Spatial homogenization in a stochastic network with mobility

Abstract

A stochastic model for a mobile network is studied. Users enter the network, and then perform independent Markovian routes between nodes where they receive service according to the Processor-Sharing policy. Once their service requirement is satisfied, they leave the system. The stability region is identified via a fluid limit approach, and strongly relies on a "spatial homogenization" property: at the fluid level, customers are instantaneously distributed across the network according to the stationary distribution of their Markovian dynamics and stay distributed as such as long as the network is not empty. In the unstable regime, spatial homogenization almost surely holds asymptotically as time goes to infinity (on the normal scale), telling how the system fills up. One of the technical achievements of the paper is the construction of a family of martingales associated to the multidimensional process of interest, which makes it possible to get crucial estimates for certain exit times.Comment: Published in at http://dx.doi.org/10.1214/09-AAP613 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Load balancing via random local search in closed and open systems

“…In the following, we prove Theorem 4.1 without the use of fluid limits, and for any random walk. Our proof is much more direct than that in [21], and hence is amenable to deal with more general cases and possible extensions. For the proof of Theorem 4.2, we use a rather classic method, i.e., we exhibit a simple Lyapunov function.…”

“…Handling this interaction turns out to be extremely difficult. Recently however, in [21], the authors were able to formally derive the system fluid limits, and analyze its stability under very specific assumptions on the client random walk (its transition matrix Q has to be diagonalizable). Their proof is quite intricate.…”

“…The following coupling initially proposed and formally justified in [21] is key to relate the open and closed systems. For n ∈ N m and ℓ, ρ ∈ R m + , denote by N n ℓ,ρ the process under RLO strategy starting in the initial state n, with arrival rate ℓ i at server i with capacity ρ i .…”

On the Transient Behavior of Ehrenfest and Engset Processes

“…Nevertheless they play a fundamental role, most of the asymptotic results obtained in this paper are based on these martingales. A more general version in a multidimensional context has been introduced by Simatos and Tibi [20].…”

“…Quite surprisingly, up to now, martingales did not play a major role in the previous studies of the Ehrenfest process. One can mention Simatos and Tibi [20] where a martingale approach is used to estimate certain exit times for multi-dimensional Ehrenfest processes. It is one of the results of this paper to show that a simple and important family of martingales allows a quite detailed investigation of this process, and also of its variants like the Engset process.…”