Queuing Theory: The M/M/1 Model Explained with Examples

Queuing theory studies waiting lines — customers at a checkout, jobs at a printer, calls at a help desk. The simplest and most important model is M/M/1: a single server, Poisson (random) arrivals, and exponential service times. This tutorial explains its formulas and works a full example.

Compute all queue metrics instantly with the free Queue Calculator.

Notation

📐 Formula

λ = arrival rate (customers per unit time) μ = service rate (customers per unit time) ρ = utilization = λ / μ (must be < 1 for a stable queue)

The "M/M/1" name means Markovian arrivals / Markovian service / 1 server.

The Key Formulas

📐 Formula

P₀ = 1 − ρ (probability the system is empty) L = ρ / (1 − ρ) (average number in the system) Lq = ρ² / (1 − ρ) (average number waiting in the queue) W = 1 / (μ − λ) (average time in the system) Wq = λ / [μ(μ − λ)] (average time waiting in the queue)

Step-by-Step Example

Customers arrive at a single-server help desk at λ = 2 per minute, and the server handles μ = 3 per minute.

Step 1: Utilization

📐 Formula

ρ = λ / μ = 2 / 3 ≈ 0.667 (the server is busy ~67% of the time)

Because ρ < 1, the queue is stable.

Step 2: Numbers in the system and queue

📐 Formula

L = 0.667 / (1 − 0.667) = 2 customers Lq = 0.667² / (1 − 0.667) ≈ 1.33 customers

Step 3: Times in the system and queue

📐 Formula

W = 1 / (3 − 2) = 1 minute Wq = 2 / [3 × (3 − 2)] ≈ 0.667 minute (40 seconds)

So on average there are 2 customers in the system, each spending 1 minute, of which 40 seconds is spent waiting.

Little's Law

All these quantities are tied together by Little's Law:

📐 Formula

L = λ · W and Lq = λ · Wq

Check: λ·W = 2 × 1 = 2 = L. ✓

Multiple Servers (M/M/c)

When there are c parallel servers sharing one queue, use the M/M/c model. The Queue Calculator supports both M/M/1 and M/M/c and shows every step.

Common Mistakes

Using the model when ρ ≥ 1 — the queue is unstable and grows without bound.
Mixing up W and Wq — W includes service time; Wq is waiting only (W = Wq + 1/μ).
Inconsistent time units — λ and μ must use the same unit.

Try It Yourself

Enter your arrival rate, service rate, and number of servers into the Queue Calculator to get utilization, L, Lq, W, Wq, and probabilities with step-by-step working. For project scheduling instead, see the PERT/CPM Solver.

Key Takeaways

M/M/1 models a single-server queue with random arrivals and service.
The queue is stable only when ρ = λ/μ < 1.
For λ = 2, μ = 3: ρ ≈ 0.67, L = 2, Lq ≈ 1.33, W = 1 min, Wq ≈ 0.67 min.
Little's Law (L = λW) links the numbers and the times.