Queuing

Queuing, or queueing, is the study of waiting lines and the behavior of queues in service and manufacturing systems. A queueing system typically consists of customers or items arriving to receive a service, a waiting line, and one or more servers. Key elements include the arrival process, the distribution of service times, the number of servers, the queueing discipline that determines service order, and the system capacity.

Queueing disciplines commonly use first-in, first-out (FIFO) ordering, but many systems employ alternatives such as priority queues or processor sharing. Models are often described using Kendall's notation, A/B/C, where A is the arrival process, B is the service time distribution, and C is the number of servers; modifiers indicate capacity or population size (for example, M/M/1/K).

Performance is summarized by metrics such as the average waiting time in queue (Wq), the time in the system (Ws), the average number in queue (Lq) and in the system (L), and the server utilization (ρ). Little's Law, L = λ W and Lq = λ Wq, relates these quantities under steady-state conditions.

Applications span call centers, computer networks, printing and checkout lines, and manufacturing assembly, where queueing theory informs capacity planning, staffing, buffering, and routing. Variants consider balking or reneging (customers leaving before joining or while waiting) and blocking when the system is full, each affecting performance predictions.

Historically, queueing theory originated with studies of telephone traffic by Agner Krarup Erlang, and the notation and methods were formalized in the mid-20th century, with ongoing development through stochastic modeling and simulation.