We consider the M/G/1 queue with an arrival rate λ that depends weakly upon time, as λ = λ(ϵt) where ϵ is a small parameter. In the asymptotic limit ϵ → 0, we construct approximations to the probability pn(t) the n customers are present at time t. We show that the asymptotics are different for several ranges of the (slow) time scale τ = ϵt. We employ singular perturbation techniques and relate the various time scales by asymptotic matching.
