Asymptotic analysis of the M/G/1 queue with a time-dependent arrival rate

Abstract

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