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We consider a heterogeneous distributed service system, consisting of $n$ servers with unknown and possibly different processing rates. Jobs with unit mean and independent processing times arrive as a renewal process of rate $\lambda n$, with $0, Comment: arXiv admin note: text overlap with arXiv:1807.02882

We consider the following distributed service model: jobs with unit mean, general distribution, and independent processing times arrive as a renewal process of rate $\lambda n$, with $0, Comment: 44 pages

We consider a multihop switched network operating under a max-weight scheduling policy and show that the distance between the queue length process and a fluid solution remains bounded by a constant multiple of the deviation of the cumulative arrival process from its average. We then exploit this result to prove matching upper and lower bounds for the time scale over which additive state space collapse (SSC) takes place. This implies, as two special cases, an additive SSC result in diffusion scaling under non-Markovian arrivals and, for the case of independent and identically distributed arrivals, an additive SSC result over an exponential time scale.

We consider an SIS-type epidemic process that evolves on a known graph. We assume that a fixed curing budget can be allocated at each instant to the nodes of the graph, towards the objective of minimizing the expected extinction time of the epidemic. We provide a lower bound on the optimal expected extinction time as a function of the available budget, the epidemic parameters, the maximum degree, and the CutWidth of the graph. For graphs with large CutWidth (close to the largest possible), and under a budget which is sublinear in the number of nodes, our lower bound scales exponentially with the size of the graph., United States. Army Research Office (Grant W911NF-12-1-0509), National Science Foundation (U.S.) (Grant CMMI-1234062)

We provide a dynamic policy for the rapid containment of a contagion process modeled as an SIS epidemic on a bounded degree undirected graph with $n$ nodes. We show that if the budget $r$ of curing resources available at each time is $\Omega (W)$ , where $W$ is the CutWidth of the graph, and also of order $\Omega (\log\; n)$ , then the expected time until the extinction of the epidemic is of order $O(n/r)$ , which is within a constant factor from optimal, as well as sublinear in the number of nodes. Furthermore, if the CutWidth increases only sublinearly with $n$ , a sublinear expected time to extinction is possible with a sublinearly increasing budget $r$ .

We study a multi-server model with n flexible servers and rn queues, connected through a fixed bipartite graph, where the level of flexibility is captured by the average degree, d(n), of the queues. Applications in content replication in data centers, skill-based routing in call centers, and flexible supply chains are among our main motivations. We focus on the scaling regime where the system size n tends to infinity, while the overall traffic intensity stays fixed. We show that a large capacity region (robustness) and diminishing queueing delay (performance) are jointly achievable even under very limited flexibility (d(n) l n). In particular, when d(n) gg ln n , a family of random-graph-based interconnection topologies is (with high probability) capable of stabilizing all admissible arrival rate vectors (under a bounded support assumption), while simultaneously ensuring a diminishing queueing delay, of order ln n/ d(n), as n-> ∞. Our analysis is centered around a new class of virtual-queue-based scheduling policies that rely on dynamically constructed partial matchings on the connectivity graph., National Science Foundation (U.S.) (Grant CMMI-1234062), MIT-Xerox Fellowship Program

