Some results on the classification of N objects in M classes with at least c objects in each class

Abstract

Summary form only given. An approach towards an information theoretic based analysis and design of cache memories is presented. Computer systems usually have a slow main memory and a faster cache memory, usually much smaller than the main memory because of its cost. A somewhat similar situation is present also in Internet servers. A miss happens whenever an item is not found in the cache memory. If so, the item is fetched from the main memory and placed in the cache. A hit is obtained whenever the item is found in the cache. Usually the cost of a miss is several times that of a hit. The goal is to find strategies for the many to one mapping of addresses of the main memory to the cache memory, as well as the replacement strategies. Usual replacement strategies are least recently used, LRU, random replacement, RR, etc. The main goal is to obtain strategies that will optimize the running time of the program under execution. Since almost all programs use branch instructions and loops, some of the information theoretic approaches previously introduced consider the prediction of the result of a branch based on its past behavior. Here a different approach is considered. In particular the opposite case is analysed, i.e. a linear loop that is executed indefinitely. In particular a combined random replacement and least recently used strategy is analyzed. It is shown that this model is equivalent to the one of classifying N objects in M classes with at least c objects in each class, and that this problem gives a generalization of the Ehrenfests' urn model used in statistical thermodynamics in connection with the Boltzmann H-theorem.