In this paper, we consider a generic model of computational grids, seen as several clusters of homogeneous processors. In such systems, a key issue when designing efficient job allocation policies is to balance the workload over the different resources.We present a Markovian model for performance evaluation of such a policy, namely work stealing (idle processors steal work from others) in large-scale heterogeneous systems. Using mean field theory, we show that when the size of the system grows, it converges to a system of deterministic ordinary differential equations that allows one to compute the expectation of performance functions (such as average response times) as well as the distributions of these functions.We first study the case where all resources are homogeneous, showing in particular that work stealing is very efficient, even when the latency of steals is large. We also consider the case where distance plays a role: the system is made of several clusters, and stealing within one cluster is faster than stealing between clusters. We compare different work stealing policies, based on stealing probabilities and we show that the main factor for deciding where to steal from is the load rather than the stealing latency.

In this paper we study the performance of a family of cache replacement algorithms. The cache is decomposed into lists. Items enter the cache via the first list. An item enters the cache via the first list and jumps to the next list whenever a hit on it occurs. The classical policies FIFO, RANDOM, CLIMB and its hybrids are obtained as special cases. We present explicit expressions for the cache content distribution and miss probability under the IRM model. We develop an algorithm with a time complexity that is polynomial in the cache size and linear in the number of items to compute the exact miss probability. We introduce lower and upper bounds on the latter that can be computed in a time that is linear in the cache size times the number of items. We further introduce a mean field model to approximate the transient behavior of the miss probability and prove that this model becomes exact as the cache size and number of items tends to infinity. We show that the set of ODEs associated to the mean field model has a unique fixed point that can be used to approximate the miss probability in case the exact computation becomes too time consuming. Using this approximation, we provide guidelines on how to select a replacement algorithm within the family considered such that a good trade-off is achieved between the cache reactivity and its steady-state hit probability. We simulate these cache replacement algorithms on traces of real data and show that they can outperform LRU. Finally, we also disprove the well-known conjecture that the CLIMB algorithm is the optimal finite-memory replacement algorithm under the IRM model.

Mean field approximation is a powerful tool to study the performance of large stochastic systems that is known to be exact as the system's size N goes to infinity. Recently, it has been shown that, when one wants to compute expected performance metric in steady-state, mean field approximation can be made more accurate by adding a term in 1/N to the original approximation. This is called the refined mean field approximation in [7]. In this paper, we show how to obtain the same result for the transient regime and we provide a further refinement by expanding the term in 1/N2 (both for transient and steady-state regime). Our derivations are inspired by moment-closure approximation. We provide a number of examples that show this new approximation is usable in practice for systems with up to a few tens of dimensions.