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

TL;DR

This paper derives geometric lower bounds for steady-state queue occupancy in processing networks with limited connectivity, revealing performance limitations compared to classic policies as network size grows.

Contribution

It introduces new geometric lower bounds based on flexibility metrics, highlighting fundamental performance constraints in constrained processing networks.

Findings

Lower bounds are geometric with ratios tied to flexibility metrics.

Performance gap persists unless flexibility metrics grow unbounded.

Classic policies like Power-of-d are asymptotically optimal only with infinite flexibility.

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.