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 first 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 infinity of index 1 - \nu if the service time distribution is regularly varying at infinity of index - \nu (\nu > 1). Furthermore, in the latter case we derive a heavy-traffic limit theorem for the sojourn time S(2) at the second queue when the traffic load $p \uparrow 1$. It states that, for a particular contraction factor $\Delta (\rho)$, the contracted sojourn time $\Delta (\rho) S^{(2)}$ converges in distribution to the limit distribution H (.) as $p \uparrow 1$ where \[ H(w)=\frac{exp\{-w^{1-\nu}\}}{1+\nu w^{1-\nu}} \]