The diffusion approximation of the Multiclass Processor Sharing queue

TL;DR

This paper studies a single-server multiclass processor-sharing queue with feedback, where jobs route between classes according to a Jackson-like mechanism and are tracked by a measure-valued state descriptor of residual service times. Under heavy-traffic diffusion scaling, the authors derive a diffusion limit for the state descriptor, showing $\hat{\mu}^r(\cdot) \Rightarrow \Delta^{\nu} W^{*}(\cdot)$, with $W^{*}$ a one-dimensional reflected Brownian motion and $\Delta^{\nu}$ the lifting map linking workload to the multiclass state. The analysis proceeds by relating the total residual service to a single-class PS diffusion, characterizing class-visit dynamics through a multiclass descriptor $\mathcal{Q}^r$, and proving a state-space collapse that yields a tractable diffusion limit for the entire multiclass network. The results extend diffusion-approximation techniques to non-head-of-the-line, multiclass PS queues with feedback, providing a rigorous diffusion-based description of the system's heavy-traffic behavior and enabling performance approximations in complex multiclass environments. The findings have implications for performance analysis and control of multiclass PS networks with feedback in communication and computing systems.

Abstract

This paper considers a multiclass processor-sharing queue with feedback. Jobs arrive according to renewal processes, and service times follow general distributions. Upon service completion, jobs may either depart the system or re-enter as a different class according to a probabilistic, Jackson-like routing mechanism. Under heavy-traffic conditions, we establish a diffusion approximation for a measure-valued process tracking the residual service times of jobs.

Theorems & Definitions (46)

• Definition 2.1: Fluid Solution Model
• Definition 3.1
• Theorem 3.1
• Lemma 4.1
• proof
• Lemma 4.2
• proof
• Remark 4.1
• Proposition 4.1
• proof