Optimal local storage policy based on stochastic intensities and its large scale behavior

TL;DR

This work develops a rigorous point-process framework for optimal local memory management, showing that the optimal causal policy depends on the stochastic intensity of requests. In the large-scale limit with catalog size $N oty$ and memory fraction $C/N o c$, the optimal policy converges to a deterministic threshold on the observed hazard rate $ heta^* = Q_ ext{∞}(1-c)$, with the limiting distribution $G_ ext{∞}(x)=\int_0^ty G(x/\lambda)L(d\lambda)$. The asymptotic miss rate is given by $\lim_{N\to\infty} m_{\mathcal{C}^*}^{(N)}/N = \int_0^ty \lambda G_0(\theta^*/\lambda)L(d\lambda) / \int_0^ty \lambda L(d\lambda)$ under uniform integrability, and the results connect threshold policies to timer-based TTL policies under monotone hazard rates, including timer-based prefetching for increasing hazards. Parametric examples with Pareto inter-arrivals and Zipf popularities, along with Erlang traffic, validate the theory and show substantial gains over conventional caching policies like LRU in appropriate regimes. The framework provides universal performance bounds and highlights practical avenues for learning traffic characteristics to approach the theoretical optimum.

Abstract

In this paper, we analyze the optimal management of local memory systems, using the tools of stationary point processes. We provide a rigorous setting of the problem, building upon recent work, and characterize the optimal causal policy that maximizes the hit probability. We specialize the result for the case of renewal request processes and derive a suitable large scale limit as the catalog size N grows to infinity, when a fixed fraction c of items can be stored. We prove that in the limiting regime, the optimal policy amounts to comparing the stochastic intensity (observed hazard rate) of the process with a fixed threshold, defined by a quantile of an appropriate limit distribution, and derive asymptotic performance metrics, as well as sharp estimates for the pre-limit case. Moreover, we establish a connection with optimal timer based policies for the case of monotonic hazard rates. We also present detailed validation examples of our results, including some close form expressions for the miss probability that are compared to simulations. We also use these examples to exhibit the significant superiority of the optimal policy for the case of regular traffic patterns.

Figures (11)

• Figure 1: Inter-arrival and age distribution of a stationary point process.
• Figure 2: Stochastic intensity of a renewal Pareto process, showing decreasing hazard rates.
• Figure 3: Distribution shapes for the Pareto renewal example with $\alpha=2, \gamma=1$, and therefore $\lambda=1$.
• Figure 4: Threshold policies in terms of timers. Right: timer-based caching for decreasing hazard rates. Left: timer-based pre-fetching for increasing hazard rates.
• Figure 5: Limit distribution $G_\infty$ and threshold behavior for the Pareto example with Zipf popularities, $\alpha=2$, $\beta=1$.

Theorems & Definitions (23)

• Definition 1
• Definition 2
• Definition 3
• Theorem
• Example 1: Renewal Pareto process
• Lemma 1
• Definition 4
• Lemma 2
• Theorem 1: Optimal causal memory management policy
• Example 2: Poisson arrivals