🎫 Queueing Theory

M/M/c is Kendall's notation for a queueing system with Markovian (Poisson) arrivals, Markovian (exponential) service times, and c parallel servers. The first M means the time between arrivals is exponentially distributed, the second M means each server's service time is exponentially distributed, and c is the number of identical servers working in parallel. This simulation implements exactly that model, running arrivals and departures as discrete random events on a simulated clock.

Utilisation ρ = λ/(cμ) is the fraction of total service capacity being used, where λ is the arrival rate and cμ is the combined service rate of all c servers. When ρ is comfortably below 1, servers can keep up and the queue stays short. As ρ approaches 1, servers are almost always busy, so any random burst of arrivals has nowhere to go and the waiting line grows increasingly long; at ρ ≥ 1, arrivals outpace capacity entirely and the queue grows without bound, which is exactly what the red load bar warns about.

Erlang-C is a closed-form formula that computes exact steady-state statistics for an M/M/c queue directly from λ, μ and c, without needing to run a simulation. It gives the probability an arriving customer has to wait at all, P(wait > 0), and from that derives the expected number waiting Lq, expected number in the whole system L, expected waiting time Wq, and expected total time in system W (via Little's law, W = Wq + 1/μ). This page shows the theoretical Erlang-C values in yellow right next to the live simulated measurements in blue, so you can see how closely random simulation matches the mathematical prediction.

The live statistics are time-weighted averages accumulated from actual random events — exponential inter-arrival and service times drawn fresh each time — so they carry statistical noise, especially soon after a reset or preset change. The Erlang-C numbers are the exact mathematical steady-state averages the system would reach given infinite time. The longer the simulation runs at a fixed λ, μ and c, the closer the blue numbers converge to the yellow ones, which is a live demonstration of the law of large numbers.

FIFO (first-in, first-out) serves customers in the order they arrived, like a typical checkout line — this is the discipline assumed by the classic Erlang-C formula. LIFO (last-in, first-out) instead serves whichever customer joined the queue most recently, which can leave early arrivals waiting indefinitely under heavy load even though the average waiting time Wq stays the same as FIFO. Switching disciplines does not change any of the Erlang-C averages, but it changes who experiences the longest individual waits, visible as the 'Longest wait' statistic.