Parallel server systems under an extended heavy traffic condition: A lower bound

TL;DR

This paper extends the heavy traffic condition for parallel server systems, allowing multiple operational modes, and proves that the workload control problem provides an asymptotic lower bound on system costs.

Contribution

It introduces an extended heavy traffic condition for PSS, allowing multiple solutions and modes, and establishes the workload control problem as a lower bound for system costs.

Findings

The workload control problem's value is an asymptotic lower bound.

Multiple operational modes are possible under the extended HTC.

The analysis generalizes the standard critical LP assumption.

Abstract

The standard setting for studying parallel server systems (PSS) at the diffusion scale is based on the heavy traffic condition (HTC), which assumes that the underlying static allocation linear program (LP) is critical and has a unique solution. This solution determines the graph of basic activities, which identifies the set of activities (i.e., class-server pairs) that are operational. In this paper we explore the extended HTC, where the LP is merely assumed to be critical. Because multiple solutions are allowed, multiple sets of operational activities, referred to as modes, are available. Formally, the scaling limit for the control problem associated with the model is given by a so called workload control problem (WCP) in which a cost associated with a diffusion process is to be minimized by dynamically switching between these modes. Our main result is that the WCP's value constitutes an asymptotic lower bound on the cost associated with the PSS model.