M/M/1 Queue Calculator

Queueing theory studies systems where work arrives over time and must wait for limited service capacity. The M/M/1 model is the simplest widely used queue: arrivals follow a Poisson process (memoryless interarrival times), service times are exponential (memoryless), and there is one server. Even with its simplicity, M/M/1 is a practical first approximation for a single help-desk agent, one checkout lane, one machine on a line, or a single API worker processing jobs one at a time.

What you enter (and units)

You provide two rates in the same time unit:

• Arrival rate λ: expected arrivals per unit time (e.g., 10 customers/hour).
• Service rate μ: expected services completed per unit time (e.g., 12 customers/hour).

Important: Keep units consistent. If λ is “per hour,” μ must also be “per hour.” The calculated times (W, Wq) will be in that same time unit (hours in this example).

Stability condition (when the formulas apply)

The classic steady-state M/M/1 results require that the server is fast enough on average:

• μ > λ (equivalently, utilization ρ < 1).

If λ ≥ μ, the system does not reach a steady state: the expected queue length and waiting time grow without bound, so metrics like L and W are not meaningful as long-run averages.

Key formulas (M/M/1)

Define utilization:

• ρ = λ/μ

Then the standard M/M/1 steady-state metrics are:

• L: average number of customers/jobs in the system (waiting + in service)
• Lq: average number waiting in line (excluding any in service)
• W: average time in the system (waiting + service time)
• Wq: average time waiting in line (excluding service time)

The relationships:

• L = ρ / (1 - ρ)
• Lq = ρ² / (1 - ρ)
• W = L / λ
• Wq = Lq / λ

Equivalent and often useful identities:

• W = 1 / (μ - λ)
• Wq = λ / (μ(μ - λ))
• W = Wq + 1/μ (time in system = time waiting + mean service time)

MathML (for unambiguous rendering)

How to interpret the results

• ρ (utilization): fraction of time the server is busy. Values near 1 imply high congestion and rapidly increasing waits. For many real operations, sustained ρ above ~0.8–0.9 can mean long and volatile queues.
• L (avg. in system): average number present, including the one possibly being served. If L = 2.5, think “about 2–3 customers/jobs in the system on average.”
• Lq (avg. in queue): average number waiting (not including the one in service). If Lq = 1.7, roughly 1–2 are waiting at any given time on average.
• W (avg. time in system): total time from arrival to completion. This includes both waiting and service time.
• Wq (avg. waiting time): time spent waiting before service begins. This excludes service time.

Because M/M/1 is nonlinear in μ - λ, small increases in service rate (or small decreases in arrival rate) can produce large reductions in W and Wq when utilization is already high.

Worked example

Suppose:

• λ = 5 arrivals/hour
• μ = 7 services/hour

1) Utilization:

• ρ = λ/μ = 5/7 ≈ 0.714

2) Average number in system:

• L = ρ/(1-ρ) = 0.714 / 0.286 ≈ 2.50

3) Average number waiting:

• Lq = ρ²/(1-ρ) = 0.714² / 0.286 ≈ 1.79

4) Average time in system and in queue:

• W = L/λ = 2.50/5 = 0.50 hours ≈ 30 minutes
• Wq = Lq/λ = 1.79/5 = 0.357 hours ≈ 21.4 minutes

Sanity check: mean service time is 1/μ = 1/7 hour ≈ 8.6 minutes. And W ≈ Wq + 1/μ ≈ 21.4 + 8.6 = 30 minutes.

Formula summary table

Assumptions and limitations

• Poisson arrivals (Markovian): interarrival times are exponential and independent; arrival rate λ is constant.
• Exponential service times (Markovian): service durations are memoryless with constant rate μ.
• Single server: exactly one job/customer can be served at a time.
• Queue discipline: typically assumes first-come, first-served (FCFS). Other disciplines can change waiting-time distributions (though some averages may coincide under specific conditions).
• Infinite queue capacity and calling population: no blocking, and potential arrivals are not limited by a finite population.
• Steady-state requirement: the closed-form averages require λ < μ.
• Real-world variability: bursty arrivals, time-varying rates, batching, priorities, or non-exponential service can make actual waits differ materially from M/M/1 predictions.

Historical note

M/M/1 analysis is rooted in early telephone-traffic engineering. Agner Krarup Erlang developed foundational queueing models to quantify congestion and staffing needs in telephone exchanges, and those ideas remain central to modern operations, from contact centers to computing systems.

Privacy: Calculations run in your browser; inputs are not sent to a server by default.