Extended Erlang C formular

The Erlang C formula was established in the early 20th century by the Danish mathematician Agner Krarup Erlang, to optimize the performance of the then manually occurring telephone connections. While the original Erlang C formula assumes that the clients are willing to wait any length of time, the here offered extended Erlang C formula also respects the impatience of the clients.

In the Erlang model it is assumed that the inter-arrival times of the clients, the service times and the waiting time tolerances of the clients are distributed exponentially. In addition, it is assumed that the system is in steady state.

To calculate the parameters of queuing system, only the arrival rate, the service rate, the average waiting time tolerance and the number of parallel operators has to be specified.

Arrival process Service process Parameter
Arrival rate (in clients per minute)
Average waiting time tolerance (in minutes)
Average service and post processing time (in minutes)
Number of agents
Waiting time (in secunds)
The probability of a client to have to wait at maximum als long as here specified will be displayed.

Calculate performance indicators

Key performance indicators of the queueing model

Erlang calculator for spreadsheets

Via the following links example tables and script allowing to calculate all common Erlang queueing models can be downloaded:

Criticism of the Erlang-C model

At its time of the early 19th century the Erlang formulas opened the first opportunity to model and calculate queueing processes analytically. However, the formulas do not take into account many of the relevant properties today:

  • The exponential distribution is used for the distribution of the service times which usualy is not very realistic.
  • The exponential distribution is used for the distribution of the waiting time tolerances which usualy is not very realistic.
  • It is assumed that the system is in steady state, i.e., that the arrival rate does not change.
  • Customers who start a repeated attempt after a waiting cancelation are not mapped in the model.
  • More complex features, such as different types of customers (and the resulting possibly of multi-skill agents), forwarding, post processing times of the agents, prioritization strategies, etc. are not mapped.

To meet these new demands, the analysis and optimization of queueing systems is now mostly done using simulation methods.

Simulation software

TU Clausthal offers a number of simulation programs via the Simulation Science Center. These are all open source programs that can be used free of charge:

Mini Simulator (Webapp)

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

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

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.


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