Click on the speaker’s name to access their abstract and their slides.

Booklet of the conference

Special talk

Slides Harrison

Slides Williams

Title: RBM and Queueing Theory: An Early History

Abstract: In this talk we will recap the development of RBM as a limit of, or approximation for, conventional discrete-flow queueing models. Emphasis will be on the speakers’ personal experiences and personal involvement in the field between 1960 and 2000.

Title: RBM meets RCM


Abstract: The random conductance model (RCM) is a well-established model for a random walk in random environment which has been object of intensive study in the last 15 years. Central questions of interest include homogenisation results such as functional central limit theorems for the random walk or local limit theorems for its heat kernel.
In this talk we will review those results where one main focus will be on invariance principles for RCMs on domains for which convergence towards reflected Brownian motion (RBM) has been established by Chen, Croydon and Kumagai (A0P 2015).
In the last part we will present a local limit theorem for the killed Green function on strongly regular domains for ergodic conductances only satisfying a moment condition. As an application we obtain scaling limits for inhomogenous discrete Gaussian free fields with random conductances.
The last part is based on a joint work in progress with Martin Slowik (Mannheim) and Anna-Lisa Sokol (Berlin).

Title: Matsumoto-Yor and Dufresne type theorems for a random walk on positive definite matrices

Abstract: We establish analogues of the geometric Pitman 2M−X theorem of Matsumoto and Yor and of the classical Dufresne identity, for a multiplicative random walk on positive definite matrices with Beta type II distributed increments. The Dufresne type identity provides another example of a stochastic matrix recursion that admits an explicit solution.

Title: A Skorohod problem for the heat equation and particle systems with selection


Abstract: Hydrodynamic limits of particle systems with selection are related to free boundary problems (FBP) associated with second order parabolic PDE. This relation has been established rigorously for some selection models while it remains open for others, where the difficulty lies in questions of regularity of the free boundary. We introduce a weak formulation of FBP in the form of a Skorohod problem for the PDE. It does not involve a free boundary at all, but at the same time reduces to a FBP provided the latter has a classical solution. The weak formulation allows us to avoid regularity questions and characterize limits in terms of the PDE in cases where classical solutions are expected to but not known to exist, as well as in cases where they are not expected to exist.

Title: Asymptotic Control of Centralized Dynamic Matching Markets with General Utilities

Abstract: We consider a matching market where buyers and sellers arrive according to Poisson processes. Participants abandon at a constant rate — independent of the arrivals. The utility for matching a given buyer with a seller is a general random variable. The utilities are i.i.d. and independent of arrivals and abandonment. We study this market in heavy-traffic and analyze two families of policies: a) the population threshold policy, matching an arriving agent to its best available mate only if the number of mates in the system is above a threshold, and b) the utility threshold policy, matching an arriving agent to its best available mate only if the corresponding utility is above a threshold. We obtain a Skorokhod problem which we use to characterize optimal (utility maximization) population threshold policies depending on the domain of attraction corresponding to the utilities. Analogous characterizations for optimal utility-based threshold policies are also discussed. This talk is based on joint work with Martin I. Reiman, Virag Shah, Lawrence M. Wein, Linjia Wu.

Title: On the stationary distribution of reflected Brownian motion in a wedge


Abstract: We consider the classical problem of determining the stationary distribution of the semimartingale reflected Brownian motion (SRBM) in a two-dimensional wedge.
We focus on the algebraic and differential nature of the Laplace transform of this stationary distribution. We derive necessary and sufficient conditions for this Laplace transform to be a rational, algebraic, differentially finite or more generally differentially algebraic function (meaning that it satisfies a polynomial differential equation). These conditions are simple explicit linear dependencies between the angles of the model. We thus recover in particular the Dieker and Moriarty conditions for rationality. In the D-algebraic case, we obtain moreover simple, explicit integral-free expression in terms of a hypergeometric function.

To prove these results, we start from a functional equation that the Laplace transform satisfies, to which we apply a variety of tools ranging from combinatorics to Galois theory.

This is a joint work with Andrew Elvey Price, Sandro Franceschi, Charlotte Hardouin and Kilian Raschel.

