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From Local to Global Stability in Stochastic Processing Networks through Quadratic Lyapunov Functions”, preprint
, 2012
"... Abstract We construct a generic, simple, and efficient scheduling policy for stochastic processing networks, and provide a general framework to establish its stability. Our policy is randomized and prioritized: with high probability it prioritizes jobs which have been least routed through the netwo ..."
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Abstract We construct a generic, simple, and efficient scheduling policy for stochastic processing networks, and provide a general framework to establish its stability. Our policy is randomized and prioritized: with high probability it prioritizes jobs which have been least routed through the network. We show that the network is globally stable under this policy if there exists an appropriate quadratic 'local' Lyapunov function that provides a negative drift with respect to nominal loads at servers. Applying this generic framework, we obtain stability results for our policy in many important examples of stochastic processing networks: open multiclass queueing networks, parallel server networks, networks of inputqueued switches, and a variety of wireless network models with interference constraints. Our main novelty is the construction of an appropriate 'global' Lyapunov function from quadratic 'local' Lyapunov functions, which we believe to be of broader interest.
HIGHER ORDER MARKOV RANDOM FIELDS FOR INDEPENDENT SETS
"... It is wellknown that if one samples from the independent sets of a large regular graph of large girth using a pairwise Markov random eld (i.e. hardcore model) in the uniqueness regime, each excluded node has a binomially distributed number of included neighbors in the limit. In this paper, motivate ..."
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It is wellknown that if one samples from the independent sets of a large regular graph of large girth using a pairwise Markov random eld (i.e. hardcore model) in the uniqueness regime, each excluded node has a binomially distributed number of included neighbors in the limit. In this paper, motivated by applications to the design of communication networks, we pose the question of how to sample from the independent sets of such a graph so that the number of included neighbors of each excluded node has a dierent distribution of our choosing. We observe that higher order Markov random elds are wellsuited to this task, and investigate the properties of these models. For the family of socalled reverse ultra logconcave distributions, which includes the truncated Poisson and geometric, we give necessary and sucient conditions for the natural higher order Markov random eld which induces the desired distribution to be in the uniqueness regime, in terms of the set of solutions to a certain system of equations. We also show that these Markov random elds undergo a phase transition, and give explicit bounds on the associated critical activity, which we prove to exhibit a certain robustness. For distributions which are small perturbations around the binomial distribution realized by the hardcore model at critical activity, we give a description of the corresponding uniqueness regime in terms of a simple polyhedral cone. Our analysis reveals an interesting nonmonotonicity with regards to biasing towards excluded nodes with no included neighbors. We conclude with a broader discussion of the potential use of higher order Markov random elds for analyzing independent sets in graphs. 1. Introduction. Recently
QueueBased . . . FLUID LIMITS AND STABILITY ISSUES
, 2013
"... We use fluid limits to explore the (in)stability properties of wireless networks with queuebased randomaccess algorithms. Queuebased randomaccess schemes are simple and inherently distributed in nature, yet provide the capability to match the optimal throughput performance of centralized schedu ..."
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We use fluid limits to explore the (in)stability properties of wireless networks with queuebased randomaccess algorithms. Queuebased randomaccess schemes are simple and inherently distributed in nature, yet provide the capability to match the optimal throughput performance of centralized scheduling mechanisms in a wide range of scenarios. Unfortunately, the type of activation rules for which throughput optimality has been established, may result in excessive queue lengths and delays. The use of more aggressive/persistent access schemes can improve the delay performance, but does not offer any universal maximumstability guarantees. In order to gain qualitative insight and investigate the (in)stability properties of more aggressive/persistent activation rules, we examine fluid limits where the dynamics are scaled in space and time. In some situations, the fluid limits have smooth deterministic features and maximum stability is maintained, while in other scenarios they exhibit random oscillatory characteristics, giving rise to major technical challenges. In the latter regime, more aggressive access schemes continue to provide maximum stability in some networks, but may cause instability in others. In order to prove that, we focus on a particular network example and conduct a detailed analysis of the fluid limit process for the associated Markov chain. Specifically, we develop a novel approach based on stopping time sequences to deal with the switching probabilities governing the sample paths of the fluid limit process. Simulation experiments are conducted to illustrate and validate the analytical results.
