Large deviations for the overflow level of G/G/1 queues in series

We present a result characterising the large deviations behaviour of the total overflow level in a cycle starting with zero customers for a system of G/G/1 queues in series. We also present large deviations results for the total overflow level as seen by a random customer and in stationarity. We prove that the large deviations behaviour of the total overflow level for all three distributions, in a cycle, as seen by a random customer and in stationarity, have the same decay rate. We find the most likely path to have overflow in the system. Based on those results we propose a state-independent importance sampling algorithm. We also give conditions under which that algorithm is asymptotically efficient. By means of numerical simulation, we provide evidence of the advantages of this algorithm.