Title: Workload analysis of a two-queue fluid polling model


Abstract: We analyze a two-queue random time-limited Markov modulated polling model. Fluid arrives at the two queues as two independent flows with deterministic rates. A single server serves both queues at constant speeds, alternatingly spending an exponentially distributed amount of time in each queue and experiencing no switchover time.
We derive a functional equation for the Laplace-Stieltjes transform of the stationary joint fluid content (workload) distribution, that leads to a Riemann-Hilbert boundary value problem (BVP). After taking a heavy-traffic limit, and restricting ourselves to the symmetric case, the boundary value problem simplifies and can be solved explicitly.
Next, allowing for more general (Levy) input processes and server switching policies, we investigate the transient process-limit of the joint workload in heavy traffic, exploiting the RBM theory. Again solving a BVP, we determine the stationary distribution of the limiting process. We show that, in the symmetric case, this distribution coincides with our earlier BVP solution, implying that in this case the two limits (stationarity and heavy traffic) commute.

Joint work with Stella Kapodistria, Offer Kella and Mayank Saxena.

Title: Positive Recurrent Semimartingale Reflecting Brownian Motion with Divergent Fluid Paths

Abstract: Fluid paths correspond to the paths of a semimartingale reflecting Brownian motion (SRBM), but with the randomness removed. Their behavior is therefore often easier to analyze than that of the corresponding SRBM. In Dupuis and Williams (1994), it was showed that, in any dimension d, if every fluid path is attracted to the origin, then the corresponding SRBM must be positive recurrent.

It remained unknown whether the converse direction always holds. In d=2, the proof that the converse holds is relatively straightforward, but the proof in d=3 is more complex. A similar approach in d=4 already becomes unmanageable because of the many possible cases that need to be analyzed.

In Bramson (2011), examples were given, in d>5, with divergent fluid paths whose corresponding SRBM are positive recurrent. In these examples, the covariance matrix for the SRBM is the identity matrix I. The existence of examples in d=4 and d=5 was left open. Heyda has recently produced examples in d=4 and d=5 with divergent fluid paths whose corresponding SRBM are positive recurrent. For the examples in d=5, the covariance matrix is I; for the examples in d=4, a different covariance matrix is needed.

In this talk, we will summarize the above results in d>5, and how related techniques can be applied in d=4 and d=5.

Title: Brownian motion with asymptotically normal reflection in unbounded domains: from transience to stability


Abstract: We quantify the asymptotic behaviour of multidimensional drifltess diffusions in domains unbounded in a single direction, with asymptotically normal reflections from the boundary. We identify the critical growth/contraction rates of the domain that separate stabil-
ity, null recurrence and transience. In the stable case we prove existence and uniqueness of the invariant distribution and establish the polynomial rate of decay of its tail. We also establish matching polynomial upper and lower bounds on the rate of convergence to stationarity in total variation. All exponents are explicit in the model parameters that determine the asymptotics of the growth rate of the domain, the interior covariance, and the reflection vector field.
Proofs are probabilistic, and use upper and lower tail bounds for additive functionals up to return times to compact sets, for which we develop novel sub/supermartingale criteria, applic- able to general continuous semimartingales. Narrowing domains fall outside of the standard literature, in part because boundary local time can accumulate arbitrarily rapidly. Establishing Feller continuity (essential for characterizing stability) thus requires an extension of the usual approach.
Our recurrence/transience classification extends previous work on strictly normal reflections, and expands the range of phenomena observed across all dimensions. For all recurrent cases, we provide quantitative information through upper and lower bounds on tails of return times to compact sets. 

(Based on joint work with Aleks Mijatovic and Andrew Wade)

Title: Invariant Measures of the Infinite Atlas Model: Domains of Attraction, Extremality, and Equilibrium Fluctuations.


