Queueing Process

A queueing process tracks the number of customers or jobs in a queueing system over time, and is most commonly modeled as a birth-death process, arrivals (births) increase the queue length, and service completions (deaths) decrease it, with rates that can depend on the current queue length. Queueing models are classified using Kendall's notation, A/S/c, where $A$ describes the arrival process (e.g. M for Markovian/Poisson arrivals), $S$ describes the service time distribution (e.g. M for exponential, D for deterministic, G for general), and $c$ is the number of servers. The canonical M/M/1 queue (Poisson arrivals at rate $\lambda$, exponential service at rate $\mu$, single server) has an explicit steady-state distribution for queue length $L$, geometric with parameter $\rho =\lambda /\mu$ (the traffic intensity), and Little's Law, $L=\lambda W$, relates the average number in the system to the average time a customer spends in it, a relationship that holds far more generally, for essentially any stable queueing system, not just M/M/1.

• Traffic intensity $\rho =\lambda /\mu$ has an outsized practical importance: a single-server queue is only stable (queue length doesn't grow without bound) when $\rho <1$, and as $\rho$ approaches 1 from below, average waiting times explode nonlinearly, a well-known and often counterintuitive phenomenon in operations and capacity planning.
• Little's Law is remarkable precisely because of how little it assumes: it holds for essentially any stable queueing system regardless of the arrival process, service distribution, or number of servers, a rare 'assumption-free' result in an area otherwise full of model-specific formulas.
• Queueing theory extends far beyond physical lines of people: internet packet routing, hospital emergency room capacity planning, call center staffing, and manufacturing production lines are all standard real-world settings modeled and optimized using this same birth-death-process queueing framework.