Last updated: 2026-06-29

M/M/1 Queue

The M/M/1 queue is the foundational model of queueing theory: a single server handling requests that arrive at random, with random service times. From just two numbers — how fast requests arrive and how fast the server can process them — it predicts the average wait, the average response time, and how long the queue grows. The calculator returns all of these along with the server's utilization.

What the name means

The notation follows Kendall's classification. The first M says arrivals are Markovian — they follow a Poisson process, meaning each arrival is independent and the count in any interval is random. The second M says service times are exponentially distributed. The 1 is the number of servers. So M/M/1 is "random arrivals, random service, one server, unlimited queue, served in order."

Utilization sets everything

The single most important quantity is utilization, the fraction of time the server is busy:

$\rho = \dfrac{\lambda}{\mu}$

where $\lambda$ is the arrival rate and $\mu$ is the service rate, both in the same time unit. The system is stable only when $\rho < 1$; if requests arrive at least as fast as they can be served, the backlog grows without bound. From $\rho$, the steady-state averages follow:

$L = \dfrac{\rho}{1 - \rho}, \qquad L_q = \dfrac{\rho^2}{1 - \rho}$

$W = \dfrac{1}{\mu - \lambda}, \qquad W_q = \dfrac{\lambda}{\mu(\mu - \lambda)}$

Here $L$ is the average number in the system, $L_q$ the average number waiting, $W$ the average response time, and $W_q$ the average wait before service.

Worked example

A server processes 10 requests per second ($\mu = 10$) and receives 8 per second ($\lambda = 8$). Utilization is $\rho = 0.8$, and:

So a request spends half a second in the system on average, and four requests are present at any moment. Even though the server is only 80% busy, the response time is five times the raw service time of 0.1 s — most of it is waiting.

The danger of high utilization

Because $(1 - \rho)$ sits in the denominator of every formula, the metrics rise sharply as the server approaches saturation. At 90% utilization the average number in the system is 9; at 95% it is 19; at 99% it is 99. This nonlinear blow-up is why capacity planners leave headroom rather than running resources near 100%. The same averaging idea underlies Little's Law Calculator, which relates concurrency, throughput, and latency without assuming any particular distribution.

The M/M/1 model assumes exponential service times and unlimited queue space, which real systems rarely match exactly. Treat its predictions as a conservative guide: when service times are more regular than exponential, actual waits are usually shorter.