Asymptotic behaviour of the tandem queueing system with identical service times at both queues

Consider a tandem queue consisting of two single server queues in series with a Poisson arrival process at the rst queue and arbitrarily distributed service times which for any customer are identical in both queues For this tandem queue we relate the tail behavior of the sojourn time distribution and the workload distribution at the second queue to that of the residual service time distribution As a by result we prove that both the sojourn time distribution and the workload distribution at the second queue are regularly varying at in nity of index if the service time distribution is regularly varying at in nity of index Furthermore in the latter case we derive a heavy tra c limit theorem for the sojourn time S at the second queue when the tra c load It states that for a particular contraction factor the contracted sojourn time S converges in distribution to the limit distribution H as where H w expf w g w AMS subject classi cation K B