 This relationship can only be solved exactly in special cases (such as the M/M/1 queue), but for any "s" the value of ϕ("s") can be calculated and by iteration with upper and lower bounds the distribution function numerically computed.Writing W*("s") for the Laplace–Stieltjes transform of the waiting time distribution, is given by the Pollaczek–Khinchine transform as
where g("s") is the Laplace–Stieltjes transform of service time probability density function.As the arrivals are determined by a Poisson process, the arrival theorem holds.