An overview of Brownian and non-Brownian FCLTs for the single-server queue

We review functional central limit theorems (FCLTs) for the queue-content p rocess in a single-server queue with finite waiting room and the first-come first-served service discipline. We emphasize alternatives to the familiar heavy-traffic FCLTs with reflected Brownian motion (RBM) limit process tha t arise with heavy-tailed probability distributions and strong dependence. Just as for the familiar convergence to RBM, the alternative FCLTs are obta ined by applying the continuous mapping theorem with the reflection map to previously established FCLTs for partial sums. We consider a discrete-time model and first assume that the cumulative net-input process has stationary and independent increments, with jumps up allowed to have infinite varianc e or even infinite mean. For essentially a single model, the queue must be in heavy traffic and the limit is a reflected stable process, whose steady- state distribution can be calculated by numerically inverting its Laplace t ransform. For a sequence of models, the queue need not be in heavy traffic, and the limit can be a general reflected Levy process. When the Levy proce ss representing the net input has no negative jumps, the steady-state distr ibution of the reflected Levy process again can be calculated by numericall y inverting its Laplace transform. We also establish FCLTs for the queue-co ntent process when the input process is a superposition of many independent component arrival processes, each of which may exhibit complex dependence. Then the limiting input process is a Gaussian process. When the limiting n et-input process is also a Gaussian process and there is unlimited waiting room, the steady-state distribution of the limiting reflected Gaussian proc ess can be conveniently approximated.