A Bayesian framework is used to calibrate a mass-action model of receptor-mediated apoptosis. Despite parameter non-identifiability and model ‘sloppiness', Bayes factor analysis discriminates between two alternative models of mitochondrial outer membrane permeabilization., Bayesian estimation returns statistically complete joint parameter distribution for mass-action models of receptor-mediated apoptosis calibrated to dynamic, live-cell data. Analysis of joint distributions reveals strong, non-linear correlations between parameters that are poorly captured by a conventional table of mean values and covariances; a high-dimensional distribution must therefore be reported as the true estimate of parameter values. Despite non-identifiablility and model ‘sloppiness,' a Bayesian framework returns probabilistic predictions for cell death dynamics that have tight confidence intervals and match experimental data. Use of a Bayesian framework to discriminate between two competing models of mitochondrial outer membrane permeabilization shows that a ‘direct' mechanism has ∼20-fold greater plausibility than an ‘indirect' mechanism, even though both models exhibit equally good fits to data for some parameters., Using models to simulate and analyze biological networks requires principled approaches to parameter estimation and model discrimination. We use Bayesian and Monte Carlo methods to recover the full probability distributions of free parameters (initial protein concentrations and rate constants) for mass-action models of receptor-mediated cell death. The width of the individual parameter distributions is largely determined by non-identifiability but covariation among parameters, even those that are poorly determined, encodes essential information. Knowledge of joint parameter distributions makes it possible to compute the uncertainty of model-based predictions whereas ignoring it (e.g., by treating parameters as a simple list of values and variances) yields nonsensical predictions. Computing the Bayes factor from joint distributions yields the odds ratio (∼20-fold) for competing ‘direct' and ‘indirect' apoptosis models having different numbers of parameters. Our results illustrate how Bayesian approaches to model calibration and discrimination combined with single-cell data represent a generally useful and rigorous approach to discriminate between competing hypotheses in the face of parametric and topological uncertainty.

We conduct a theoretical study of various solution methods for the adaptive fractionation problem. The two messages of this paper are as follows: (i) dynamic programming (DP) is a useful framework for adaptive radiation therapy, particularly adaptive fractionation, because it allows us to assess how close to optimal different methods are, and (ii) heuristic methods proposed in this paper are near-optimal, and therefore, can be used to evaluate the best possible benefit of using an adaptive fraction size. The essence of adaptive fractionation is to increase the fraction size when the tumor and organ-at-risk (OAR) are far apart (a 'favorable' anatomy) and to decrease the fraction size when they are close together. Given that a fixed prescribed dose must be delivered to the tumor over the course of the treatment, such an approach results in a lower cumulative dose to the OAR when compared to that resulting from standard fractionation. We first establish a benchmark by using the DP algorithm to solve the problem exactly. In this case, we characterize the structure of an optimal policy, which provides guidance for our choice of heuristics. We develop two intuitive, numerically near-optimal heuristic policies, which could be used for more complex, high-dimensional problems. Furthermore, one of the heuristics requires only a statistic of the motion probability distribution, making it a reasonable method for use in a realistic setting. Numerically, we find that the amount of decrease in dose to the OAR can vary significantly (5–85%) depending on the amount of motion in the anatomy, the number of fractions and the range of fraction sizes allowed. In general, the decrease in dose to the OAR is more pronounced when: (i) we have a high probability of large tumor–OAR distances, (ii) we use many fractions (as in a hyper-fractionated setting) and (iii) we allow large daily fraction size deviations. General scientific summary: Conventional radiation therapy procedures deliver an equal dose to the tumor every day over the course of 30 to 40 days. In this paper, we consider the effect of delivering a different dose or fraction size based on the changes observed in the patient anatomy. Given that a fixed prescription dose must be delivered to the tumor over the course of the treatment, we find that adaptively varying the fraction size results in a lower cumulative dose to the healthy organ-at-risk. The two messages of this paper are: (i) dynamic programming is a useful framework for adaptive radiation therapy, particularly adaptive fractionation, because it allows us to measure the proximity of different methods to the optimal one, and (ii) heuristic methods proposed in this paper are near-optimal, and therefore, can be used to evaluate the best possible benefit of using an adaptive fraction size., Siemens Aktiengesellschaft

