on Mathematical and Computing Sciences

We consider the least-recently-used (LRU) cache replacement rule with a Zipf-type page request distribution and investigate the asymptotic property of the fault probability according to the increase of cache size. We first derive the asymptotics of the fault probability for the i.i.d. request model and then extend it to a general dependent request model, where our result shows that, under some weak assumptions, the fault probability is asymptotically invariant from the dependence in the page request process. An existing result has derived a similar result by applying a Poisson embedding technique, that is, it has given a continuous-time proof with some assumptions based on a continuous-time modeling. However, the Poisson embedding is just a technique for the proof and the problem is essentially on discrete-time basis, and thus, it is preferable to make assumptions, if any, directly in the discrete-time setting. We here consider a general dependent request process model and give a direct discrete-time proof under different assumptions. A key to the proof is that the numbers of requests for respective items construct conditionally negatively associated random variables.