Erlang-C

Erlang-C returns P(wait > 0) in an M/M/c queue with arrival rate λ, service rate μ = 1/AHT, and n parallel servers. Utilisation ρ = λ/(nμ) must be below 1 or the queue blows up. Given a service-level target 'X percent of calls answered within T seconds' (e.g., 80/20), the minimum n is solved. It is the industry standard for call-center staffing. Its limitation: it assumes infinitely patient customers — no abandonment. As a result, in the real world it tends to over-staff calm hours and under-staff spikes. Erlang-A (Garnett et al. 2002) adds a patience distribution and is more realistic. Erlang-B (the loss formula) applies when waiting is not allowed and blocked calls are lost (circuit switching).