We propose and analyze a multi-server model that captures a performance trade-off between centralized and distributed processing. In our model, a fraction p of an available resource is deployed in a centralized manner (e.g., to serve a most loaded station) while the remaining fraction 1-p is allocated to local servers that can only serve requests addressed specifically to their respective stations. Using a fluid model approach, we demonstrate a surprising phase transition in steady-state delay, as p changes: in the limit of a large number of stations, and when any amount of centralization is available (p>0), the average queue length in steady state scales as log [subscript 1/1-p] 1/1-λ when the traffic intensity λ goes to 1. This is exponentially smaller than the usual M/M/1-queue delay scaling of 1/1-λ, obtained when all resources are fully allocated to local stations (p=0). This indicates a strong qualitative impact of even a small degree of centralization. We prove convergence to a fluid limit, and characterize both the transient and steady-state behavior of the finite system, in the limit as the number of stations N goes to infinity. We show that the queue-length process converges to a unique fluid trajectory (over any finite time interval, as N → ∞), and that this fluid trajectory converges to a unique invariant state v[superscript I], for which a simple closed-form expression is obtained. We also show that the steady-state distribution of the N-server system concentrates on v[superscript I] as N goes to infinity., Irwin Mark Jacobs and Joan Klein Jacobs Presidential Fellowship, Xerox Fellowship Program, Thomas and Stacey Siebel Foundation (Scholarship), National Science Foundation (U.S.) (Grant CCF-0728554)

In distributed consensus and averaging algorithms, processors exchange and update certain values ("estimates" or "opinions") by forming a local average with the values of their neighbors. Under suitable conditions, such algorithms converge to consensus (every processor ends up holding the same value) or even average-consensus (consensus is achieved on the average of the initial values held by the processors). We first review some convergence rate results. We then abstract the problem by introducing a model of deterministic distributed function computation by a network of identical and anonymous nodes, with limited communication and storage capabilities, and where each node only knows its neighbors, not the entire graph. Our goal is to characterize the class of functions that can be computed within this model. In our main result, we provide a necessary condition for computability which we show to be nearly sufficient, in the sense that every function that satisfies this condition can at least be approximated. The problem of computing (suitably rounded) averages in a distributed manner plays a central role in our development, and we provide an algorithm that solves it in time that grows at most quadratically with the size of the network. (Joint work with J. Hendrickx and A. Olshevsky)., National Science Foundation (U.S.) (Grant EECS-0701623)

We consider finite horizon Markov decision processes under performance measures that involve both the mean and the variance of the cumulative reward. We show that either randomized or history-based policies can improve performance. We prove that the complexity of computing a policy that maximizes the mean reward under a variance constraint is NP-hard for some cases, and strongly NP-hard for others. We finally offer pseudopolynomial exact and approximation algorithms., National Science Foundation (U.S.) (CMMI-0856063)

We consider the decentralized Bayesian binary hypothesis testing problem in feedback architectures, in which the fusion center broadcasts information based on the messages of some sensors to some or all sensors in the network. We show that the asymptotically optimal detection performance (as quantified by error exponents) does not benefit from the feedback messages. In addition, we determine the corresponding optimal error exponents., National Science Foundation (U.S.) (Grant ECCS-0701623)

We consider a flow-level model of a network operating under an $\alpha$-fair bandwidth sharing policy (with $\alpha>0$) proposed by Roberts and Massouli\'{e} [Telecomunication Systems 15 (2000) 185-201]. This is a probabilistic model that captures the long-term aspects of bandwidth sharing between users or flows in a communication network. We study the transient properties as well as the steady-state distribution of the model. In particular, for $\alpha\geq1$, we obtain bounds on the maximum number of flows in the network over a given time horizon, by means of a maximal inequality derived from the standard Lyapunov drift condition. As a corollary, we establish the full state space collapse property for all $\alpha\geq1$. For the steady-state distribution, we obtain explicit exponential tail bounds on the number of flows, for any $\alpha>0$, by relying on a norm-like Lyapunov function. As a corollary, we establish the validity of the diffusion approximation developed by Kang et al. [Ann. Appl. Probab. 19 (2009) 1719-1780], in steady state, for the case where $\alpha=1$ and under a local traffic condition., Comment: Published in at http://dx.doi.org/10.1214/12-AAP915 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)

