Load Balancing Under Strict Compatibility Constraints

Abstract

Consider a system with N identical single-server queues and M(N ) task types, where each server is able to process only a small subset of possible task types. Arriving tasks select d ≥ 2 random compatible servers, and join the shortest queue among them. The compatibility constraints are captured by a fixed bipartite graph G N between the servers and the task types. When G N is complete bipartite, the meanfield approximation is accurate. However, such dense compatibility graphs are infeasible for large-scale implementation. We characterize a class of sparse compatibility graphs for which the meanfield approximation remains valid. For this, we introduce a novel notion, called proportional sparsity, and establish that systems with proportionally sparse compatibility graphs asymptotically match the performance of a fully flexible system. Furthermore, we show that proportionally sparse random compatibility graphs can be constructed, which reduce the server-degree almost by a factor N /ln(N ) compared to the complete bipartite compatibility graph.

Optimal Load Balancing in Bipartite Graphs

“…Contemporaneous to our work, in [41], the authors study the waiting time of JSQ(d) policies in bipartite graphs in the limit as the size of the graph goes to infinity. While the papers are motivated by related problems, the models and routing policies studied, and the results in the two papers are different.…”

“…While the papers are motivated by related problems, the models and routing policies studied, and the results in the two papers are different. The authors in [41] consider the case of homogeneous servers with infinite buffers, and show that the performance of JSQ(d) in a bipartite graph with limited connectivity converges to the performance of the fully flexible system in terms of queue length (or waiting time) under appropriate connectivity conditions. In addition, they prove that the occupancy in steady state of the limited-connectivity system converges to the steady state of the fully flexible system.…”

“…Additionally, the techniques used in the two papers are different. We use the drift method to obtain performance bounds for finite-sized systems while [41] uses process-level convergence techniques.…”

Large-System Insensitivity of Zero-Waiting Load Balancing Algorithms

“…Significant processes have been made over the past few years on understanding achieving asymptotic zero-waiting (as the system size approaches infinity) in a large-scale data center with distributed queues, including the classic supermarket model [14,8,32,17,3,4,30,24,25,23,22,45,9], models with data locality [40,31] and models where each job consists of parallel tasks [39,37,19], etc.…”

“…However, almost all these results assume exponential service time distributions. While each of these results [14,8,32,17,29,30,24,25,23,22,40,31,39,37,19,45,9] provided important insights of achieving zero-waiting in a practical system, theoretically, it is not clear whether these principles hold for general service times. This is a very important question to answer because it is well-known that service time distributions in real-world systems are not exponential.…”