Queue Depth Calculator
Calculate M/M/1 and M/M/c queueing metrics: queue length, wait times, utilization, and Erlang C probability.
Queue Parameters
Unstable Queue (utilization ≥ 1)
Traffic Intensity ρ
—
λ / μ
Utilization
—
ρ (= ρ/c for c=1)
P(0) — empty system
—
prob. no customers
P(wait) Erlang C
—
= 1 − P0 for c=1
Lq — in queue
—
avg customers waiting
L — in system
—
avg customers total
Wq — queue wait
—
avg time waiting
W — system time
—
avg time in system
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
Calculate M/M/1 and M/M/c queueing metrics: queue length, wait times, utilization, and Erlang C probability.
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: 2026-07-01 ·
Reviewed by Nham Vu