We investigate the asymptotic behavior of the steady-state queue length distribution under generalized max-weight scheduling in the presence of heavy-tailed traffic. We consider a system consisting of two parallel queues, served by a single server. One of the queues receives heavy-tailed traffic, and the other receives light-tailed traffic. We study the class of throughput optimal max-weight-α scheduling policies, and derive an exact asymptotic characterization of the steady-state queue length distributions. In particular, we show that the tail of the light queue distribution is heavier than a power-law curve, whose tail coefficient we obtain explicitly. Our asymptotic characterization also shows that the celebrated max-weight scheduling policy leads to the worst possible tail of the light queue distribution, among all non-idling policies. Motivated by the above `negative' result regarding the max-weight-α policy, we analyze a log-max-weight (LMW) scheduling policy. We show that the LMW policy guarantees an exponentially decaying light queue tail, while still being throughput optimal., National Science Foundation (U.S.) (grant CNS-0626781), National Science Foundation (U.S.) (grant CNS-0915988), National Science Foundation (U.S.) (grant CCF-0728554), United States. Army Research Office (muri grant W911NF-08-1-0238)

We propose a model for deterministic distributed function computation by a network of identical and anonymous nodes. In this model, each node has bounded computation and storage capabilities that do not grow with the network size. Furthermore, each node only knows its neighbors, not the entire graph. Our goal is to characterize the class of functions that can be computed within this model. In our main result, we provide a necessary condition for computability which we show to be nearly sufficient, in the sense that every function that violates this condition can at least be approximated. The problem of computing (suitably rounded) averages in a distributed manner plays a central role in our development; we provide an algorithm that solves it in time that grows quadratically with the size of the network., National Science Foundation (U.S.) (Graduate Fellowship), National Science Foundation (U.S.) (Grant ECCS-0701623), Belgian American Educational Foundation, inc. (Postdoctoral Fellowship), Belgian National Foundation for Scientific Research (Postdoctoral Fellowship)

We consider the decentralized binary hypothesis testing problem in networks with feedback, where some or all of the sensors have access to compressed summaries of other sensors' observations. We study certain two-message feedback architectures, in which every sensor sends two messages to a fusion center, with the second message based on full or partial knowledge of the first messages of the other sensors. Under either a Neyman-Pearson or a Bayesian formulation, we show that the asymptotically optimal (in the limit of a large number of sensors) detection performance (as quantified by error exponents) does not benefit from the feedback messages.

We study a decentralized detection architecture in which each of a set of sensors transmits a highly compressed summary of its observations (a binary message) to a fusion center, which then decides on one of two alternative hypotheses. In contrast to the star (or “parallel”) architecture considered in most of the literature, we allow a subset of the sensors to both transmit their messages to the fusion center and to also broadcast them to the remaining sensors. We focus on the following architectural question: is there a significant performance improvement when we allow such a message broadcast? We consider the error exponent (asymptotically, in the limit of a large number of sensors) for the Neyman-Pearson formulation of the detection problem. We prove that the sharing of messages does not improve the optimal error exponent., National Science Foundation (U.S.) (grant ECCS-0701623)

We consider bandit problems involving a large (possibly infinite) collection of arms, in which the expected reward of each arm is a linear function of an r-dimensional random vector Z ∈ ℝ(superscript r), where r ≥ 2. The objective is to minimize the cumulative regret and Bayes risk. When the set of arms corresponds to the unit sphere, we prove that the regret and Bayes risk is of order Θ(r √T), by establishing a lower bound for an arbitrary policy, and showing that a matching upper bound is obtained through a policy that alternates between exploration and exploitation phases. The phase-based policy is also shown to be effective if the set of arms satisfies a strong convexity condition. For the case of a general set of arms, we describe a near-optimal policy whose regret and Bayes risk admit upper bounds of the form O(r √T log(superscript 3/2)T)., National Science Foundation (U.S.) (grant DMS-0732196), National Science Foundation (U.S.) (grant ECCS-0701623), National Science Foundation (U.S.) (grant CMMI-0856063), National Science Foundation (U.S.) (grant CMMI-0855928)

We study the detection performance of large scale sensor networks, configured as trees with bounded height, in which information is progressively compressed as it moves towards the root of the tree. We show that, under a Bayesian formulation, the error probability decays exponentially fast, and we provide bounds for the error exponent. We then focus on the case where the tree has certain symmetry properties. We derive the form of the optimal exponent within a restricted class of easily implementable strategies, as well as optimal strategies within that class. We also find conditions under which (suitably defined) majority rules are optimal. Finally, we provide evidence that in designing a network it is preferable to keep the branching factor small for nodes other than the neighbors of the leaves.

