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Queueing simulation

Queueing simulation

Queues occur in many situations in production and logistics. They always occur when the customer arrival flow cannot be exactly matched with the available operating capacity:

  • At inbound call centers, the times at which customers call cannot be predicted exactly, so that either agents have to be kept on standbyor customers may have to wait.
  • The exact arrival times of aircrafts at the destination airport depend on various weather conditions, especially on wind direction and strength. Therefore, an exact prearrangement of landing time slots is only possible to a limited extent or aircrafts have to fly holding patterns or wait on the ground for a take-off clearance, if a delayed arriving aircraft has to get a landing clearance first.
  • In multi-variant industrial production, the operating times at individual stations vary depending on the specific properties of the respective workpiece. Therefore, a fully clocked production is not always possible, especially with batch size 1. If, however, a high utilization of the machines is to be ensured, this can only be achieved by buffer stocks in front of the machines.

In general, queues are usually undesirable, as they either lead to unnecessary time loss (call center queue), directly result in unnecessary costs (kerosene consumption due to waiting in queues before landing) or indirectly cause costs (capital tie-up due to stock, long and poorly predictable delivery dates). The goal of the queueing theory is therefore to design operating systems in such a way that the additional costs, e.g. due to more operators, and the financial advantages due to short queues are in an optimal ratio.

Industrial projects

Leitwarte Sinteranlage, Photo: Uwe Braun, SHS – Stahl-Holding-Saar

The methods of queueing theory and event-oriented stochastic simulation described on this page have been used in a number of industrial projects. The following list contains some examples of corresponding projects:

Queueing theory and simulation

Queueing theory as a subfield of stochastics was founded more than 100 years ago by Agner Krarup Erlang. Originally, the main field of application was personnel requirements planning in manual telephone switching centers. Since the 1950s, research has increasingly focused on questions from industrial production. Based on the analytical queueing theory, waiting time distributions for simple models can be determined exactly and characteristics for more complex models and queueing networks can be approximated.

However, the more complex the models to be considered are, the further the quality of the respective models decreases. With the availability of more and more computing power, the simulation of queueing models has therefore become the focus of research since the 1980s. In principle, simulation models can be used to illustrate all conceivable models and questions. The quality of the results is only limited by the available computing power. While the necessary computing times in the last century were still in the range of hours and days, today a few seconds up to a few minutes are common for simulation. - For simple models, the computing power of a smartphone is already sufficient.


The following 4 simulation tools have been developed within the research at SWZ. The three desktop programs are available as open source for free download. The webapp is available for free use on the SWZ web pages:

The Mini Simulator is a web app fully implemented in Javascript that can be run in any modern browser (including tablets and smartphones). The app maps a G/G/c/K+M model, i.e. a model consisting of a queue and a operating station. Batch arrivals, batch operations, customer impatience, repeaters and forwarding can be mapped.

Mini Callcenter Simulator essentially reproduces the same G/G/c/K+G model that the webapp contains. However, it has much more probability distributions that can be used for inter-arrival times, service times, post-processing times, waiting time tolerances and repeat distance distributions. In addition, considerably more characteristics are recorded and various export options are available for the simulation results. Furthermore, the simulation results can be directly compared to corresponding Erlang-C results and explanations can be displayed why deviations occur at which points.

Mini Callcenter Simulator requires a Java runtime environment and has been published as open source.

Callcenter Simulator is designed to map real call center systems consisting of several sub-call centers, different caller groups, different agent groups (with different skill levels and different shift plans), complex assignment rules, etc. It can be used directly for staff requirements planning and for the analysis of possible control strategies in large call center networks. In addition to pure simulation, the program also provides functions for automatic optimization of the number of agents.

Callcenter Simulator requires a Java runtime environment and has been published as open source.

Warteschlangensimulator allows the simulation of any complex queueing network. The models are defined in Warteschlangensimulator in the form of flowcharts. Optionally, an animation can be displayed during the simulation of the models to illustrate the movement of the customers through the system. For the automated examination of different models, parameter series can be created automatically and an optimizer is also available. Furthermore, external data sources can be connected directly during the simulation of models and (partial) results can also be transferred directly to external programs (e.g. databases).

Warteschlangensimulator requires a Java runtime environment and was published as open source.


Dr. Alexander Herzog

Managing Director of Simulationswissenschaftliches Zentrum
Research group Stochastische Modelle in den Ingenieurwissenschaften
Institute of Mathematics
Clausthal University of Technology

Room 313, Arnold-Sommerfeld-Straße 6, 38678 Clausthal-Zellerfeld
Room 115, Erzstraße 1, 38678 Clausthal-Zellerfeld
E-Mail: Alexander.Herzog@tu-clausthal.de
Phone: +49 5323 72-2966, Fax: +49 5323 72-2304