Cache Networks of Counting Queues

Abstract

We consider a cache network in which intermediate nodes equipped with caches can serve content requests. We model this network as a universally stable queuing system, in which packets carrying identical responses are consolidated before being forwarded downstream. We refer to resulting queues as <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathtt {M/M/1c}$ </tex-math></inline-formula> or <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">counting queues</i> , as consolidated packets carry a counter indicating the packet’s multiplicity. Cache networks comprising such queues are hard to analyze; we propose two approximations: one via <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathtt {M/M/\infty }$ </tex-math></inline-formula> queues, and one based on <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathtt {M/M/1c}$ </tex-math></inline-formula> queues under the assumption of Poisson arrivals. We show that, in both cases, the problem of jointly determining (a) content placements and (b) service rates admits a poly-time, <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$1-1/e$ </tex-math></inline-formula> approximation algorithm. We also show that our analysis, with respect to both algorithms and associated guarantees, extends to (a) counting queues over items, rather than responses, as well as to (b) queuing at nodes and edges, as opposed to just edges. Numerical evaluations indicate that our proposed approximation algorithms yield good solutions in practice, significantly outperforming competitors.