Tightness for a family of recursion equations

In this paper we study the tightness of solutions for a family of recursion equations. These equations arise naturally in the study of random walks on tree-like structures. Examples include the maximal displacement of a branching random walk in one dimension and the cover time of a symmetric simple random walk on regular binary trees. Recursion equations associated with the distribution functions of these quantities have been used to establish weak laws of large numbers. Here, we use these recursion equations to establish the tightness of the corresponding sequences of distribution functions after appropriate centering. We phrase our results in a fairly general context, which we hope will facilitate their application in other settings.Comment: Published in at http://dx.doi.org/10.1214/08-AOP414 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

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

Shortest spanning trees and a counterexample for random walks in random environments

We construct forests that span $\mathbb{Z}^d$, $d\geq2$, that are stationary and directed, and whose trees are infinite, but for which the subtrees attached to each vertex are as short as possible. For $d\geq3$, two independent copies of such forests, pointing in opposite directions, can be pruned so as to become disjoint. From this, we construct in $d\geq3$ a stationary, polynomially mixing and uniformly elliptic environment of nearest-neighbor transition probabilities on $\mathbb{Z}^d$, for which the corresponding random walk disobeys a certain zero--one law for directional transience.Comment: Published at http://dx.doi.org/10.1214/009117905000000783 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Proportional switching in FIFO networks

We consider a family of discrete time multihop switched queueing networks where each packet movesalong a xed route. In this setting, BackPressure is the canonical choice of scheduling policy; this policy hasthe virtues of possessing a maximal stability region and not requiring explicit knowledge of tra c arrival rates.BackPressure has certain structural weaknesses because implementation requires information about each route,and queueing delays can grow super-linearly with route length. For large networks, where packets over manyroutes are processed by a queue, or where packets over a route are processed by many queues, these limitationscan be prohibitive.In this article, we introduce a scheduling policy for FIFO networks, the Proportional Scheduler, which isbased on the proportional fairness criterion. We show that, like BackPressure, the Proportional Scheduler hasa maximal stability region and does not require explicit knowledge of tra c arrival rates. The ProportionalScheduler has the advantage that information about the network's route structure is not required for scheduling,which substantially improves the policy's performance for large networks. For instance, packets can be routedwith only next-hop information and new nodes can be added to the network with only knowledge of thescheduling constraintsThe research of the rst author was partially supported by NSF grants DMS-1105668 and DMS-1203201. The research of the second author was partially supported by the Spanish Ministry of Economy and Competitiveness Grants MTM2013-42104-P via FEDER funds; he thanks the ICMAT (Madrid, Spain) Research Institute that kindly hosted him while developing this project

Stability and Instability of the MaxWeight Policy

Consider a switched queueing network with general routing among its queues. The MaxWeight policy assigns available service by maximizing the objective function $\sum_j Q_j \sigma_j$ among the different feasible service options, where $Q_j$ denotes queue size and $\sigma_j$ denotes the amount of service to be executed at queue $j$. MaxWeight is a greedy policy that does not depend on knowledge of arrival rates and is straightforward to implement. These properties, as well as its simple formulation, suggest MaxWeight as a serious candidate for implementation in the setting of switched queueing networks; MaxWeight has been extensively studied in the context of communication networks. However, a fluid model variant of MaxWeight was shown by Andrews--Zhang (2003) not to be maximally stable. Here, we prove that MaxWeight itself is not in general maximally stable. We also prove MaxWeight is maximally stable in a much more restrictive setting, and that a weighted version of MaxWeight, where the weighting depends on the traffic intensity, is always stable.Comment: Now includes addendum on longest-queue-first-serve

Decay of tails at equilibrium for FIFO join the shortest queue networks

In join the shortest queue networks, incoming jobs are assigned to the shortest queue from among a randomly chosen subset of $D$ queues, in a system of $N$ queues; after completion of service at its queue, a job leaves the network. We also assume that jobs arrive into the system according to a rate-$\alpha N$ Poisson process, $\alpha<1$, with rate-1 service at each queue. When the service at queues is exponentially distributed, it was shown in Vvedenskaya et al. [Probl. Inf. Transm. 32 (1996) 15-29] that the tail of the equilibrium queue size decays doubly exponentially in the limit as $N\rightarrow\infty$. This is a substantial improvement over the case D=1, where the queue size decays exponentially. The reasoning in [Probl. Inf. Transm. 32 (1996) 15-29] does not easily generalize to jobs with nonexponential service time distributions. A modularized program for treating general service time distributions was introduced in Bramson et al. [In Proc. ACM SIGMETRICS (2010) 275-286]. The program relies on an ansatz that asserts, in equilibrium, any fixed number of queues become independent of one another as $N\rightarrow\infty$. This ansatz was demonstrated in several settings in Bramson et al. [Queueing Syst. 71 (2012) 247-292], including for networks where the service discipline is FIFO and the service time distribution has a decreasing hazard rate. In this article, we investigate the limiting behavior, as $N\rightarrow \infty$, of the equilibrium at a queue when the service discipline is FIFO and the service time distribution has a power law with a given exponent $-\beta$, for $\beta>1$. We show under the above ansatz that, as $N\rightarrow\infty$, the tail of the equilibrium queue size exhibits a wide range of behavior depending on the relationship between $\beta$ and $D$. In particular, if $\beta>D/(D-1)$, the tail is doubly exponential and, if $\beta<D/(D-1)$, the tail has a power law. When $\beta=D/(D-1)$, the tail is exponentially distributed.Comment: Published in at http://dx.doi.org/10.1214/12-AAP888 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org