Queueing M/M/s Calculator

An M/M/s system assumes Poisson arrivals at rate lambda, exponentially distributed service times at rate mu per server, s identical servers drawing from one shared line, unlimited waiting room and first-come first-served order. The offered load r equals lambda divided by mu and is expressed in Erlangs; utilisation rho equals r divided by s and must be strictly below 1 or the queue grows without limit. The probability of an empty system P0 comes from normalising the birth-death balance equations, and the Erlang C expression built on it gives the chance an arriving customer finds every server busy. Mean queue length Lq follows, and Little law - L equals lambda times W - converts it into mean waiting time Wq by simple division. The practical lesson is that congestion is non-linear in rho. At 70 percent utilisation waits are modest, at 90 percent they are several times longer, and beyond about 95 percent small forecast errors produce very large swings, which is why contact centres and clinics generally plan to 70 to 85 percent occupancy rather than to full capacity. Pooling matters too: one queue feeding six servers outperforms three separate queues of two. The main caveats are the distributional assumptions. Exponential service is optimistic for tasks with low variability, such as scripted transactions, and pessimistic for heavy-tailed work; real arrival rates vary by hour, so the model must be applied to short stationary intervals; and it ignores abandonment, retrials, priority classes and finite waiting room, all of which need extensions such as Erlang A.

Queueing theory begins with Agner Krarup Erlang, a Danish mathematician employed by the Copenhagen Telephone Company. His 1909 paper on the theory of probabilities and telephone conversations established that call arrivals could be modelled as a Poisson process, and his 1917 work produced the loss formula now called Erlang B, used to size trunk groups where blocked calls are simply lost. The delay formula for systems where blocked calls wait in line - Erlang C, the basis of the waiting probability computed here - grew out of the same programme, and the unit of offered traffic was later named the Erlang in his honour. The field was generalised after the Second World War. David Kendall introduced the compact A/B/c notation in 1953, which is why the model is written M/M/s with M standing for the memoryless, or Markovian, assumption on arrivals and service. John D. C. Little published the proof of the relationship L equals lambda times W in 1961, giving the field a result that holds regardless of the arrival or service distribution. Textbooks by Frederick Hillier and Gerald Lieberman and by Leonard Kleinrock carried the models into operations research and computer science teaching, and Kleinrock work in the 1960s applied queueing analysis to packet-switched networks. Modern workforce management software for call centres still computes staffing from Erlang C, usually augmented with abandonment models.