Lumpability in LRU stack with Markovian page references

The dynamic behavior of programs is studied through LRU (Least-Recently-Used) stack distribution assuming Markovian page references. We propose a new analysis method to calculate steady state probabilities for the LRU stack. We first prove that a Markov chain which describes the behavior of an LRU stack of full-length is lumpable to a Markov chain having a smaller transition matrix than that in the original Markov chain. Based on this lumpability, we propose an algorithm to calculate the invariant vector for the transition matrix of the Markov chain for a full-length LRU stack. We show numerical examples calculated with the proposed algorithm, and evaluate computational efficiency of the proposed method.