A Restless Time-Fractional Multiclass Queue

Abstract

We study a single-server priority queue with a finite number of classes, in which the arrivals follow a fractional Poisson process of index $$\alpha \in (0,1]$$ α ∈ ( 0 , 1 ] and the service completions are triggered by an independent fractional Poisson process of index $$\beta \in (0,1]$$ 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 $$\alpha $$ 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 $$\alpha \le \beta $$ α ≤ β , which does include the critical case $$\alpha = \beta $$ α = β . A final example shows how the model can be extended to a continuum of classes.