One of the classical problems in queuing theory is to schedule customers/jobs in an optimal way. These problems are known as the scheduling problems. They arise in a wide variety of applications, in particular, whenever there are different customer classes present competing for the same resources. In a recent work “Ergodic control of multi-class M/M/N+M queues in the Halfin-Whitt regime”, Ari Arapostathis, Anup Biswas and Guodong Pang solved an ergodic control problem for multi-class many server queuing networks. The optimal solution of the queuing control problem can be approximated by that of the corresponding ergodic diffusion control problem in the limit. The proof technique introduces a new method of spatial truncation for the diffusion control problem.
Junse Lee, Xinchen Zhang and François Baccelli proposed new models for analyzing spatially correlated shadowing fields. These models allow one to analyze the interference field created by a wireless infrastructure through the walls and floors of a building with variable size rooms. These models provide a mathematical characterization of the interference distribution, which further leads to closed-from expressions for the coverage probability in cellular networks. Three network scenarios are studied: 2-D outdoor, 2-D indoor, and 3-D inbuilding.
Stochastic geometry provides a natural way of defining and computing macroscopic properties of classical channels of multiuser information theory. These macroscopic properties are obtained by some averaging over all node patterns found in a large random network of the Euclidean plane. One of the most important geometric objects are the coverage regions of a transmitter or a set of transmitters. This domain of research is jointly studied by Jeffrey Andrews, François Baccelli, Gustavo de Veciana, Robert Heath and Sanjay Shakkottai. Most of the initial steps are based on Poisson point processes. Lately, this continued with Yingzhe Li (Simons PhD student, ECE, UT Austin) to the case of determinantal point processes. Another line of work on studying cell-association problems in multi-technology cellular networks was carried out in this paper by Abishek Sankararaman, Jeong-woo Cho and François Baccelli.
Rachel Ward and collaborator Giang Tran (former Bing Instructor in the UT Mathematics department) are investigating the identification of a dynamical system (say, within the class of polynomial systems of ordinary differential equations) given snapshots of the system in time. Such problems prove challenging when there is a high level of noise on the data. In the paper Exact recovery of Chaotic systems from highly corrupted data, they show that if the underlying trajectory exhibits ergodicy or chaos, and if the underlying dynamics have a sparse representation with respect to the polynomial basis, then a LASSO / l1 type algorithm will exactly recover the underlying dynamics even when most of the data is highly corrupted. This establishes a new link between the areas of dynamical systems and machine learning / sparse recovery, and many interesting questions remain.
The stochastic block model (aka. planted partition model) is a popular model for representing networks with communities. Elchanan Mossel, Joe Neeman, and Allan Sly have been investigating algorithms and fundamental limits for detecting and recovering these communities. They established sharp transitions for the problem of extracting non-trivial information and the problem of exactly recovering communities. They also gave a new algorithm that obtains provably optimal accuracy for the problem of detecting communities in “Consistency thresholds for the planted bisection model” and “Belief propagation, robust reconstruction, and optimal recovery of block models“.
François Baccelli and Jae Oh Woo initiated a study on the entropy and mutual information of point processes (On the Entropy and Mutual Information of Point Processes). The main new mathematical objects are the relative entropy rate and the mutual information rate of two stationary point processes. They also derived expression of the mutual information rate in the case of a homogeneous point process and its displacement. This machinery is used to revisit the Gaussian noise channel in the Shannon-Poltyrev regime recently introduced in Capacity and error exponents of stationary point processes under random additive displacements.
Anup Biswas and François Baccelli studied the scaling limit of a class of shot-noise fields defined on an independently marked stationary Poisson point process and with a power law response function. Under appropriate conditions, they showed that the shot-noise field can be scaled suitably to have a non degenerate alpha-stable limit, as the intensity of the underlying point process goes to infinity. More precisely, finite dimensional distributions converge and the finite dimensional distributions of the limiting random field have i.i.d. stable random components. This limit is hence called the alpha- stable white noise field. Analogous results are also obtained for the extremal shot-noise field which converges to a Fréchet white noise field.
