In this work, we propose general, practical and accurate models to analyze the performance of multi-cache systems, in which a cache forwards its miss stream (i.e., requests which have not found the target item) to other caches. We extend a miss stream modeling technique originally known as Melazzi's approximation, which provides a simple but accurate approximate analysis for caches with cascade configurations. We consider several practical replication strategies, which have been commonly adopted in the context of ICN, taking into account the effects of temporal locality. Also, we capture the existing state correlations between neighboring caches by exploiting the cache eviction time. Our proposed models to handle traffic patterns allow us to go beyond the standard Poisson approximation under Independent Reference Model. Our results, validated against simulations, provide interesting insights into the performance of multi-cache systems with different replication strategies.