# Kelly Cache Networks

## TL;DR

This paper investigates cache placement in queueing networks to optimize objectives like delay and congestion, proposing efficient algorithms and extending results to various queue types.

## Contribution

It formulates cache placement as a submodular maximization problem and introduces a fast continuous greedy algorithm with near-optimal approximation guarantees.

## Key findings

- The optimization problem is NP-hard but approximable.
- The continuous greedy algorithm achieves near 63% of optimal.
- Results extend to M/M/k and M/D/1 queue networks.

## Abstract

We study networks of M/M/1 queues in which nodes act as caches that store objects. Exogenous requests for objects are routed towards nodes that store them; as a result, object traffic in the network is determined not only by demand but, crucially, by where objects are cached. We determine how to place objects in caches to attain a certain design objective, such as, e.g., minimizing network congestion or retrieval delays. We show that for a broad class of objectives, including minimizing both the expected network delay and the sum of network queue lengths, this optimization problem can be cast as an NP- hard submodular maximization problem. We show that so-called continuous greedy algorithm attains a ratio arbitrarily close to $1 - 1/e \approx 0.63$ using a deterministic estimation via a power series; this drastically reduces execution time over prior art, which resorts to sampling. Finally, we show that our results generalize, beyond M/M/1 queues, to networks of M/M/k and symmetric M/D/1 queues.