queueingteoria

Queueing theory is the mathematical study of waiting lines. It analyzes how random arrivals of customers or jobs and the time required to serve them interact to determine performance measures such as waiting times, queue lengths, and server utilization. The field provides models and methods to predict system behavior under uncertainty and to guide design and operations decisions.

The subject originated with A. K. Erlang’s early 20th‑century work on telephone traffic, and it expanded in the mid‑century with formal notation and analysis developed by D. G. Kendall and others. Kendall’s notation classifies queueing models by the arrival process, service process, number of servers, system capacity, and queue discipline, written as A/B/c/K/π. For example, M/M/1 denotes Poisson arrivals, exponential service, and a single server, while FCFS indicates first‑come, first‑served discipline.

Common models include M/M/1, M/M/c, M/G/1, and G/G/1, among others. Parameters such as L (average number in the system), Lq (in the queue), W (average time in the system), Wq (in the queue), and ρ (utilization) are used to summarize performance. Little’s law, L = λ W, links these quantities in a general, model‑independent form. Many results exist in closed form for Markovian models, while general or complex systems often require approximations or simulation.

Queueing networks extend single queues to interacting servers and systems, including open and closed networks and product‑form models like Jackson networks. Applications span telecommunications, data centers, manufacturing, healthcare, and service operations. Limitations include assumptions of independence, stationary processes, and well‑defined distributions, which may not hold in all real systems.