M/M/1 Queue

The engineering

The M/M/1 queue models a single server fed by memoryless (Poisson) arrivals at rate λ, each taking an exponentially-distributed service time at mean rate μ. It's the workhorse first approximation for a single-threaded worker, a lock, a disk, or one CPU core — enter arrival and service rates in the same units and read the mean latency and backlog straight off.

The whole story lives in the utilization ρ: latency blows up as ρ → 1, not linearly but as 1/(1−ρ). Push a server to 90% busy and mean wait is 10× the service time; at 95% it's 20×. That knee is why capacity planners keep steady-state utilization well below 1 and why the last 10% of headroom is the most expensive.

Where this math comes from

Queueing theory was born at a telephone exchange. Agner Krarup Erlang, working for the Copenhagen Telephone Company, published his analysis of call congestion in 1909 and gave engineers their first tools for sizing trunk lines against random demand — the Poisson-arrival assumption came straight from watching real switchboards.

The compact M/M/1 results engineers use today were consolidated by David Kendall, who introduced the A/B/c shorthand notation in 1953, and above all by Leonard Kleinrock, whose early-1960s MIT work applied queueing theory to message-switched networks — the theoretical seed of packet switching and the ARPANET. His textbooks made these formulas standard equipment for anyone sizing a system.

1. 1909A. K. ErlangPublishes the first queueing analysis of telephone traffic congestion.
2. 1953David G. KendallIntroduces the A/B/c notation that names the M/M/1 queue.
3. 1961John D. C. LittleProves L = λW, the law tying queue length to wait time.
4. 1964Leonard KleinrockApplies queueing theory to data networks, seeding packet switching.

See the full timeline of the math behind every calculator →

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