A restless time-fractional multiclass queue

TL;DR

The paper addresses a non-Markovian, multiclass single-server queue in which both arrivals and services follow independent fractional Poisson processes with indices $α$ and $β$, respectively. It leverages a time-change representation via inverse stable subordinators to obtain a multinomial thinning structure for class arrivals, derives a functional LLN and FCLT for the arrivals, and uses Skorokhod reflection to obtain scaling limits for aggregated and per-class queue lengths. The key contributions include an explicit thinned-FPP representation, process-level limit theorems for the arrival process, detailed queue-length scaling limits under various regimes of $α$ and $β$, and recurrence/transience characterizations, with an extension to a continuum of classes. These results illuminate how heavy-tailed, non-Markovian dynamics and a shared random clock induce positive cross-class dependence and nontrivial queue behavior, with practical implications for staffing and tail-risk assessment in congested systems.

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 $α\in (0,1]$ and the service completions are triggered by an independent fractional Poisson process of index $β\in (0,1]$. 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 $α\le β$, which does include the critical case $α= β$. A final example shows how the model can be extended to a continuum of classes.

Theorems & Definitions (7)

• proof : Proof of Theorem \ref{['thm:babyarrival']}
• proof : Proof of Corollary \ref{['cor:babyarrival']}
• proof : Proof of Theorem \ref{['thm:LLN']}
• proof : Proof of Theorem \ref{['thm:fclt']}
• proof : Proof of Theorem \ref{['thm:QConvergence']}
• proof : Proof of Corollary \ref{['cor:qclt']}
• proof : Proof of Theorem \ref{['thm:Rec+Trans']}