ASYMPTOTIC ANALYSIS OF THE M G/1 QUEUE WITH A TIME-DEPENDENT ARRIVAL RATE/

We consider the M/G/1 queue with an arrival rate lambda that depends w eakly upon time, as lambda = lambda(epsilon t) where epsilon is a smal l parameter. In the asymptotic limit epsilon --> 0, we construct appro ximations to the probability p(n)(t) that n customers are present at t ime t. We show that the asymptotics are different for several ranges o f the (slow) time scale tau = epsilon t. We employ singular perturbati on techniques and relate the various time scales by asymptotic matchin g.