arXiv subjects

John N. Tsitsiklis

Publications and source records attributed to John N. Tsitsiklis.

At least 19 recordsLinked to original sources

arXiv2023

Jumping Fluid Models and Delay Stability of Max-Weight Dynamics under Heavy-Tailed Traffic

We say that a random variable is $light$-$tailed$ if moments of order $2+ε$ are finite for some $ε>0$; otherwise, we say that it is $heavy$-$tailed$. We study queueing networks that operate under the Max-Weight scheduling policy, for the case where some queues receive heavy-tailed and some receive light-tailed traffic. Queues with light-tailed arrivals are often delay stable (that is, expected queue sizes are uniformly bounded over time) but can also become delay unstable because of resource-sharing with other queues that receive heavy-tailed arrivals. Within this context, and for any given "tail exponents" of the input traffic, we develop a necessary and sufficient condition under which a queue is robustly delay stable, in terms of $jumping$ $fluid$ models - an extension of traditional fluid models that allows for jumps along coordinates associated with heavy-tailed flows. Our result elucidates the precise mechanism that leads to delay instability, through a coordination of multiple abnormally large arrivals at possibly different times and queues and settles an earlier open question on the sufficiency of a particular fluid-based criterion. Finally, we explore the power of Lyapunov functions in the study of delay stability.

eess.SY↗

arXiv2020

Stability, memory, and messaging tradeoffs in heterogeneous service systems

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 $λn$, with $0<λ<1$, to the system. Incoming jobs are immediately dispatched to one of several queues associated with the $n$ servers. We assume that the dispatching decisions are made by a central dispatcher endowed with a finite memory, and with the ability to exchange messages with the servers. We study the fundamental resource requirements (memory bits and message exchange rate) in order for a dispatching policy to be {\bf maximally stable}, i.e., stable whenever the processing rates are such that the arrival rate is less than the total available processing rate. First, for the case of Poisson arrivals and exponential service times, we present a policy that is maximally stable while using a positive (but arbitrarily small) message rate, and $\log_2(n)$ bits of memory. Second, we show that within a certain broad class of policies, a dispatching policy that exchanges $o\big(n^2\big)$ messages per unit of time, and with $o(\log(n))$ bits of memory, cannot be maximally stable. Thus, as long as the message rate is not too excessive, a logarithmic memory is necessary and sufficient for maximal stability.

cs.PF↗

arXiv2020

Private Sequential Learning

We formulate a private learning model to study an intrinsic tradeoff between privacy and query complexity in sequential learning. Our model involves a learner who aims to determine a scalar value, $v^*$, by sequentially querying an external database and receiving binary responses. In the meantime, an adversary observes the learner's queries, though not the responses, and tries to infer from them the value of $v^*$. The objective of the learner is to obtain an accurate estimate of $v^*$ using only a small number of queries, while simultaneously protecting her privacy by making $v^*$ provably difficult to learn for the adversary. Our main results provide tight upper and lower bounds on the learner's query complexity as a function of desired levels of privacy and estimation accuracy. We also construct explicit query strategies whose complexity is optimal up to an additive constant.

cs.LG↗

arXiv2019

Blind identification of stochastic block models from dynamical observations

We consider a blind identification problem in which we aim to recover a statistical model of a network without knowledge of the network's edges, but based solely on nodal observations of a certain process. More concretely, we focus on observations that consist of single snapshots taken from multiple trajectories of a diffusive process that evolves over the unknown network. We model the network as generated from an independent draw from a latent stochastic block model (SBM), and our goal is to infer both the partition of the nodes into blocks, as well as the parameters of this SBM. We discuss some non-identifiability issues related to this problem and present simple spectral algorithms that provably solve the partition recovery and parameter estimation problems with high accuracy. Our analysis relies on recent results in random matrix theory and covariance estimation, and associated concentration inequalities. We illustrate our results with several numerical experiments.

cs.LG↗

arXiv2019

Fluctuation Bounds for the Max-Weight Policy, with Applications to State Space Collapse

We consider a multi-hop switched network operating under a Max-Weight (MW) 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 i.i.d. arrivals, an additive SSC result over an exponential time scale.

cs.NI↗

arXiv2019

Nonexpansive Piecewise Constant Hybrid Systems are Conservative

Consider a partition of $R^n$ into finitely many polyhedral regions $D_i$ and associated drift vectors $μ_i\in R^n$. We study ``hybrid'' dynamical systems whose trajectories have a constant drift, $\dot x=μ_i$, whenever $x$ is in the interior of the $i$th region $D_i$, and behave consistently on the boundary between different regions. Our main result asserts that if such a system is nonexpansive (i.e., if the Euclidean distance between any pair of trajectories is a nonincreasing function of time), then the system must be conservative, i.e., its trajectories are the same as the trajectories of the negative subgradient flow associated with a potential function. Furthermore, this potential function is necessarily convex, and is linear on each of the regions $D_i$. We actually establish a more general version of this result, by making seemingly weaker assumptions on the dynamical system of interest.

