Section: New Results

Asymptotic Models

• Mean field approximation is a popular means to approximate large and complex stochastic models that can be represented as $N$ interacting objects. The idea of mean field approximation to study the limit of this system as $N$ goes to infinity.

  In [18], we study how accurate is mean field approximation as $N$ goes to infinity. We show that under very general conditions the expectation of any performance indicator converges at rate $O(1/N)$ to its mean field approximation. In [7] we continue this analysis and establish a result that expresses the constant associated with this $1/N$ term. This allows us to propose what we call a refined mean field approximation. By considering a variety of applications, we illustrate that the proposed refined mean field approximation is significantly more accurate that the classic mean field approximation for small and moderate values of $N$: the relative errors of this refined approximation is often below 1% for systems with $N=10$.

• Computer system and network performance can be significantly improved by caching frequently used information. When the cache size is limited, the cache replacement algorithm has an important impact on the effectiveness of caching. In [8] we introduce time-to-live (TTL) approximations to determine the cache hit probability of two classes of cache replacement algorithms: h-LRU and LRU(m). Using a mean field approach, we provide both numerical and theoretical support for the claim that the proposed TTL approximations are asymptotically exact. We use this approximation and trace-based simulation to compare the performance of h-LRU and LRU(m). First, we show that they perform alike, while the latter requires less work when a hit/miss occurs. Second, we show that as opposed to LRU, h-LRU and LRU(m) are sensitive to the correlation between consecutive inter-request times. Last, we study cache partitioning. In all tested cases, the hit probability improved by partitioning the cache into different parts—each being dedicated to a particular content provider. However, the gain is limited and the optimal partition sizes are very sensitive to the problem's parameters.

• Mean field approximation is often use to characterize the transient or steady state performance of a stochastic system. In [6], we use this approach to compute absorbing times. We use mean field approximation to provide an asymptotic expansion of this absorbing time that uses the spectral decomposition of the kernel of the original chains. Our results rely on extreme values theory. We show the applicability of this approach with three different problems: the coupon collector, the erasure channel lifetime and the coupling times of random walks in high dimensional spaces.