Abstract: The infinite Atlas model describes a countable system of competing Brownian particles where the lowest particle gets a unit upward drift and the rest evolve as standard Brownian motions. The stochastic process of gaps between the particles in this model is described by an infinite dimensional Skorohod problem. This Markov process does not have a unique stationary distribution and in fact there is a one parameter family {p(a),a>= 0} of product form mutually singular stationary distributions.
We say that an initial distribution of gaps is in the weak domain of attraction of the stationary measure p(a) if the time averaged laws of the stochastic process of the gaps, when initialized using that distribution, converge to p(a) weakly in the large time limit. We provide general sufficient conditions on the initial gap distribution of the Atlas particles for it to lie in the weak domain of attraction of p(a) for each a. Results on extremality and ergodicity of p(a) will also be presented. Finally, we will introduce a SPDE that describes equilibrium fluctuations associated with p(a). This is based on joint works with Sayan Banerjee and Peter Rudzis.

Title: On the spine of Fleming-Viot process


Abstract: A Fleming-Viot process is a system of n particles driven by independent copies of a driving Markov process. When one of the particles hits the boundary of the domain, it is killed, and some other particle branches. There is only one infinite path (spine) in the branching structure. Under some assumptions, when n goes to infinity, the distribution of the spine converges to the distribution of the driving Markov process conditioned to avoid the boundary of the domain forever. Based on joint articles with M. Bieniek, J. Englander, and T. Tadic.

Title: Asymptotic product-form stationary distributions for reflected Brownian motions


Abstract: We prove that a sequence of multi-scaled stationary
distributions of reflected Brownian motions (RBMs) has a product-form limit. Each component in the limit is an exponential
distribution. The multi-scaling corresponds to the « multi-scale heavy
traffic » recently advanced in Dai, Glynn and Xu (2023) for generalized Jackson networks. The proof utilizes the basic adjoint relationship (BAR) first introduced in Harrison and Williams (1987) that characterizes the stationary distribution of an RBM. This is a joint work with Jin Guang at CUHK-Shenzhen, Jim Dai at Cornell, and Peter Glynn at Stanford.

Title: Convergence of reflected Browinian motions on generalized Sierpinski carpets


Abstract: Let F be a generalized Sierpinski carpet inside a d-dimensional unit hypercube with d\geq 2

and F_n be its n-stage approximation. Denote by d_w and L\geq 3 the walk dimension and the length scale of the carpet F.  Let X^n be the normally reflected Brownian motion on F_n running at speed L^{(d_w-2)n}}. In this talk, we show that X^n converges weakly to a Brownian motion on F. We further show that the effective resistance between two opposite faces of F_n with respect to X^n converges to a positive constant as n tends to infinity. This gives a positive answer to an open problem of Barlow and Bass (1990). 

Based on a joint work with Shiping Cao.

Title: Obliquely reflecting diffusions in curved, nonsmooth domains


Abstract: Obliquely reflecting diffusions in curved, nonsmooth domains, arise, for instance, in diffusion approximation of certain stochastic networks (switched networks, networks operating under resource sharing). In a polyhedral domain with constant direction of reflection on each face, Dai and Williams (1996) find very general conditions on the directions of reflection under which reflecting Brownian motion can be constructed and uniquely characterized. These conditions are necessary for simple polyhedrons. For a curved, nonsmooth domain and varying direction of reflection, there were several important results in the 1980s and 1990s, but all of them under conditions restrictive in some respect.
In this talk, after briefly reviewing the main results in the literature, I will discuss a series of recent joint papers with T.G. Kurtz (published and unpublished) that present a new construction of semimartingale reflecting diffusions and new uniqueness results in some types of domains. In particular, we obtain existence and uniqueness in a 2-dimensional piecewise smooth domain, under very general, geometric, easily verifiable conditions. In fact our conditions are optimal in the sense that they reduce to those of Dai and Williams (1996) in the case of a polyhedron with constant direction of reflection on each face. Moreover our conditions allow for cusps in the boundary of the domain. The construction part of our work is based on constrained martingale problems, while the uniqueness part is based on a new reverse ergodic theorem for killed, inhomogeneous Markov chains that, in a sense, generalizes the Krein-Rutman theorem used in Kwon and Williams (1991).

Title: Asymptotic steady-state independence for generalized Jackson networks in multi-scale heavy traffic


Abstract: We prove that under a multi-scale heavy traffic condition, the stationary distribution of the scaled queue length process in any generalized Jackson network has a product-form limit. Each component in the product-form has an exponential distribution, corresponding to the Brownian approximation of a single station queue. Each “single station” can be constructed precisely and its parameters have a good intuitive interpretation. The proof employs a basic adjoint relationship (BAR)-approach that has been advanced recently by Braverman, Dai, and Miyazawa (2017, 2023). This is a joint work with Peter Glynn at Stanford and Yaosheng Xu at Amazon.