eess.SY↗

arXiv2018

Sensitivity to Cumulative Perturbations for a Class of Piecewise Constant Hybrid Systems

We consider a class of continuous-time hybrid dynamical systems that correspond to subgradient flows of a piecewise linear and convex potential function with finitely many pieces, and which include the fluid-level dynamics of the Max-Weight scheduling policy as a special case. We study the effect of an external disturbance/perturbation on the state trajectory, and establish that the magnitude of this effect can be bounded by a constant multiple of the integral of the perturbation.

eess.SY↗

arXiv2018

A lower bound on the queueing delay in resource constrained load balancing

We consider the following distributed service model: jobs with unit mean, general distribution, and independent processing times arrive as a renewal process of rate $λn$, with $0<λ<1$, and are immediately dispatched to one of several queues associated with $n$ identical servers with unit processing rate. We assume that the dispatching decisions are made by a central dispatcher endowed with a finite memory, and with the ability to exchange messages with the servers. We study the fundamental resource requirements (memory bits and message exchange rate), in order to drive the expected queueing delay in steady-state of a typical job to zero, as $n$ increases. We develop a novel approach to show that, within a certain broad class of "symmetric" policies, every dispatching policy with a message rate of the order of $n$, and with a memory of the order of $\log n$ bits, results in an expected queueing delay which is bounded away from zero, uniformly as $n\to\infty$.

math.PR↗

arXiv2017

Delay, memory, and messaging tradeoffs in distributed service systems

We consider the following distributed service model: jobs with unit mean, exponentially distributed, and independent processing times arrive as a Poisson process of rate $λn$, with $0<λ<1$, and are immediately dispatched by a centralized dispatcher to one of $n$ First-In-First-Out queues associated with $n$ identical servers. The dispatcher is endowed with a finite memory, and with the ability to exchange messages with the servers. We propose and study a resource-constrained "pull-based" dispatching policy that involves two parameters: (i) the number of memory bits available at the dispatcher, and (ii) the average rate at which servers communicate with the dispatcher. We establish (using a fluid limit approach) that the asymptotic, as $n\to\infty$, expected queueing delay is zero when either (i) the number of memory bits grows logarithmically with $n$ and the message rate grows superlinearly with $n$, or (ii) the number of memory bits grows superlogarithmically with $n$ and the message rate is at least $λn$. Furthermore, when the number of memory bits grows only logarithmically with $n$ and the message rate is proportional to $n$, we obtain a closed-form expression for the (now positive) asymptotic delay. Finally, we demonstrate an interesting phase transition in the resource-constrained regime where the asymptotic delay is non-zero. In particular, we show that for any given $α>0$ (no matter how small), if our policy only uses a linear message rate $αn$, the resulting asymptotic delay is upper bounded, uniformly over all $λ<1$; this is in sharp contrast to the delay obtained when no messages are used ($α= 0$), which grows as $1/(1-λ)$ when $λ\uparrow 1$, or when the popular power-of-$d$-choices is used, in which the delay grows as $\log(1/(1-λ))$.

math.PR↗

arXiv2017

Flexible Queueing Architectures

We study a multi-server model with $n$ flexible servers and $n$ queues, connected through a bipartite graph, where the level of flexibility is captured by the graph's average degree, $d_n$. 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 and an asymptotically vanishing queueing delay are simultaneously achievable even under limited flexibility ($d_n \ll n$). Our main results demonstrate that, when $d_n\gg \ln n$, a family of expander-graph-based flexibility architectures has a capacity region that is within a constant factor of the maximum possible, while simultaneously ensuring a diminishing queueing delay for all arrival rate vectors in the capacity region. Our analysis is centered around a new class of virtual-queue-based scheduling policies that rely on dynamically constructed job-to-server assignments on the connectivity graph. For comparison, we also analyze a natural family of modular architectures, which is simpler but has provably weaker performance.

math.PR↗

arXiv2015

When is a network epidemic hard to eliminate?

We consider the propagation of a contagion process (epidemic) on a network and study the problem of dynamically allocating a fixed curing budget to the nodes of the graph, at each time instant. For bounded degree graphs, we provide a lower bound on the expected time to extinction under any such dynamic allocation policy, in terms of a combinatorial quantity that we call the resistance of the set of initially infected nodes, the available budget, and the number of nodes n. Specifically, we consider the case of bounded degree graphs, with the resistance growing linearly in n. We show that if the curing budget is less than a certain multiple of the resistance, then the expected time to extinction grows exponentially with n. As a corollary, if all nodes are initially infected and the CutWidth of the graph grows linearly, while the curing budget is less than a certain multiple of the CutWidth, then the expected time to extinction grows exponentially in n. The combination of the latter with our prior work establishes a fairly sharp phase transition on the expected time to extinction (sub-linear versus exponential) based on the relation between the CutWidth and the curing budget.

