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M/M/1 Queue Calculator

Enter the arrival rate λ and service rate μ (λ < μ) to get the M/M/1 queue metrics.

How it is calculated

Arrival rate λ and service rate μ are counts per the same unit of time, for example customers per hour. The tool first finds the utilization ρ = λ ÷ μ, then L = ρ ÷ (1 − ρ) customers in the system, Lq = ρ² ÷ (1 − ρ) customers waiting in line, W = 1 ÷ (μ − λ) time in the system and Wq = λ ÷ (μ × (μ − λ)) time waiting before service starts. W and Wq come out in the inverse of your rate unit, so rates per hour give hours. Results are shown with two decimals and utilization as a percentage. If λ is equal to or larger than μ the queue grows without limit and the tool returns no result, and so it does for μ at or below zero or a negative λ.

Worked example

Customers arrive at 8 per hour and one server handles 10 per hour. Utilization: 8 ÷ 10 = 80.00%. In the system: 0.8 ÷ 0.2 = 4.00 customers. In the queue: 0.64 ÷ 0.2 = 3.20 customers. Time in the system: 1 ÷ (10 − 8) = 0.50 hours, which is 30 minutes. Wait before service: 8 ÷ (10 × 2) = 0.40 hours, or 24 minutes.

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Common mistakes

  • Entering rates in different time units, such as arrivals per hour and service per minute. Both must use the same unit, or every result is wrong.
  • Entering the average service time instead of the service rate. If one customer takes 6 minutes, μ is 10 per hour, not 6.
  • Expecting a result when λ equals μ. A server that is 100% busy gets an infinitely long queue on average, so the tool shows nothing.
  • Reading the wait as a guarantee. These are long-run averages; individual customers can wait far more or far less.
  • Applying it to systems with several servers or scheduled arrivals. The model assumes one server, random arrivals and exponentially varied service times.

About this calculator

The M/M/1 queue is the foundational model of queueing theory: a single server, Markovian (Poisson) arrivals at rate λ, and Markovian (exponential) service times at rate μ — the "M/M/1" name comes directly from Kendall's notation for this arrival/service/server-count combination. This calculator takes λ and μ (with λ < μ required for a stable, non-exploding queue) and returns the standard performance metrics.

It computes utilization ρ = λ/μ, the average number of customers in the system L = ρ/(1−ρ) and in the queue Lq = ρ²/(1−ρ), and the average time in the system W = 1/(μ−λ) and waiting time Wq = ρ/(μ−λ) — via Little's Law, L = λW. Operations researchers, call-center planners and computer-network engineers use these formulas to predict wait times and congestion from nothing more than an arrival rate and a service rate.

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Comments (1)

  • Editorial team

    Try nudging the arrival rate up in small steps and watch how the queue reacts. At 90 percent utilization the average number in the system is 9; at 95 percent it is 19, and it climbs steeply as the rate approaches the service rate. That is why a server only slightly busier on paper can feel much slower. Keep both rates in the same time unit. What are you modelling: a help desk, a checkout or a network link?

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