waitingline

Waitingline, often equivalent to a queue, is the arrangement of customers, jobs, or items waiting for service or processing in a system. It forms whenever service capacity is limited and demand varies over time, and it is a central subject of queueing theory within operations research.

A waitingline is described by several elements: the arrival process (how entities enter the line, frequently modeled as a Poisson process), the service process (time required to serve each entity, which may have any distribution), the number of servers, the service discipline or queue policy (for example, first-come, first-served), and system capacity. Other factors include balking or reneging, wherein customers leave the line if it is too long. Key performance measures include the average queue length, the average waiting time, the probability that an entity will wait, and the system utilization.

Common models use Kendall’s notation, such as M/M/1 for a single-server system with Poisson arrivals and exponential service times, or M/M/c for multiple servers. Other models include M/G/1 and G/G/1. These models provide analytic or approximate formulas for performance metrics under assumed distributions and traffic conditions, often with concepts like Little’s Law, which relates the average number in the system to the arrival rate and average time spent.

Applications span retail, call centers, healthcare, manufacturing, and transportation. Management strategies include staffing, appointment systems, virtual queues, and priority rules to balance wait times against service capacity and customer satisfaction.

Limitations arise from simplifying assumptions (independent arrivals and services, stationary processes) and sensitivity to distributional assumptions. The field has historical roots in telephony research and developments in queueing notation and theory.