Validity of heavy traffic steady-state approximations in generalized Jackson Networks

We consider a single class open queueing network, also known as a generalized Jackson network (GJN). A classical result in heavy-traffic theory asserts that the sequence of normalized queue length processes of the GJN converge weakly to a reflected Brownian motion (RBM) in the orthant, as the traffic intensity approaches unity. However, barring simple instances, it is still not known whether the stationary distribution of RBM provides a valid approximation for the steady-state of the original network. In this paper we resolve this open problem by proving that the re-scaled stationary distribution of the GJN converges to the stationary distribution of the RBM, thus validating a so-called ``interchange-of-limits'' for this class of networks. Our method of proof involves a combination of Lyapunov function techniques, strong approximations and tail probability bounds that yield tightness of the sequence of stationary distributions of the GJN.Comment: Published at http://dx.doi.org/10.1214/105051605000000638 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Optimal Paths in Large Deviations of Symmetric Reflected Brownian Motion in the Octant

We study the variational problem that arises from consideration of large deviations for semimartingale reflected Brownian motion (SRBM) in the positive octant. Due to the difficulty of the general problem, we consider the case in which the SRBM has rotationally symmetric parameters. In this case, we are able to obtain conditions under which the optimal solutions to the variational problem are paths that are gradual (moving through faces of strictly increasing dimension) or that spiral around the boundary of the octant. Furthermore, these results allow us to provide an example for which it can be verified that a spiral path is optimal. For rotationally symmetric SRBM's, our results facilitate the simplification of computational methods for determining optimal solutions to variational problems and give insight into large deviations behavior of these processes

Positive recurrence of reflecting Brownian motion in three dimensions

Consider a semimartingale reflecting Brownian motion (SRBM) $Z$ whose state space is the $d$-dimensional nonnegative orthant. The data for such a process are a drift vector $\theta$, a nonsingular $d\times d$ covariance matrix $\Sigma$, and a $d\times d$ reflection matrix $R$ that specifies the boundary behavior of $Z$. We say that $Z$ is positive recurrent, or stable, if the expected time to hit an arbitrary open neighborhood of the origin is finite for every starting state. In dimension $d=2$, necessary and sufficient conditions for stability are known, but fundamentally new phenomena arise in higher dimensions. Building on prior work by El Kharroubi, Ben Tahar and Yaacoubi [Stochastics Stochastics Rep. 68 (2000) 229--253, Math. Methods Oper. Res. 56 (2002) 243--258], we provide necessary and sufficient conditions for stability of SRBMs in three dimensions; to verify or refute these conditions is a simple computational task. As a byproduct, we find that the fluid-based criterion of Dupuis and Williams [Ann. Probab. 22 (1994) 680--702] is not only sufficient but also necessary for stability of SRBMs in three dimensions. That is, an SRBM in three dimensions is positive recurrent if and only if every path of the associated fluid model is attracted to the origin. The problem of recurrence classification for SRBMs in four and higher dimensions remains open.Comment: Published in at http://dx.doi.org/10.1214/09-AAP631 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

A Numerical Scheme for Invariant Distributions of Constrained Diffusions

Reflected diffusions in polyhedral domains are commonly used as approximate models for stochastic processing networks in heavy traffic. Stationary distributions of such models give useful information on the steady state performance of the corresponding stochastic networks and thus it is important to develop reliable and efficient algorithms for numerical computation of such distributions. In this work we propose and analyze a Monte-Carlo scheme based on an Euler type discretization of the reflected stochastic differential equation using a single sequence of time discretization steps which decrease to zero as time approaches infinity. Appropriately weighted empirical measures constructed from the simulated discretized reflected diffusion are proposed as approximations for the invariant probability measure of the true diffusion model. Almost sure consistency results are established that in particular show that weighted averages of polynomially growing continuous functionals evaluated on the discretized simulated system converge a.s. to the corresponding integrals with respect to the invariant measure. Proofs rely on constructing suitable Lyapunov functions for tightness and uniform integrability and characterizing almost sure limit points through an extension of Echeverria's criteria for reflected diffusions. Regularity properties of the underlying Skorohod problems play a key role in the proofs. Rates of convergence for suitable families of test functions are also obtained. A key advantage of Monte-Carlo methods is the ease of implementation, particularly for high dimensional problems. A numerical example of a eight dimensional Skorohod problem is presented to illustrate the applicability of the approach

Analysis of Large Unreliable Stochastic Networks

In this paper a stochastic model of a large distributed system where users' files are duplicated on unreliable data servers is investigated. Due to a server breakdown, a copy of a file can be lost, it can be retrieved if another copy of the same file is stored on other servers. In the case where no other copy of a given file is present in the network, it is definitively lost. In order to have multiple copies of a given file, it is assumed that each server can devote a fraction of its processing capacity to duplicate files on other servers to enhance the durability of the system. A simplified stochastic model of this network is analyzed. It is assumed that a copy of a given file is lost at some fixed rate and that the initial state is optimal: each file has the maximum number $d$ of copies located on the servers of the network. Due to random losses, the state of the network is transient and all files will be eventually lost. As a consequence, a transient $d$-dimensional Markov process $(X(t))$ with a unique absorbing state describes the evolution this network. By taking a scaling parameter $N$ related to the number of nodes of the network. a scaling analysis of this process is developed. The asymptotic behavior of $(X(t))$ is analyzed on time scales of the type $t\mapsto N^p t$ for $0\leq p\leq d{-}1$. The paper derives asymptotic results on the decay of the network: Under a stability assumption, the main results state that the critical time scale for the decay of the system is given by $t\mapsto N^{d-1}t$. When the stability condition is not satisfied, it is shown that the state of the network converges to an interesting local equilibrium which is investigated. As a consequence it sheds some light on the role of the key parameters $\lambda$, the duplication rate and $d$, the maximal number of copies, in the design of these systems

Sensitivity analysis for diffusion processes constrained to an orthant

This paper studies diffusion processes constrained to the positive orthant under infinitesimal changes in the drift. Our first main result states that any constrained function and its (left) drift-derivative is the unique solution to an augmented Skorohod problem. Our second main result uses this characterization to establish a basic adjoint relationship for the stationary distribution of the constrained diffusion process jointly with its left-derivative process.Comment: Published in at http://dx.doi.org/10.1214/13-AAP967 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org