PollaczekGeiringer

Pollaczek-Geiringer is a mathematical formula used in queueing theory to calculate the expected waiting time in a queueing system. It is named after Ladislaus von Bortkiewicz and Alfred Geiringer, who independently derived the formula in the early 20th century. The formula is particularly useful in analyzing systems where customers arrive randomly and are served one at a time, such as in call centers, computer networks, and traffic management.

The Pollaczek-Geiringer formula is derived from the Pollaczek-Khinchine formula, which provides the expected waiting time in a queueing system with a single server and infinite waiting room. The Pollaczek-Geiringer formula extends this by considering the variability of service times, which can significantly impact the expected waiting time. It is expressed as:

E[W] = (lambda * E[S^2]) / (2 * (1 - rho))

where E[W] is the expected waiting time, lambda is the arrival rate, E[S^2] is the variance of the service time, and rho is the server utilization factor, defined as lambda * E[S], where E[S] is the expected service time.

The Pollaczek-Geiringer formula is a fundamental tool in queueing theory and has been widely used in various fields to model and analyze queueing systems. It provides valuable insights into the performance of these systems and helps in designing and optimizing them to improve efficiency and reduce waiting times.