We study a simple continuous-time multi-agent system related to Krause's model of opinion dynamics: each agent holds a real value, and this value is continuously attracted by every other value differing from it by less than 1, with an intensity proportional to the difference. We prove convergence to a set of clusters, with the agents in each cluster sharing a common value, and provide a lower bound on the distance between clusters at a stable equilibrium, under a suitable notion of multi-agent system stability. To better understand the behavior of the system for a large number of agents, we introduce a variant involving a continuum of agents. We prove, under some conditions, the existence of a solution to the system dynamics, convergence to clusters, and a non-trivial lower bound on the distance between clusters. Finally, we establish that the continuum model accurately represents the asymptotic behavior of a system with a finite but large number of agents., 25 pages, 2 figures, 11 tex files and 2 eps files

We propose a model for deterministic distributed function computation by a network of identical and anonymous nodes, with bounded computation and storage capabilities that do not scale with the network size. Our goal is to characterize the class of functions that can be computed within this model. In our main result, we exhibit a class of non-computable functions, and prove that every function outside this class can at least be approximated. The problem of computing averages in a distributed manner plays a central role in our development.

We study a model of opinion dynamics introduced by Krause: each agent has an opinion represented by a real number, and updates its opinion by averaging all agent opinions that differ from its own by less than one. We give a new proof of convergence into clusters of agents, with all agents in the same cluster holding the same opinion. We then introduce a particular notion of equilibrium stability and provide lower bounds on the inter-cluster distances at a stable equilibrium. To better understand the behavior of the system when the number of agents is large, we also introduce and study a variant involving a continuum of agents, obtaining partial convergence results and lower bounds on inter-cluster distances, under some mild assumptions.

We consider the problem of allocating a fixed amount of an infinitely divisible resource among multiple competing, fully rational users. We study the efficiency guarantees that are possible when we restrict to mechanisms that satisfy certain scalability constraints motivated by large scale communication networks; in particular, we restrict attention to mechanisms where users are restricted to one-dimensional strategy spaces. We first study the efficiency guarantees possible when the mechanism is not allowed to price differen- tiate. We study the worst-case efficiency loss (ratio of the utility associated with a Nash equilibrium to the maximum possible utility), and show that the proportional allocation mechanism of Kelly (1997) minimizes the efficiency loss when users are price anticipating. We then turn our attention to mechanisms where price differentiation is permitted; using an adaptation of the Vickrey-Clarke-Groves class of mechanisms, we con- struct a class of mechanisms with one-dimensional strategy spaces where Nash equilibria are fully efficient. These mechanisms are shown to be fully efficient even in general convex environments, under reasonable assumptions. Our results highlight a fundamental insight in mechanism design: when the pricing flexibility available to the mechanism designer is limited, restricting the strategic flexibility of bidders may actually improve the efficiency guarantee., National Science Foundation, Army Research Office, DARPA - Next Generation Internet Initiative, National Science Foundation Graduate Research Fellowship

We consider a convex unconstrained optimization problem that arises in a network of agents whose goal is to cooperatively optimize the sum of the individual agent objective functions through local computations and communications. For this problem, we use averaging algorithms to develop distributed subgradient methods that can operate over a time-varying topology. Our focus is on the convergence rate of these methods and the degradation in performance when only quantized information is available. Based on our recent results on the convergence time of distributed averaging algorithms, we derive improved upper bounds on the convergence rate of the unquantized subgradient method. We then propose a distributed subgradient method under the additional constraint that agents can only store and communicate quantized information, and we provide bounds on its convergence rate that highlight the dependence on the number of quantization levels.

