Large deviations of the sojourn time for queues in series

We consider an open queueing network consisting of an arbitrary number of queues in series. We assume that the arrival process into the first queue and the service processes at the individual queues are jointly stationary and ergodic, and that the mean inter-arrival time exceeds the mean service time at each of the queues. Starting from Lindley's recursion for the waiting time, we obtain a simple expression for the total delay (sojourn time) in the system. Under some mild additional assumptions, which are satisfied by many commonly used models, we show that the delay distribution has an exponentially decaying tail and compute the exact decay rate. We also find the most likely path leading to the build-up of large delays. Such a result is of relevance to communication networks, where it is often necessary to guarantee bounds on the probability of large delays. Such bounds are part of the specification of the quality of service desired by the network user.