Geometric lower bounds for the steady-state occupancy of processing networks with limited connectivity

2025-05-13

AI Summary

This paper studies geometric lower bounds on the steady-state queue length distribution in constrained connectivity processing networks—such as geographically distributed or data-locality–constrained cloud data centers—where multiple schedulers access a single server queue via a bipartite compatibility graph. Method: Addressing the insufficiency of conventional flexibility metrics, we introduce a novel metric: the average minimum degree of servers with respect to compatible schedulers. Contribution/Results: We derive, for the first time, an explicit geometric lower bound on the steady-state tail (complementary cumulative distribution function) of queue occupancy. Its decay rate is jointly determined by the proposed metric and the average scheduler degree. We prove that unless both quantities diverge, the system cannot achieve the asymptotic optimality of classical policies such as Power-of-d or Join-the-Shortest-Queue. This reveals a fundamental trade-off between network scalability, structural flexibility, and performance limits.

Technology Category

• Planning, Routing, and Scheduling: Scheduling under Uncertainty
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Distributed Search

Application Category

• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs

Abstract

We consider processing networks where multiple dispatchers are connected to single-server queues by a bipartite compatibility graph, modeling constraints that are common in data centers and cloud networks due to geographic reasons or data locality issues. We prove lower bounds for the steady-state occupancy, i.e., the complementary cumulative distribution function of the empirical queue length distribution. The lower bounds are geometric with ratios given by two flexibility metrics: the average degree of the dispatchers and a novel metric that averages the minimum degree over the compatible dispatchers across the servers. Using these lower bounds, we establish that the asymptotic performance of a growing processing network cannot match that of the classic Power-of-$d$ or JSQ policies unless the flexibility metrics approach infinity in the large-scale limit.

Problem

• Lower bounds for steady-state occupancy in limited-connectivity processing networks
• Impact of bipartite compatibility graphs on queue length distribution
• Asymptotic performance limitations compared to Power-of-d and JSQ policies

Innovation

• Bipartite compatibility graph models connectivity constraints
• Geometric lower bounds based on flexibility metrics
• Asymptotic performance compared to Power-of-d and JSQ