Section: New Results

Asymptotic Models

Analyzing a set of $n$ stochastic entities interacting with each others can be particularly difficult but the mean field approximation is a very effective technique to characterize the probability distribution of such systems when the number of entities $n$ grows very large. The limit system is generally deterministic and characterized by a differential equation that is more amenable to analysis and optimization. Such approximation however typically requires that the dynamics of the entities depend only on their state (the state space of each object does not scale with $n$ the number of objects) but neither on their identity nor on their spatial location.

• In [28] , we analyze a family of list-based cache replacement algorithms. We present explicit expressions for the cache content distribution and miss probability under some assumptions and 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 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

• For distributed systems where /locality/ is essential in the dynamics the mean-field approach requires to resort to discretization of space into a finite number of cells to fit in the classical framework. Such approach not only scales badly but also requires that spatial interactions are weak. One of the tool to tackle this difficult problem comes from statistical physics and is popular in biology: pair approximation. In [26] , we successfully apply this approach to the "Power of Two Choice" load balancing paradigm: each incoming task is allocated to the least loaded of two servers picked at random among a collection of $n$ servers. We study the power of two-choice in a setting where the two servers are not picked independently at random but are connected by an edge in an underlying graph. Our problem is motivated by systems in which choices are geometrically constrained (e.g., a bike-sharing system). We study a dynamic setting in which jobs leave the system after being served by a server to which is was allocated. Our focus is when each server has few neighbors (typically 2 to 4) for which an mean-field approximation is not accurate. We build the pair-approximation equations and show that they describe accurately the steady-state of the system. Our results show that, even in a graph of degree 2, choosing between two neighboring improve dramatically the performance compared to a random allocation.

• In [8] , we consider a queueing system composed of a dispatcher that routes deterministically jobs to a set of non-observable queues working in parallel. In this setting, the fundamental problem is which policy should the dispatcher implement to minimize the stationary mean waiting time of the incoming jobs. We present a structural property that holds in the classic scaling of the system where the network demand (arrival rate of jobs) grows proportionally with the number of queues. Assuming that each queue of type $r$ is replicated $k$ times, we consider a set of policies that are periodic with period $k{\sum }_r{p}_r$ and such that exactly $p_r$ jobs are sent in a period to each queue of type $r$. When $k\to \infty$, our main result shows that all the policies in this set are equivalent, in the sense that they yield the same mean stationary waiting time, and optimal, in the sense that no other policy having the same aggregate arrival rate to all queues of a given type can do better in minimizing the stationary mean waiting time. Furthermore, the limiting mean waiting time achieved by our policies is a convex function of the arrival rate in each queue, which facilitates the development of a further optimization aimed at solving the fundamental problem above for large systems.