Mean Field Analysis of Join-Below-Threshold Load Balancing for Resource Sharing Servers

Load balancing plays a crucial role in many large scale computer systems. Much prior work has focused on systems with First-Come-First-Served (FCFS) servers. However, servers in practical systems are more complicated. They serve multiple jobs at once, and their service rate can depend on the number of jobs in service. Motivated by this, we study load balancing for systems using Limited-Processor-Sharing (LPS). Our model has heterogeneous servers, meaning the service rate curve and multiprogramming level (limit on the number of jobs sharing the processor) differs between servers. We focus on a specific load balancing policy: Join-Below-Threshold (JBT), which associates a threshold with each server and, whenever possible, dispatches to a server which has fewer jobs than its threshold. Given this setup, we ask: how should we configure the system to optimize objectives such as mean response time? Configuring the system means choosing both a load balancing threshold and a multiprogramming level for each server. To make this question tractable, we study the many-server mean field regime. In this paper we provide a comprehensive study of JBT in the mean field regime. We begin by developing a mean field model for the case of exponentially distributed job sizes. The evolution of our model is described by a differential inclusion, which complicates its analysis. We prove that the sequence of stationary measures of the finite systems converges to the fixed point of the differential inclusion, provided a unique fixed point exists. We derive simple conditions on the service rate curves to guarantee the existence of a unique fixed point. We demonstrate that when these conditions are not satisfied, there may be multiple fixed points, meaning metastability may occur. Finally, we give a simple method for determining the optimal system configuration to minimize the mean response time and related metrics. While our theoretical results are proven for the special case of exponentially distributed job sizes, we provide evidence from simulation that the system becomes insensitive to the job size distribution in the mean field regime, suggesting our results are more generally applicable.