What does M/M/1 mean?

M/M/1 is a queueing model where arrivals follow a Markov (Poisson) process, service times are Markov (exponential), and there is 1 server. M/M/c extends this to c identical parallel servers.

Erlang C (C(c,ρ)) is the probability that an arriving customer finds all c servers busy and must wait. It is the fundamental result of M/M/c analysis and drives Wq and Lq.

Why does utilization ≥ 1 break the calculator?

When arrival rate equals or exceeds total service capacity (λ ≥ c·μ), the queue never drains — it grows to infinity. The steady-state metrics are undefined. You must add servers or reduce arrival rate.

What time unit should I use?

λ and μ must use the same time unit. If λ is requests/second, μ must also be requests/second. Results for Wq and W will then be in seconds. Use per-minute values for slower systems.

How does adding a second server compare to doubling service rate?

Two servers at rate μ each (M/M/2) behaves very differently from one server at 2μ (M/M/1). M/M/2 typically gives shorter queues under load because work can proceed in parallel, while M/M/1 at 2μ has no queueing when a single fast server is free.

Queue Depth Calculator

Calculate M/M/1 and M/M/c queueing metrics: queue length, wait times, utilization, and Erlang C probability.

Queue Parameters

Arrival Rate λ (customers/unit)

Service Rate μ (per server/unit)

Number of Servers c

Metric Reference

Symbol	Name	Meaning
λ	Arrival rate	Customers arriving per time unit (Poisson process).
μ	Service rate	Customers one server completes per time unit (exponential service times).
c	Servers	Number of parallel identical servers.
ρ	Traffic intensity	λ / μ. Total offered load. Must be < c for a stable queue.
u	Utilization	ρ / c. Fraction of time each server is busy. Must be < 1.
P0	Idle probability	Probability the system has zero customers.
C(c,ρ)	Erlang C	Probability an arriving customer waits (all c servers busy).
Lq	Queue length	Average number of customers waiting in the queue (not being served).
L	System length	Average total customers in the system: Lq + ρ (Little's Law: L = λ × W).
Wq	Queue wait	Average time a customer spends waiting before service begins.
W	System time	Average time in the system: Wq + 1/μ. (Little's Law: W = L / λ).

Summary

Queue Depth Calculator applies M/M/1 (single-server) and M/M/c (multi-server) queueing theory models to compute steady-state queue metrics from three inputs: arrival rate (λ), service rate (μ), and number of servers (c). Results include traffic intensity, server utilization, average number of customers in queue (Lq) and system (L), average wait time in queue (Wq) and system (W), probability of empty system (P0), and Erlang C — the probability all servers are busy. A utilization ≥ 1 means the queue grows without bound.

How it works

Enter the arrival rate (λ) in customers per second (or per minute).
Enter the service rate (μ) — how many customers one server handles per the same time unit.
Set the number of servers (c) to 1 for M/M/1 or higher for M/M/c.
The calculator computes traffic intensity ρ = λ/μ and server utilization u = ρ/c.
If utilization ≥ 1 the queue is unstable; a warning is shown and infinite-queue metrics are not defined.
For stable queues, Erlang C gives P(waiting) and all Lq, L, Wq, W metrics follow from Little's Law.

Use cases

Capacity planning: find how many worker threads or pods are needed to keep wait time under a target SLA.
Database connection pool sizing: compute queue buildup when request rate approaches pool capacity.
Call center staffing: model agent count versus caller wait time using Erlang C.
Message queue monitoring: set alert thresholds for Lq before latency degrades.
Cloud autoscaling policy: determine the utilization trigger point where adding a server halves queue length.

Frequently Asked Questions

Last updated: 2026-06-19 · Reviewed by Nham Vu