Queueing theory continues to be one of the most researched areas of operational research, and has generated numerous review papers over the years. The phrase 'queue modelling' is used in the title to indicate a more practical emphasis. This paper uses work taken predominantly from the last 50 years of pages of the Operational Research Quarterly and the Journal of the Operational Research Society to offer a commentary on attempts of operational researchers to tackle real queueing problems, and on research foci past and future. A new discipline of 'queue modelling' is proposed, drawing upon the combined strengths of analytic and simulation approaches with the responsibility to derive meaningful insights for managers.

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Reflections on queue modelling from the last 50 years

Waiting time, busy periods and output models of a server analyzed via Wiener-Hopf factorization

Since Erlang did pioneering work in the application of queues to telephony, a lot has been written about queues in continuous time (see, for example [As-mussel, 1987, Bacelli and Bremaud, 1994, Bhat and Basawa, 1992, Boxma and Syski, 1988, Bunday, 1986, Bunday, 1996, Chaudhry and Templeton, 1983, Cohen, 1982, Cooper, 1981, Daigle, 1992, Gnedenko and Kovalenko, 1989, Gross and Harris, 1985, Kalashnikov, 1994, Kashyap and Chaudhry, 1988, Kleinrock, 1975, Lipsky, 1992, Medhi, 1991, Prabhu, 1965, Robertazzi, 1990, Srivastava and Kashyap, 1982, Tijms, 1986, White et al., 1975]). In comparison to that large body of literature, not much has been written about queues in discrete time.

On Numerical Computations of Some Discrete-Time Queues

Semi?Markovian processes (SMP) serve as a versatile means to model various traffic types generated by sources or superposed from links in ATM networks. An efficient discrete time analysis method is proposed for SMP/G/1 queueing systems to evaluate ATM switches with non?renewal input. Some basic results are derived for the autocorrelation function of semi?Markovian processes and for the effect of state space reduction on autocorrelation, which show how to represent correlated traffic with given characteristic by an adequate SMP model of limited size.

Semi-Markovian modelling and performance analysis of variable rate traffic in ATM networks

We study analysis methods for the dimensioning of switched or routed networks to determine sufficient link capacity that can guarantee small loss probabilities as a main quality of service requirement. The bufferless case is considered, motivated by a far simpler analysis as compared to queueing models but also by the inefficiency of buffering to achieve low cell or packet loss rates in the presence of real-time constraints and/or long range dependencies in the development of the traffic load. Statistical multiplexing assures quality of service at reasonable bandwidth utilization with a minimum of buffering under fairly general conditions. For voice traffic we show that at most a few percent of bandwidth is gained by buffering, as also expected for data applications revealing self-similar properties over several time scales.

The relevance of the bufferless analysis for traffic management in telecommunication networks

This paper deals with approximate analysis methods for open queueing networks. External and internal flows from and to the nodes are characterized by renewal processes with discrete time distributions of their interarrival times. Stationary distributions of the waiting time, the queue size and the interdeparture times are obtained using efficient discrete time algorithms for single server (GI/G/1) and multi-server (GI/D/c) nodes with deterministic service. The network analysis is extended to semi-Markovian representations of each flow among the nodes, which include parameters of the autocorrelation function.

https://link.springer.com/content/pdf/10.1057%2Fjors.1996.76.pdf

Analysis of Open Discrete Time Queueing Networks: A Refined Decomposition Approach

Es wird ein Ansatz zur Berechnung der Verteilungen der Wartezeit und der Kundenanzahl an den Stationen eines offenen Netzes mit beliebig gegebenen Verzweigungswahrscheinlichkeiten vorgestellt. Die Analyse erfolgt durch ein Iterationsverfahren und geht von einer Dekomposition des Netzwerks in einzelne Knoten aus, wobei insbesondere Einbedienerstationen (GI/G/1-Systeme) betrachtet werden. Die Ankunfte und Abgange an den Netzknoten werden dabei als Erneuerungsprozesse im diskreten Zeitbereich beschrieben. Allgemein konnen beliebige Netzknoten oder aggregierte Teile eines Netzes mit einbezogen werden, wenn zu gegebenen Ankunftsprozessen auch ihre Abgangsprozesse zumindest naherungsweise in der Form unabhangiger Erneuerungsprozesse darstellbar sind.

Erik S. Rieger

Papers

Analysis of Open Discrete Time Queueing Networks: A Refined Decomposition Approach

Modellierung offener Warteschlangennetzwerke durch Erneuerungsprozesse im diskreten Zeitbereich