A restless time-fractional multiclass queue

We study a single-server priority queue with a finite number of classes, in which the arrivals follow a fractional Poisson process of index α ∈ (0, 1] and the service completions are triggered by an independent fractional Poisson process of index β ∈ (0, 1]. Each of the customers arriving is assigned at random to one of the priority classes. This assignment is independent of the rest of the system and follows a fixed probability distribution. Using a time-change representation of a fractional Poisson process, we first give a multinomial thinning decomposition: The total number of arrivals in each class are independent standard Poisson processes of appropriate intensities, time-changed by a common independent random clock that is the inverse of an α-stable subordinator. This yields a process-level law of large numbers and a functional central limit theorem for the process of arrivals. For the queueing system itself, we identify process-level scaling limits for the cumulative and individual queue lengths of the classes. We also prove that the queue gets empty infinitely often when α ≤ β, which does include the critical case α = β. A final example shows how the model can be extended to a continuum of classes.