# Queueing Theory Calculator

URL: https://speedycalculator.com/calculators/math/queueing-theory/
Last updated: 2026-10-30

> The queueing theory calculator analyses M/M/1 and M/M/s queues: utilisation, probability that the system is empty, probability of waiting (Erlang C), average numbers in the queue and in the system, average waiting and system times, and the probability of n customers.

## Inputs

- Queue model
- Arrival rate λ (customers per time unit)
- Service rate μ per server
- Number of servers s
- Probability of exactly n customers: n =

## Results

- Utilisation ρ
- Probability the system is empty P₀
- Probability an arrival waits (Erlang C)
- Average number waiting L_q
- Average number in the system L
- Average waiting time W_q
- Average time in the system W
- Probability of exactly n customers

## Formula

```
ρ = λ/(sμ),  a = λ/μ
P₀ = [Σₙ₌₀^{s−1} aⁿ/n! + aˢ/(s!(1 − ρ))]⁻¹,  L_q = P₀·aˢ·ρ / (s!(1 − ρ)²)
L = L_q + a,  W_q = L_q/λ,  W = W_q + 1/μ
```

## Explanation

Queueing theory predicts waiting lines at shop tills, call centres, web servers, hospital emergency rooms and toll booths. With random (Poisson) arrivals and exponential service times, a few formulas tell you how long people wait and how many servers you need.

### How to use the Queueing Theory Calculator

- **Queue model**
- **Arrival rate λ (customers per time unit)**
- **Service rate μ per server**
- **Number of servers s**
- **Probability of exactly n customers: n =**

Results update as you type, and everything is calculated in your browser.

### How it works

Enter the arrival rate λ and the service rate μ per server in the same time unit, and the number of servers s. The queue is stable only if λ < s·μ (utilisation below 100%).

### Inputs explained

FieldWhat to enterQueue modelChoose one: M/M/1 (one server), M/M/s (several servers)Arrival rate λ (customers per time unit)A numberService rate μ per serverA numberNumber of servers sChoose one: 1, 2, 3, 4 …Probability of exactly n customers: n =A number

### What each result means

- **Utilisation ρ** = [Arrival rate λ (customers per time unit)]÷([servers]×[Service rate μ per server])×100
- **Probability an arrival waits (Erlang C)** = [Probability the system is empty P₀]×([Arrival rate λ (customers per time unit)]÷[Service rate μ per server])^[servers]÷(fact([servers])×(1 - [Arrival rate λ (customers per time unit)]÷([servers]×[Service rate μ per server])))
- **Average number waiting L_q** = [Probability the system is empty P₀]×([Arrival rate λ (customers per time unit)]÷[Service rate μ per server])^[servers]×([Arrival rate λ (customers per time unit)]÷([servers]×[Service rate μ per server]))÷(fact([servers])×(1 - [Arrival rate λ (customers per time unit)]÷([servers]×[Service rate μ per server]))^2)
- **Average number in the system L** = [Average number waiting L_q] + [Arrival rate λ (customers per time unit)]÷[Service rate μ per server]
- **Average waiting time W_q** = [Average number waiting L_q]÷[Arrival rate λ (customers per time unit)]
- **Average time in the system W** = [Average waiting time W_q] + 1÷[Service rate μ per server]

### Example values

How the main result changes when **Arrival rate λ** changes and everything else stays at the example values above:

Arrival rate λ (customers per time unit)Average waiting time W_q40.066760.1580.412-0.616-0.2667

## Worked example

Inputs — Queue model: M/M/1 (one server); Arrival rate λ (customers per time unit): 8; Service rate μ per server: 10; Probability of exactly n customers: n =: 3.

Results — Utilisation ρ = 80 %; Probability the system is empty P₀ = 0.2; Probability an arrival waits (Erlang C) = 0.8; Average number waiting L_q = 3.2; Average number in the system L = 4; **Average waiting time W_q = 0.4**; Average time in the system W = 0.5; Probability of exactly n customers = 0.1024.

## Key terms

- **Utilisation**: The fraction of time servers are busy.
- **Erlang C**: The probability that an arriving customer has to wait.

## FAQ

**What does M/M/1 mean?**
Markovian (Poisson) arrivals, Markovian (exponential) service, one server.

**What happens when utilisation approaches 100%?**
Queues and waits grow without bound.

**Why does adding a server help so much?**
Waiting time falls sharply as utilisation drops below about 80%.

**What is Little's law?**
L = λW: average number in the system equals arrival rate times average time spent.


## Sources

- [Wolfram MathWorld – Queuing theory](https://mathworld.wolfram.com/QueuingTheory.html)
- [MIT OpenCourseWare – Queueing Systems](https://ocw.mit.edu/courses/6-262-discrete-stochastic-processes-spring-2011/)