Engset formula

Engset formula is a fundamental concept in the field of telecommunications engineering and traffic engineering, used to calculate the probability of blocking in a telecommunications system. The formula is named after its inventor, T. O. Engset, a Norwegian engineer and mathematician who introduced this method in the early 20th century. The Engset formula is particularly useful in scenarios where the number of sources (such as telephone lines or communication channels) is limited and known, making it a critical tool for the design and analysis of telecommunication networks.

Overview

The Engset formula is applied to systems where the user population is finite, a condition that differentiates it from the Erlang B formula, which assumes an infinite number of sources. This makes the Engset calculation more applicable to private branch exchange (PBX) systems and small networks where the number of users is restricted and usually does not exceed the system's capacity.

Mathematical Formulation

The Engset formula calculates the blocking probability (the probability that a call will be blocked due to all channels being occupied) in a system with a finite number of sources. The formula is given by:

\[ P_b = \frac{{\binom{N-1}{Y} \cdot (\frac{A}{Y+1})^Y}}{{\sum_{k=0}^{Y} \binom{N-1}{k} \cdot (\frac{A}{Y+1})^k }} \]

where:

• \(P_b\) is the blocking probability,
• \(N\) is the total number of sources (users),
• \(Y\) is the number of available channels,
• \(A\) is the traffic intensity offered to the system, measured in Erlangs,
• \(\binom{N-1}{Y}\) and \(\binom{N-1}{k}\) are binomial coefficients.

Applications

The Engset formula is widely used in the planning and optimization of telecommunications networks, especially in environments with a finite and relatively small user group. It helps in determining the required number of channels or lines to achieve a desired quality of service, characterized by a specific blocking probability. This is crucial in the design of circuit-switched networks, PBX systems, and other telecommunications systems where capacity and quality of service are of paramount importance.

Comparison with Other Formulas

While the Engset formula is suitable for systems with a finite number of sources, the Erlang B formula and the Erlang C formula are used in scenarios with different assumptions. The Erlang B formula is ideal for calculating blocking probabilities in systems with an infinite number of sources and no waiting queues, whereas the Erlang C formula is used for systems where callers are queued when all channels are busy. The choice between these formulas depends on the specific characteristics of the system being analyzed.

Limitations

The main limitation of the Engset formula is its assumption of a finite number of sources, which may not be applicable to large-scale public networks. Additionally, the formula assumes that all sources generate traffic with the same statistical properties, which may not always be the case in real-world scenarios.

Conclusion

The Engset formula remains a vital tool in the field of telecommunications for the analysis and design of networks with a finite number of users. Its ability to accurately predict blocking probabilities helps engineers optimize network capacity and ensure a high quality of service.