Title: Extremes of reflected Gaussian processes


Abstract: We derive exact asymptotics of the tail distribution of extreme-type functionals of reflected (at 0) Gaussian processes with stationary increments. Then, we turn to the multidimensional case for which we analyse asymptotic properties of the probability of hitting the upper orthant.

Title: A Pitman Theorem for a Brownian motion in the interval

Abstract: We will present a Pitman theorem for a Brownian motion in the interval and will explain why this theorem provides a very good example of a fruitful dialogue between probability theory and representation theory.

Title: Multidimensional singular control and related Skorokhod problem: sufficient conditions for the characterization of optimal controls


Abstract: We characterize the optimal control for a class of singular stochastic control problems as the unique solution to a related Skorokhod reflection problem. The optimization problems concern the minimization of a discounted cost over an infinite time-horizon through a process of bounded variation affecting an Itô-diffusion. The setting is multidimensional, the drift of the state equation and the costs are convex, the volatility matrix can be constant or linear in the state. Our result applies to a relevant class of linear-quadratic models and it allows to construct the optimal control in degenerate and non degenerate settings considered in the literature.

Title: On the diameter of the stopped spider process

Abstract: We consider the « Brownian spider » (Walsh Brownian motion), which is constructed as a set of half-lines, or “ribs,” meeting at a common point O. A Brownian motion on a spider starting at zero may be constructed from a standard reflecting Brownian motion by assigning an integer uniformly and independently to each excursion which is then transferred to an excursion on rib.

In the early 2000s, Lester Dubins posed the following problem: how can one design a stopping time to maximize the coverage of Brownian motion on the spider for a given expected time? Dubins et al. (2009) provided a solution to the optimal strategy for n=1,2 ribs, and Ernst (2016) gave an alternative solution to that presented by Dubins et al. (2009). However, neither Dubins et al. (2009) nor Ernst (2016) succeeded in providing a solution to the general question.

Over the past few years, we studied an alternative version of Dubins’ question in which the coverage or size of the spider process is measured differently. Instead of measuring the spider as the sum of the lengths of its ribs, we consider the ‘true’ diameter, that is, the sum of the two longest ribs. The resulting optimal stopping problem can, rather surprisingly, be completely solved. This work (to appear, Mathematics of Operations Research, 2023) is joint with E. Bednarz and A. Osekowski.

The solution to the original formulation of the Dubins’ problem remains open.

Title: Load Balancing in Parallel Queues and Rank-based Diffusions


Abstract: We analyze heavy traffic limits and stationary distributions of rank-based routing policies where incoming jobs are routed to servers with probability depending on their ranked queue-lengths. The limiting reflected diffusions are connected to the gaps between particles in rank-based diffusions like the Atlas model. We discuss in particular the special case where a small fraction of the incoming jobs is channeled to the shortest queue and the rest are uniformly routed, which we call the marginal join-the-shortest-queue (MJSQ) policy, and compare this to well-known load balancing schemes. This policy significantly reduces overhead communication costs compared to the asymptotically optimal join-the-shortest-queue policy, but we have shown that it nevertheless achieves comparable performance. Even when the whole system is unstable, we have shown that the gap between the maximum and minimum queue length under MJSQ is tight, exhibiting a novel form of load balancing. The stationary distributions of the limiting diffusion processes (for the general rank-based routing policies) are identified as product laws of Exponential random variables. The associated statistics of the pre-limiting process have been shown to converge to those of the limiting diffusion as the heavy traffic parameter goes to infinity.

Joint work with Sayan Banerjee and Amarjit Budhiraja.

Title: Reflected Brownian motions and random growth models


Abstract: I will discuss an identity between the invariant measure of a reflected system of Brownian motions and a vector of point-to-line last passage percolation times. By time reversal, this describes the distribution of the maximum over all time of the largest eigenvalue in Dyson Brownian motion with negative drifts. A positive temperature version relates the point-to-line partition functions of two directed polymers in an inverse-gamma and Brownian environment, and generalises Dufresne’s identity.