The most basic capacity and error exponent questions of information theory can be expressed in terms of random geometric objects living in Euclidean spaces with dimensions tending to infinity. This approach was introduced by Venkat Anantharam and François Baccelli to evaluate random coding error exponents in channels with additive stationary and ergodic noise. More generally, the analysis of stochastic geometry in the Shannon regime leads to new high dimension stochastic geometry questions that are currently investigated. Eliza O’Reilly and and Francois Baccelli have also studied determinantal point processes in high dimensions. This work describes the strength and reach of repulsion of a typical point of certain parametric families of determinantal point process in the Shannon regime.
Recent advances in neuroscience provide theoretical neuroscientists with a vast wealth of new data and open questions related to information theory, high-dimensional geometry of representation and computation, and dynamics in the brain. The groups of Ila Fiete, Ngoc Mai Tran and Thibaud Taillefumier study these questions from analytical and numerical perspectives. Fiete and Tran have recently studied the learning capacity of neural networks (see “A binary Hopfield network with 1/\log(n) information rate and applications to grid cell decoding“, “ Robust exponential memory in Hopfield networks“, and “ Associative content-addressable networks with exponentially many robust stable states“).
Avhishek Chatterjee, François Baccelli and Sriram Vishwanath proposed a stochastic extension of the bounded confidence model where opinions take their values in the Euclidean space and where friendship and interactions are dynamically defined through time varying and random neighborhoods. Two basic sub-models are defined: the influencing model where each agent is an attractor to the opinions of its neighbors and the listening model where each agent gathers information from others to update its own opinions. The general model contains a rich set of variants for which they proposed a classification. They analyzed the stability of its dynamics. The analysis highlights the need of certain leaders with heavy tailed neighborhoods for stability to hold. See Pairwise Stochastic Bounded Confidence Opinion Dynamics: Heavy Tails and Stability
In a paper entitled Metastability of Queuing Networks with Mobile Servers, A. Rybko, S. Shlosman, A. Vladimorov and F. Baccelli study symmetric queuing networks with moving servers and FIFO service discipline. The mean-field limit dynamics demonstrates unexpected behavior which is attributed to the meta-stability phenomenon. Large enough finite symmetric networks on regular graphs are proved to be transient for arbitrarily small inflow rates. However, the limiting non-linear Markov process possesses at least two stationary solutions. The mean-field analysis is based on the Non Linear Markov Process developed for this type of queuing networks in Queuing Networks with Varying Topology – A Mean-Field Approach.
With Pranav Madadi, F. Baccelli, and G. de Veciana analyzed the temporal variations in the Shannon rate experienced by a user moving along a straight line in a cellular network represented by a Poisson-Voronoi tessellation. We consider a network that is shared by static users distributed as a Poisson point process and analyzed the time series of the final shared rate and the number of users sharing the network. The first paper On Shared Rate Time Series for Mobile Users in Poisson Networks was focused on the noise limited case. The ongoing research is focused on the general case, with both interference and thermal noise taken into account.
François Baccelli, Sriram Vishwanath and Jae Oh Woo proposed a computational framework for continuous time opinion dynamics with additive noise. They derived a non-local partial differential equation for the distribution of opinions differences. They used Mellin transforms to solve the stationary solution of this equation in closed form. This approach can be applied both to linear dynamics on an interaction graph and to bounded confidence dynamics in the Euclidean space. To the best of our knowledge, the closed form expression on the stationary distribution of the bounded confidence model is the first quantitative result on the equilibria of this class of models.
High throughput DNA sequencing technology has greatly increased the speed and reduced the cost of genome sequencing. The process is divided into to two steps: generating a library of short reads and reassembling those reads into the original genome. Eliza O’Reilly, François Baccelli, Gustavo de Veciana, and Haris Vikalo have worked on modeling this process using stochastic geometry and queueing theory in order to optimize the output of correct reads and the probability of successful reassembly (see End-to-End Optimization of High Throughput DNA Sequencing).