cs.SI↗

arXiv2015

A lower bound on the performance of dynamic curing policies for epidemics on graphs

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.

cs.SI↗

arXiv2015

Optimization of Radiation Therapy Fractionation Schedules in the Presence of Tumor Repopulation

We analyze the effect of tumor repopulation on optimal dose delivery in radiation therapy. We are primarily motivated by accelerated tumor repopulation towards the end of radiation treatment, which is believed to play a role in treatment failure for some tumor sites. A dynamic programming framework is developed to determine an optimal fractionation scheme based on a model of cell kill due to radiation and tumor growth in between treatment days. We find that faster tumor growth suggests shorter overall treatment duration. In addition, the presence of accelerated repopulation suggests larger dose fractions later in the treatment to compensate for the increased tumor proliferation. We prove that the optimal dose fractions are increasing over time. Numerical simulations indicate potential for improvement in treatment effectiveness.

physics.med-ph↗

arXiv2014

An efficient curing policy for epidemics on graphs

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 $Ω(W)$, where $W$ is the CutWidth of the graph, and also of order $Ω(\log n)$, 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 $r$.

cs.SI↗

arXiv2013

Convergence of type-symmetric and cut-balanced consensus seeking systems (extended version)

We consider continuous-time consensus seeking systems whose time-dependent interactions are cut-balanced, in the following sense: if a group of agents influences the remaining ones, the former group is also influenced by the remaining ones by at least a proportional amount. Models involving symmetric interconnections and models in which a weighted average of the agent values is conserved are special cases. We prove that such systems always converge. We give a sufficient condition on the evolving interaction topology for the limit values of two agents to be the same. Conversely, we show that if our condition is not satisfied, then these limits are generically different. These results allow treating systems where the agent interactions are a priori unknown, e.g., random or determined endogenously by the agent values. We also derive corresponding results for discrete-time systems.

eess.SY↗

arXiv2012

Delay Stability Regions of the Max-Weight Policy under Heavy-Tailed Traffic

We carry out a delay stability analysis (i.e., determine conditions under which expected steady-state delays at a queue are finite) for a simple 3-queue system operated under the Max-Weight scheduling policy, for the case where one of the queues is fed by heavy-tailed traffic (i.e, when the number of arrivals at each time slot has infinite second moment). This particular system exemplifies an intricate phenomenon whereby heavy-tailed traffic at one queue may or may not result in the delay instability of another queue, depending on the arrival rates. While the ordinary stability region (in the sense of convergence to a steady-state distribution) is straightforward to determine, the determination of the delay stability region is more involved: (i) we use "fluid-type" sample path arguments, combined with renewal theory, to prove delay instability outside a certain region; (ii) we use a piecewise linear Lyapunov function to prove delay stability in the interior of that same region; (iii) as an intermediate step in establishing delay stability, we show that the expected workload of a stable M/GI/1 queue scales with time as $\mathcal{O}(t^{1/(1+γ)})$, assuming that service times have a finite $1+γ$ moment, where $γ\in (0,1)$.

eess.SY↗

arXiv2012

Pricing of Fluctuations in Electricity Markets

In an electric power system, demand fluctuations may result in significant ancillary cost to suppliers. Furthermore, in the near future, deep penetration of volatile renewable electricity generation is expected to exacerbate the variability of demand on conventional thermal generating units. We address this issue by explicitly modeling the ancillary cost associated with demand variability. We argue that a time-varying price equal to the suppliers' instantaneous marginal cost may not achieve social optimality, and that consumer demand fluctuations should be properly priced. We propose a dynamic pricing mechanism that explicitly encourages consumers to adapt their consumption so as to offset the variability of demand on conventional units. Through a dynamic game-theoretic formulation, we show that (under suitable convexity assumptions) the proposed pricing mechanism achieves social optimality asymptotically, as the number of consumers increases to infinity. Numerical results demonstrate that compared with marginal cost pricing, the proposed mechanism creates a stronger incentive for consumers to shift their peak load, and therefore has the potential to reduce the need for long-term investment in peaking plants.

math.OC↗

arXiv2012

Efficiency Loss in a Cournot Oligopoly with Convex Market Demand

We consider a Cournot oligopoly model where multiple suppliers (oligopolists) compete by choosing quantities. We compare the social welfare achieved at a Cournot equilibrium to the maximum possible, for the case where the inverse market demand function is convex. We establish a lower bound on the efficiency of Cournot equilibria in terms of a scalar parameter derived from the inverse demand function, namely, the ratio of the slope of the inverse demand function at the Cournot equilibrium to the average slope of the inverse demand function between the Cournot equilibrium and a social optimum. Also, for the case of a single, monopolistic, profit maximizing supplier, or of multiple suppliers who collude to maximize their total profit, we establish a similar but tighter lower bound on the efficiency of the resulting output. Our results provide nontrivial quantitative bounds on the loss of social welfare for several convex inverse demand functions that appear in the economics literature.

math.OC↗