PollaczekKhinchineformeln

Pollaczek–Khinchin formula is a fundamental result in queueing theory that provides key performance measures for an M/G/1 queue, where arrivals are Poisson with rate λ and service times have a general distribution with mean E[S] and second moment E[S^2]. The utilization factor is ρ = λ E[S], which must be less than 1 for the system to be stable. The formulas assume a single server and a first-come, first-served discipline.

The main results describe the average waiting time in the queue and related quantities. The average waiting time in the queue is Wq = λ E[S^2] / (2(1−ρ)). The average time in the system, including service, is W = E[S] + Wq. The average number in the queue is Lq = λ Wq = (λ^2 E[S^2]) / (2(1−ρ)), and the average number in the system is L = ρ + Lq.

Equivalently, using E[S^2] = Var(S) + (E[S])^2, the waiting time in queue can be expressed as Wq = λ (Var(S) + (E[S])^2) / (2(1−ρ)). The Pollaczek–Khinchin formulas separate the impact of arrival rate, service-time variability, and utilization, highlighting how variability in service times increases delays.

Named after Felix Pollaczek and A. Khintchine, the formulas are widely applied in telecommunications, computer networks, and service system design to assess performance and scalability under varying traffic and service characteristics.