Dense phases of emergent systems, such as constrained complex networks,
exhibit distinct characteristics, the most studied being broken symmetry.
However for practical purposes “rigidity”, the resistance to change, is
also of wide interest. There are difficulties analyzing rigidity since
when perturbed a system can easily move out of its phase. A new approach to
overcome this contradiction has been initiated by David Aristoff and
Charles Radin: see the discussion in Quanta Magazine. Another characteristic of dense phases are their nonspherical `Wulff’ shapes, polyhedral for ordinary crystals. This is examined in this expository paper by Charles Radin. In a different direction, the process of
nucleation is a dynamical signature of the creation of a dense phase, which can appear even in systems far removed from ordinary atomic materials, as shown in this paper by Frank Rietz, Charles Radin, Harry Swinney and Matthias Schroeter. All the above characteristics make essential use of finite systems, a nonstandard approach to understanding emergent phases.
In the phenomenon of emergence, a system of many interacting objects
exhibits the collective behavior of one or more “phases”, which are
only detectable or even meaningful for the system as a whole. This is
an organizing principle widely used in biology, physics and indeed all
the sciences: crystals, hurricanes, animal flocking etc. One wants to
understand the spontaneous appearance of phases in systems of large
size, in particular to determine a mechanism of some generality. A
convenient framework for such an analysis is large networks with
constraints. Such an analysis has been undertaken by the group of
Richard Kenyon, Charles Radin, Kui Ren and Lorenzo Sadun, on entropy singularities, the edge/triangle system I, the edge/triangle system II, multipodal structure,
order-disorder transitions and oversaturated networks. There is also a related asymptotic analysis of large permutations undertaken by the group of Richard Kenyon, Daniel Kral, Charles Radin and Peter Winkler: permutations with fixed pattern densities, and a review of phases in general combinatorial systems.
A compatible point-shift f maps, in a translation invariant way, each point of a stationary point process N to some point of N. It is fully determined by its associated point-map, g, which gives the image of the origin by f. The initial question studied by Mir-Omid Haji-Mirsadeghi and François Baccelli is whether there exist probability measures which are left invariant by the translation of -g. The point map probabilities of N are defined from the action of the semigroup of point-map translations on the space of Palm probabilities, and more precisely from the compactification of the orbits of this semigroup action. If the point-map probability is uniquely defined, and if it satisfies certain continuity properties, it then provides a solution to the initial question. Point-map probabilities are shown to be a strict generalization of Palm probabilities: when the considered point-shift f is bijective, the point-map probability of N boils down to the Palm probability of N. When it is not bijective, there exist cases where the point-map probability of N is absolutely continuous with respect to its Palm probability, but there also exist cases where it is singular with respect to the latter.
Each such point-shift defines a random graph on the points of the point process. The connected components of this graph can be split into a collection of foils, which are the analogue of the stable manifold of the point-shift dynamics.
The same authors give a general classification of point-shifts in terms of the cardinality of the foils of these connected components. There are three types: F/F, I/F and I/I as shown in the paper Point-Shift Foliation of a Point Process.
Using the framework of Günter Last, James Murphy has extended the cardinality classification to the case of point processes on unimodular groups. J. Murphy has studied point-shifts of point processes on topological groups at length.
Consider a queue where the server is the Euclidean space, the customers are random closed sets (RACS) of the Euclidean space. These RACS arrive according to a Poisson rain and each of them has a random service time (in the case of hail falling on the Euclidean plane, this is the height of the hailstone, whereas the RACS is its footprint). The Euclidean space serves customers at speed 1. The service discipline is a hard exclusion rule: no two intersecting RACS can be served simultaneously and service is in the First In First Out order (only the hailstones in contact with the ground melt at speed 1, whereas the other ones are queued; a tagged RACS waits until all RACS arrived before it and intersecting it have fully melted before starting its own melting). We prove that this queue is stable for a sufficiently small arrival intensity, provided the typical diameter of the RACS and the typical service time have finite exponential moments.
