindleys equation for the D/G/1 queue with discrete service-time distribution using a moment-iteration method. Numerical results show excellent performance of the method.

Simple analysis of a fluid queue driven by an M/M/1 queue

In this note we consider the fluid queue driven by anM/M/1 queue as analysed by Virtamo and Norros [Queueing Systems 16 (1994) 373–386]. We show that the stationary buffer content in this model can be easily analysed by looking at embedded time points. This approach gives the stationary buffer content distribution in terms of the modified Bessel function of the first kind of order one. By using a suitable integral representation for this Bessel function we show that our results coincide with the ones of Virtamo and Norros.

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.

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.

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.

Delay models for contention trees in closed populations

In this paper we consider some models for contention resolution in cable networks, in case the contention pertains to requests from a finite population of stations and is carried out by means of contention trees. More specifically, we study a number of variants of the standard machine repair model, that differ in the service order at the repair facility. Considered service orders are first come first served, random order of service, and gated random order of service. For these variants, we study the sojourn time at the repair facility. In the case of the free access protocol for contention trees, the first two moments of the access delay in contention are accurately represented by those of the sojourn time at the repair facility under random order of service. In the case of the blocked access protocol, gated random order of service is shown to be more appropriate.

A two-level traffic shaper for an on-off source

Abstract This paper studies a two-level traffic shaper. The cell arrival process is modelled by an on–off fluid source. The output rate of the shaper is regulated by the content of a finite token bank. Our interest focuses on the stationary joint distribution of the content of the cell buffer and the token bank, from which various performance measures such as the steady-state cell delay and burst duration can be obtained. The system is analysed using an approximative model based on a stochastic discretization of the content of the token bank. We believe that this discretization technique has much wider applicability. Numerical results show the quality of the approximation and demonstrate the effect of the shaping mechanism.

Mean value analysis of single server retrial queues

Mean value analysis is an elegant tool for determining mean performance measures in queueing models. In this paper we show how mean value analysis can be applied to retrial queues. First, we illustrate the technique for the standard M/G/1 retrial queue with exponential retrial times. After that we show how the relations can be adapted to obtain mean performance measures in more advanced M/G/1-type retrial queues.

Performance analysis of small non-uniform packet switches

Packet switches have been studied extensively as part of ATM and LAN networks under the assumption that the number of input ports N tends to infinity. Our study of packet switches is motivated by networks on chips, where N is usually 4 or 5 and asymptotic models lead to inaccurate results. We consider small non-uniform switches and accurately approximate stability conditions and throughput. In addition to this, we approximate the mean waiting time in the switch by that in a ./Geo/1 queue.