It was recently proved by Jelenković and Radovanović (2004) that the least-recently-used (LRU) caching policy, in the presence of semi-Markov-modulated requests that have a generalized Zipf's law popularity distribution, is asymptotically insensitive to the correlation in the request process. However, since the previous result is asymptotic, it remains unclear how small the cache size can become while still retaining the preceding insensitivity property. In this paper, assuming that requests are generated by a nearly completely decomposable Markov-modulated process, we characterize the critical cache size below which the dependency of requests dominates the cache performance. This critical cache size is small relative to the dynamics of the modulating process, and in fact is sublinear with respect to the sojourn times of the modulated chain that determines the dependence structure.
