Understanding Queueing Theory: Core Models, Rules, and Key Metrics

1 Queueing Theory Model

Queueing theory describes the phenomenon where people, objects, or information must wait for service.

Physical queues, e.g., buying tickets or waiting at a gas station.

Virtual queues, e.g., telephone switch calls or computer processing requests.

For convenience, we refer to all entities that wait as customers and all service providers (people or machines) as servers .

Queueing theory studies the stochastic behavior of queueing systems under various arrival and service time distributions, aiming to build mathematical models that support optimal design and operation decisions.

2 Mathematical Model of Queueing Systems

All queueing systems consist of three basic components: input process , queueing rule , and service rule .

2.1 Input Process

The input process describes the source of customers and the pattern of their arrivals. Typical aspects include:

Whether the number of potential customers is finite or infinite.

Whether arrivals occur singly or in batches.

Whether arrivals are independent (no influence between successive arrivals) or correlated.

Whether inter‑arrival times follow a deterministic or random distribution.

Whether the process is stationary or non‑stationary.

In the following discussion we assume independent arrivals and a stationary input process.

2.2 Service Facility and Service Rule

The service facility is described by:

Number of service stations (servers) : one or many.

Configuration of stations : single‑queue‑single‑server, single‑queue‑multiple‑servers, multiple queues‑multiple servers, series, or hybrid arrangements.

Service mode : serving customers individually or in batches (this article considers only individual service).

Service time : deterministic or random; if random, its distribution (often exponential) must be known.

Assumption : at least one of inter‑arrival time or service time is random.

We consider only single‑customer service in this discussion.

2.3 Queueing and Service Rules

Common queueing disciplines include:

Waiting discipline (customers wait when all servers are busy) with service orders such as: First‑Come‑First‑Served (FCFS) Last‑Come‑First‑Served (LCLS) Random Service (SIRO) Priority Service (PR)

Loss discipline (customers leave immediately if all servers are busy).

Hybrid discipline (combines waiting and loss, e.g., finite queue length, limited waiting time, or limited total time in system).

3 Kendall Notation for Queueing Models

The general form is A/S/c/K/N/D where:

A – inter‑arrival time distribution

S – service time distribution

c – number of servers

K – system capacity

N – size of the customer population

D – service discipline

For example, M/M/1/∞/∞/FCFS denotes a single‑server system with exponential inter‑arrival and service times, unlimited capacity, unlimited population, and first‑come‑first‑served discipline.

4 Main Performance Indicators of Queueing Systems

Key metrics used to evaluate system performance include:

Queue length (L) : expected number of customers in the system (waiting + being served).

Waiting length (Lq) : expected number of customers waiting in line.

Waiting time (W) and service time (Ws) : average times spent waiting and being served.

Busy period : continuous time during which a server remains occupied.

Idle period : continuous time a server remains free.

Server utilization : proportion of time the server is busy.

Customer loss rate : proportion of customers that leave without receiving service.

These indicators help assess efficiency, determine optimal parameters, and guide system design improvements.

References

ThomsonRen GitHub: https://github.com/ThomsonRen/mathmodels