---
title: 'Queueing theory: reading route'
url: https://doc.liz6.com/en/theory/03-queueing-theory
locale: en
area: theory
tags:
- Queueing theory
- Theory
date: 2026-09-10
modified: 2026-09-12
description: Queueing theory studies arrivals, waiting and service at limited resources. This series starts with hand-calculable request timelines, develops Little’s law and classical stochastic queues, then applies them to scheduling, networks, measurement and recovery. Assumptions accompany formulas, and means remain distinct from tails.
---

# Queueing theory: reading route

Queueing theory studies arrivals, waiting and service at limited resources. This series starts with hand-calculable request timelines, develops Little’s law and classical stochastic queues, then applies them to scheduling, networks, measurement and recovery. Assumptions accompany formulas, and means remain distinct from tails.

## Chapters

1. [System boundaries and queueing timelines](/en/theory/03-queueing-theory/01-boundaries-and-timelines) — Separate waiting, service and occupancy; calculate a FIFO trace.
2. [Little’s law and measurement boundaries](/en/theory/03-queueing-theory/02-littles-law) — Derive the area identity and account for cutoff boundaries.
3. [Arrival processes and model assumptions](/en/theory/03-queueing-theory/03-arrival-processes) — Understand Poisson, exponential gaps, variability and Kendall notation.
4. [M/M/1: utilization and tail latency](/en/theory/03-queueing-theory/04-mm1-and-tail-latency) — Derive stationary probabilities, means and model-specific p99.
5. [Variability, M/G/1 and Kingman’s approximation](/en/theory/03-queueing-theory/05-variability-and-mg1) — Explain second moments, residual service and approximation limits.
6. [Multiple servers and resource pooling](/en/theory/03-queueing-theory/06-multiple-servers) — Compare pooling and random splitting using Erlang C.
7. [Scheduling, head-of-line blocking and fairness](/en/theory/03-queueing-theory/07-scheduling-and-fairness) — Compare per-job gains and costs, fairness and starvation.
8. [Finite queues and admission control](/en/theory/03-queueing-theory/08-finite-queues-and-admission) — Jointly assess rejection, admission, utilization and admitted delay.
9. [Queueing networks and bottlenecks](/en/theory/03-queueing-theory/09-networks-and-bottlenecks) — Find visit counts and bottleneck bounds; separate serial and fan-out tails.
10. [Measurement, load testing and discrete-event simulation](/en/theory/03-queueing-theory/10-measurement-and-simulation) — Check open/closed load, censoring, warmup and sampling error.
11. [Capacity planning and overload recovery](/en/theory/03-queueing-theory/11-capacity-and-overload-recovery) — Budget recovery using net headroom, retries and scaling delays.

## Choose a route

- First encounter: 01 → 02 → 03 → 04, then continue in order.
- Service capacity and tails: 01 → 02 → 04 → 05 → 06 → 10 → 11; add 03 if distributions are unfamiliar.
- Messaging and multitenancy: 01 → 02 → 07 → 08 → 09 → 11, using 10 to check observation bias.
- Modeling: 03 → 04 → 05 → 06 → 09 → 10, after understanding the boundaries in 01–02.

01–02 need algebra and an intuition for area. 03–06 use expectation, variance, exponentials and series; 09 adds linear traffic equations. The code in 10 uses arrays and loops. Time is in seconds unless an example explicitly converts to milliseconds.

## Notation

| Symbols | Meaning |
| --- | --- |
| $a_i,b_i,d_i$ | Arrival, service start and departure timestamps |
| $S_i,W_{q,i},T_i$ | Service, waiting and system time; $T_i=W_{q,i}+S_i$ |
| $N(t),Q(t),B(t)$ | Jobs in system, waiting and in service |
| $L,L_q$ | Long-run mean system and waiting counts |
| $\lambda,\lambda_{\mathrm{eff}},X$ | Offered arrival, effective admission and actual completion rates |
| $\mu,c,K$ | Per-server capacity, server count and total capacity including service |
| $\rho$ | Per-server utilization in homogeneous infinite-capacity models; finite queues use r for the offered ratio |
| $C_a^2,C_s^2$ | Squared coefficients of variation of gaps and service |
| $V_j,D_j$ | Visits and service demand per business request |

$A(\Delta)$ in 03 counts arrivals in a window; $A(t)$ in 01 is cumulative arrivals from the origin. Chapter 11 redefines $A$ as attempts per business request. Erlang C and fluid capacity C are also local notation.

## Match a conclusion to its question

| Tool | What it provides | What does not follow directly |
| --- | --- | --- |
| Area identity / Little's law | Consistent flow, mean occupancy and mean time | p99 or an instantaneous capacity guarantee |
| Stationary M/M/1, M/M/c, M/M/1/K | Exact metrics under each model's assumptions | The same results for arbitrary real traffic |
| M/G/1 PK | Exact FCFS means with finite second moments | The entire tail distribution |
| Kingman | GI/GI/1 mean-wait approximation | A universal upper bound or p99 |
| Finite traces / simulation | Policy comparisons and sensitivity checks | Proof of stationarity or statistical precision from one trace |
| Fluid work balance | Backlog trends and recovery from net headroom | Stochastic request tails |

## Use experiments to test a prediction

Predict, vary one parameter, then explain the change. Each chapter includes two expandable reasoning answers. Derivations, hand calculations and summaries remain readable without scripts. Initial state, units, scheduling and censoring are explicit. Identical parameters reproduce identical results; a fixed seed does not remove statistical error.

## Sources and connections

Scope is informed by [CMU performance modeling](https://www.cs.cmu.edu/~harchol/Tools/class.html) and the [MIT queueing lecture](https://web.mit.edu/1.041/www/lectures/L8-queuing-models-2026sp.pdf). Chapters cite original Google SRE, AWS Builders' Library, k6 and HdrHistogram material for practice. Examples and interactions are independently designed, without universal utilization targets or production capacity promises.

Combine this with [control theory](/en/theory/02-control-theory) to connect the creation of waiting with resource adjustment, and [message semantics](/en/distributed-systems/07-messages-and-streams/01-message-semantics) for durability and retries. Information theory studies informational uncertainty; queueing theory studies residence at limited resources. Their metrics are not interchangeable.