CSMA using the Bethe Approximation: Scheduling and Utility Maximization
, 2013
"... CSMA (Carrier Sense Multiple Access), which resolves contentions over wireless networks in a fully distributed fashion, has recently gained a lot of attentions since it has been proved that appropriate control of CSMA parameters guarantees optimality in terms of stability (i.e., scheduling) and syst ..."
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CSMA (Carrier Sense Multiple Access), which resolves contentions over wireless networks in a fully distributed fashion, has recently gained a lot of attentions since it has been proved that appropriate control of CSMA parameters guarantees optimality in terms of stability (i.e., scheduling) and systemwide utility (i.e., scheduling and congestion control). Most CSMAbased algorithms rely on the popular MCMC (Markov Chain Monte Carlo) technique, which enables one to find optimal CSMA parameters through iterative loops of ‘simulationandupdate’. However, such a simulationbased approach often becomes a major cause of exponentially slow convergence, being poorly adaptive to flow/topology changes. In this paper, we develop distributed iterative algorithms which produce approximate solutions with convergence in polynomial time for both stability and utility maximization problems. In particular, for the stability problem, the proposed distributed algorithm requires, somewhat surprisingly, only one iteration among links. Our approach is motivated by the Bethe approximation (introduced by Yedidia, Freeman and Weiss in 2005) allowing us to express approximate solutions via a certain nonlinear system with polynomial size. Our polynomial convergence guarantee comes from directly solving the nonlinear system in a distributed manner, rather than multiple simulationandupdate loops in existing algorithms. We provide numerical results to show that the algorithm produces highly accurate solutions and converges much faster than the prior ones.
Scheduling in CyberPhysical Systems
, 2012
"... Cyberphysical systems (CPS) refer to a promising class of systems featuring intimate coupling between the ‘cyber’ intelligence and the ‘physical ’ world. Enabled by the ubiquitous availability of computation and communication capabilities, such systems are widely envisioned to redefine the way that ..."
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Cyberphysical systems (CPS) refer to a promising class of systems featuring intimate coupling between the ‘cyber’ intelligence and the ‘physical ’ world. Enabled by the ubiquitous availability of computation and communication capabilities, such systems are widely envisioned to redefine the way that people interact with the physical world, similar to the revolutionary role of internet in transforming how people interact with each other. As the whole society becomes increasingly dependent on such systems, it is crucial to develop a theory to understand and optimize the CPS in a systematic manner. This thesis contributes to the foundations of CPS by identifying and addressing a general class of schedulingtype applications for a vital class of CPS, the physical networks (PhyNets). Different from the abstract CPS, a PhyNet has a graphtype physical part, which represents the local interactions among users in the system, as specified by certain wellknown physical laws. Thus, it is very promising to develop efficient distributed algorithms in PhyNets with proper communication infrastructure and protocols, due to the physical graph structure. The ‘scheduling’ refers to the applications where joint actions of all users are coordinated, in order to allocate system resources
Lingering Issues in Distributed Scheduling
"... Recent advances have resulted in queuebased algorithms for medium access control which operate in a distributed fashion, and yet achieve the optimal throughput performance of centralized scheduling algorithms. However, fundamental performance bounds reveal that the “cautious ” activation rules inv ..."
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Recent advances have resulted in queuebased algorithms for medium access control which operate in a distributed fashion, and yet achieve the optimal throughput performance of centralized scheduling algorithms. However, fundamental performance bounds reveal that the “cautious ” activation rules involved in establishing throughput optimality tend to produce extremely large delays, typically growing exponentially in 1/(1−ρ), with ρ the load of the system, in contrast to the usual linear growth. Motivated by that issue, we explore to what extent more “aggressive ” schemes can improve the delay performance. Our main finding is that aggressive activation rules induce a lingering effect, where individual nodes retain possession of a shared resource for excessive lengths of time even while a majority of other nodes idle. Using central limit theorem type arguments, we prove that the idleness induced by the lingering effect may cause the delays to grow with 1/(1−ρ) at a quadratic rate. To the best of our knowledge, these are the first mathematical results illuminating the lingering effect and quantifying the performance impact. In addition extensive simulation experiments are conducted to illustrate and validate the various analytical results.
2 From Local to Global Stability in Stochastic Processing Networks through Quadratic Lyapunov Functions
, 2014
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