We provide an example proving that there exists no quadratic Lyapunov function for a certain class of linear agreement/consensus algorithms, a fact that had been numerically verified in [5]. We also briefly discuss sufficient conditions for the existence of such a Lyapunov function.

We study the convergence speed of distributed iterative algorithms for the consensus and averaging problems, with emphasis on the latter. We first consider the case of a fixed communication topology. We show that a simple adaptation of a consensus algorithm leads to an averaging algorithm. We prove lower bounds on the worst-case convergence time for various classes of linear, time-invariant, distributed consensus methods, and provide an algorithm that essentially matches those lower bounds. We then consider the case of a time-varying topology, and provide a polynomial-time averaging algorithm.

We consider a resource allocation problem where individual users wish to send data across a network to maximize their utility, and a cost is incurred at each link that depends on the total rate sent through the link. It is known that as long as users do not anticipate the effect of their actions on prices, a simple proportional pricing mechanism can maximize the sum of users' utilities minus the cost (called aggregate surplus). Continuing previous efforts to quantify the effects of selfish behavior in network pricing mechanisms, we consider the possibility that users anticipate the effect of their actions on link prices. Under the assumption that the links' marginal cost functions are convex, we establish existence of a Nash equilibrium. We show that the aggregate surplus at a Nash equilibrium is no worse than a factor of 4*sqrt{2} - 5 times the optimal aggregate surplus; thus, the efficiency loss when users are selfish is no more than approximately 34%., Originally Laboratory for Information and Decision Systems (MIT) Publication 2605

In this paper we study problems such as: given a discrete time dynamical system of the form x(t + 1)= f(x(t)) where f: R-n --> R-n is a piecewise affine function, decide whether all trajectories converge to 0. We show in our main theorem that this Attractivity Problem is undecidable as soon as n greater than or equal to2. The same is true of two related problems: Stability (is the dynamical system globally asymptotically stable?) and Mortality (do all trajectories go through 0?). We then show that AM-activity and Stability become decidable in dimension 1 for continuous functions. (C) 2001 Elsevier Science B.V. All rights reserved.

Colloque avec actes et comité de lecture. internationale.; International audience; We prove that several global properties (global convergence, global asymptotic stability, mortality, and nilpotence) of particular classes of discrete time dynamical systems are undecidable. Such results had been known only for point-to-point properties. We prove these properties undecidable for saturated linear dynamical systems, and for continuous piecewise affine dynamical systems in dimension three. We also describe some consequences of our results on the possible dynamics of such systems.

We consider open and closed multiclass queueing networks, with Poisson arrivals (for open networks), exponentially distributed class dependent service times and class dependent deterministic or probabilistic routing. The performance objective is to minimize, over all sequencing and routing policies, a weighted sum of the expected response times of different classes. Using a powerful technique involving quadratic or higher order potential functions, we propose methods for deriving polyhedral and nonlinear sets that contain the set of achievable response times under stable and preemptive scheduling policies. By optimizing over these sets, we obtain lower bounds on achievable performance. In the special case of single station networks (multiclass queues and Klimov's model) and homogeneous multiclass networks, the polyhedron derived is exactly equal to the achievable region. Consequently, the proposed method can be viewed as the natural extension of conservation laws to multiclass queueing networks. We apply the same approach to closed networks to obtain upper bounds on the optimal throughput. We check the tightness of our bounds by simulating heuristic policies and we find that the first order approximation of our method is at least as good as simulation-based existing methods. In terms of computational complexity and in contrast to simulation-based existing methods, the calculation of our first order bounds consists of solving a linear programming problem with a number of variables and constraints that is polynomial (quadratic) in the number of classes in the network. The $i$th order approximation leads to a convex programming problem in dimension $O(R^{i+1})$, where $R$ is the number of classes in the network, and can be solved efficiently using techniques from semidefinite programming.