A Fluid Limit for Cache Algorithms with General Request Processes

We introduce a formal limit, which we refer to as a fluid limit, of scaled stochastic models for a cache managed with the Least-Recently-Used algorithm when requests are issued according to general stochastic point processes, which may be non-stationary. We define our fluid limit as a superposition of dependent replications of the original system with smaller item sizes as the number of replications approaches infinity. We derive the average probability that a requested item is not in a cache (average miss probability) in the fluid limit. The usefulness of the fluid limit is demonstrated in two ways. First, our numerical experiments show that, when items are requested according to inhomogeneous Poisson processes, the average miss probability in the fluid limit closely approximates that in the original system as long as there are sufficient number of items. Second, we show that the asymptotic characteristics of the average miss probability as the cache size approaches infinity are often preserved in the fluid limit. This preservation is attractive since the asymptotic analysis in the fluid limit appears to be simpler than that in the original system. In addition, we show that the average miss probability in the fluid limit is asymptotically insensitive to particular dependencies in the requests when the request rates have a light tail, a property not known for the original system.