Erlangformler

Erlangformler are a set of mathematical formulas used in queueing theory to analyze traffic and staffing in service systems. Named after the Danish mathematician Agner Krarup Erlang, these formulas originated from early work on telephone networks and are widely used today in call centers, data centers, and other environments where multiple servers handle random arrivals. They typically assume Poisson arrivals and exponential service times (an M/M/s model) and focus on probabilities of blocking and waiting, as well as mean delays.

Erlang B gives the blocking probability when there are s servers and no waiting space. Let λ be the arrival rate, μ the service rate per server, and a = λ/μ the offered traffic in Erlangs. The blocking probability is B(s,a) = (a^s / s!) / Σ_{k=0}^{s} (a^k / k!).

Erlang C, used for systems with waiting rooms, provides the probability that an arriving customer must wait. With a = λ/μ and ρ = a/s, the wait probability is P_wait = [ a^s / (s! (1 - ρ)) ] / [ Σ_{k=0}^{s-1} (a^k / k!) + a^s / (s! (1 - ρ)) ]. The mean waiting time in queue is Wq = P_wait / (sμ - λ), and the total time in the system is W = Wq + 1/μ.

These formulas are exact for the M/M/s model and provide practical guidance for capacity planning, staffing, and service level targets. They assume constant arrival rates and service times, independence, and no variability beyond exponential service, and they are less applicable to heavily time-varying or non-Markovian systems.