queueingtheory

Queueing theory is a branch of mathematics that studies waiting lines, or queues. It provides theoretical frameworks for analyzing systems in which customers or tasks arrive, wait for service, and depart. The theory applies to a wide range of fields such as telecommunications, manufacturing, healthcare, transportation, and computer science. By modeling arrival processes, service disciplines, and system capacity, queueing theory can predict performance measures like average wait time, queue length, and server utilization.

Key models include the M/M/1 queue, which assumes Markovian (Poisson) arrivals and exponentially distributed service times with a single server. Variations such as M/M/c (multiple servers), M/G/1 (general service times), and G/G/1 (general arrivals and services) extend its applicability. The discipline of service can be first‑come first‑served, priority, or random, and these choices significantly impact system metrics. Queueing theory also addresses closed networks, where customers circulate among multiple service nodes, and open networks, where customers enter and leave the system.

Analytical solutions often rely on probability generating functions, Laplace transforms, and balance equations. Because exact solutions can be intractable for complex systems, approximation techniques and simulation play important roles. Queueing theory informs design decisions—such as how many servers to deploy, how to prioritize tasks, and how to balance load—to optimize cost and customer satisfaction.