The Study of Dynamic Caching via State Transition Field -- the Case of Time-Varying Popularity

In the second part of this two-part paper, we extend the study of dynamic caching via state transition field (STF) to the case of time-varying content popularity. The objective of this part is to investigate the impact of time-varying content popularity on the STF and how such impact accumulates to affect the performance of a replacement scheme. Unlike the case in the first part, the STF is no longer static over time, and we introduce instantaneous STF to model it. Moreover, we demonstrate that many metrics, such as instantaneous state caching probability and average cache hit probability over an arbitrary sequence of requests, can be found using the instantaneous STF. As a steady state may not exist under time-varying content popularity, we characterize the performance of replacement schemes based on how the instantaneous STF of a replacement scheme after a content request impacts on its cache hit probability at the next request. From this characterization, insights regarding the relations between the pattern of change in the content popularity, the knowledge of content popularity exploited by the replacement schemes, and the effectiveness of these schemes under time-varying popularity are revealed. In the simulations, different patterns of time-varying popularity, including the shot noise model, are experimented. The effectiveness of example replacement schemes under time-varying popularity is demonstrated, and the numerical results support the observations from the analytic results.