Cached Files Updating Revisited: The Distribution of Popularity-Weighted Average File Age

Low-latency information transmission is required in 5G communication networks, where the cache is commonly used to speed up the data access. Meanwhile, a large number of IoT applications need real-time information exchange, resulting in the information freshness more important. In this article, under the discrete-time model, we consider the average age of all files in a cached-files-updating system. To keep the cached files fresh, in each time slot the server refreshes files with certain probabilities. The age of one file or its age of information (AoI) is defined as the time the file stays in cache since it was sent to cache last time. Assume that each file in cache has its own request popularity, we determine the probability distribution of the popularity-weighted average file age. For the random age of single file, both the mean and its probability distribution can be obtained by establishing a simple Markov chain. Following the same line of thinking, we show that an N -dimensional stochastic process can be constituted to characterize the changes of N file ages simultaneously. By solving the steady state of the resulting process, we obtain the explicit expression of stationary probability for an arbitrary state vector. Then, the distribution of the popularity-weighted average file age can be found by mergering a proper set of stationary probabilities, which gives the complete description of the average age. We also provide the numerical results of the distribution for some simple cases. The age-distribution can be utilized to compute certain stochastic performance indices, such as the AoI violation probability of some status updating systems.