In Shape Theorems For Poisson Hail on a Bivariate Ground, H. Chang, S. Foss and F. Baccelli have extended this Poisson Hail model to the situation where the service speed is either zero or infinity at each point of the Euclidean space. Tools pertaining to sub-additive ergodic theory are used to establish shape theorems for the growth of the ice-heap under light tail assumptions on the hailstone characteristics. The asymptotic shape depends on the statistics of the hailstones, the intensity of the underlying Poisson point process and on the geometrical properties of the zero speed set.
Ergodic spectral efficiency quantifies the achievable Shannon transmission rate per unit area, and captures the effects of rate adaptation techniques. Junse Lee, Namyoon Lee and François Baccelli studied the benefits of multiple antenna communication in ad-hoc networks using this metric. In this work, the primary finding is that, with knowledge of channel state information between a receiver and its associated transmitter, the ergodic spectral efficiency can be made to scale linearly with both 1) the minimum of the number of transmit and receive antennas and 2) the density of nodes. This scaling law is achieved when the multiple transmit antennas send multiple data streams and the multiple receive antennas are leveraged to cancel interference. Spatial multiplexing transmission methods are shown to be essential for obtaining better and eventually optimal scaling laws in such random wireless networks.
Spatial point processes involving birth and death dynamics are ubiquitous in networks. Such dynamics are particularly important in peer-to-peer networks and in wireless networks. In the paper “Mutual Service Processes in R^d, Existence and Ergodicity”, Fabien Mathieu (Bell Laboratories), Ilkka Norros (VTT) and François Baccelli proposed a way to analyze the long term behavior of such dynamics on Euclidean spaces using coupling techniques. This line of though is continued by Mayank Manjrekar on other classes of processes like hard core spatial birth and death processes.
Neural systems propagate information via neuronal networks that transform sensory input into distributed spiking patterns, and dynamically process these patterns to generate behaviorally relevant responses. The presence of noise at every stage of neural processing imposes serious limitation on the coding strategies of these networks. In particular, coding information via spike timings, which presumably achieves the highest information transmission rate, requires neural assemblies to exhibit high level of synchrony. Thibaud Taillefumier and collaborators are interested in understanding how synchronous activity emerges in modeled populations of spiking neurons, focusing on the interplay between driving inputs and network structure. Their approach relies on methods from Markov chain, point processes, and diffusion processes theories, in combination with exact event-driven simulation techniques. The ultimate goal is two-fold: 1) to identify the input/structure relations that optimize information transmission capabilities and 2) to characterize the “physical signature’’ of such putative optimal tunings in recorded spiking activity.
Ali Khezeli, Mir-Omid Haji-Mirsadeghi and François Baccelli studied dynamics on the vertices of a random graph in the paper Dynamics on Unimodular Graphs. The first result of the paper is a classification of vertex-shifts on unimodular random networks. Each such vertex-shift partitions the vertices into a collection of connected components and foils, as in the case of point-shifts on point processes.
The classification is based on the cardinality of the connected components and foils. Up to an event of zero probability, there are three types of foliations in a connected component: F/F (with finitely many finite foils), I/F (infinitely many finite foils), and I/I (infinitely many infinite foils).
An infinite connected component of the graph of a vertex-shift on a random network forms an infinite directed tree with one selected end which is referred to as an Eternal Family Tree. An Eternal Family Tree contains a subtree which is a stochastic generalization of a branching process. In a unimodular Eternal Family Tree, the subtree in question is a generalization of a critical branching process. In a $\sigma$-invariant Eternal Family Tree, the subtree is a generalizisation of a non-necessarily critical branching process. The latter trees allow one to analyze dynamics on networks which are not necessarily unimodular.
Ngoc Mai Tran and François Baccelli derived a tropical version of the result of Kac on the zeros of polynomials with random coefficients (Zeros of Random Tropical Polynomials, Random Polygons and Stick-Breaking).
The common roots of tropical of a class of random polynomials in two variables is analyzed in a recent work by the same authors in Iterated Gilbert Mosaics and Poisson Tropical Plane Curves using a stochastic geometry approach based on iterated random tessellations.