Title: Lévy driven non-linear Langevin type equations


Abstract: We will study a one-dimensional kinetic stochastic model driven by a stable Lévy process, and having a non-linear time-inhomogeneous drift. More precisely, a process $(V,X)$ is considered, where $X$ is the position of a particle having velocity $V$ the solution of a stochastic differential equation with a drift of the form $t^{-\beta}F(V)$ and driven by a stable process. The behaviour of the process $(V,X)$ will be described for two different situations : when the noise is small, or in large time but with fixed noise.

Title: Uniform Moment Bounds for Generalized Jackson Networks in Multi-scale Heavy Traffic


Abstract: We establish uniform moment bounds for steady-state queue lengths of generalized Jackson networks (GJN) in multi-scale heavy traffic as recently proposed by Dai et al. [2022]. Uniform moment bounds lay the foundation for further analysis of the limit stationary distribution. Our result can be used to verify the crucial moment state space collapse (SCC) assumption in Dai et al. [2022] to establish a product-form limit of GJN in the multi-scale heavy traffic regime. Our proof utilizes a novel basic adjoint relationship (BAR) approach to characterize the stationary distribution directly.

Title: Drift control of high-dimensional RBM: A computational method based on neural networks

Abstract: We consider a stochastic control problem whose state space is the d-dimensional non-negative orthant. The controlled process Z evolves as a reflected Brownian motion whose covariance matrix is exogenously specified, as are its directions of reflection from the orthant’s boundary surfaces. A controlling agent chooses a drift vector at each time t based on the history of Z, and the cost rate at time t depends on both Z(t) and the drift rate chosen. In our initial problem formulation, the objective is to minimize expected discounted cost over an infinite planning horizon, after which we treat the corresponding ergodic control problem. Extending earlier work by Han et al. (Proceedings of the National Academy of Sciences, 2018, 8505-8510), we develop and illustrate a simulation-based computational method that relies heavily on deep neural network technology. Our method is accurate to within a fraction of one percent, and is computationally feasible in dimensions up to at least d=20. (Based on joint work with Baris Ata and Nian Si.)

Title: An inverse Pitman’s theorem for a space-time brownian motion in a type A^1_1 Weyl chamber


Abstract: We present an inverse Pitman’s theorem for a space-time Brownian
motion conditioned in Doob’s sense to remain in an affine Weyl chamber. Our theorem provides a way to recover an unconditioned spacetime Brownian motion from a conditioned one applying a sequence of path transformations.

Title: Extended Lévy’s Theorem for a Two-Sided Reflection


Abstract: We aim to set forth an extension of the result found in paper [2], which finds an explicit realisation of a reflecting Brownian motion with drift −μ , started at x , reflecting above zero, and its local time at zero. In this paper we find a corresponding realisation for a reflecting Brownian motion with drift −μ , started at x , reflected both above zero and below one, along with a corresponding expression in terms of associated local times, namely as the difference between the local time at zero and the local time at one.

Title: Degenerate Competing Three-Particle Systems


Abstract: We study systems of three interacting particles, in which drifts and variances are assigned by rank. These systems are « degenerate »: the variances corresponding to one or two ranks can vanish, so the corresponding ranked motions become ballistic rather than diffusive. Depending on which ranks are allowed to « go ballistic », the systems exhibit markedly different behavior which we study in some detail. Also studied are stability properties for the resulting planar process of gaps between successive ranks.

Joint work with Ioannis Karatzas.

Title: The moments of polynomial functionals in Levy-driven queues


Abstract: Consider the problem of computing the moments of the area beneath the workload process of a stable M/G/1 queue during a busy period. This problem was introduced by Iglehart (1971) who solved the first moment. Later, Cohen (1978) solved the second moment and there are no existing results about higher moments. In this work, we solve this long-standing problem in the setup of a Levy-driven queue. Specifically, the main result of the current work is a new recursive formulae which yield all moments (including joint ones) of polynomial functionals of the workload process. This is a joint work with Peter Glynn and Michel Mandjes.

Title: Competing diffusive particle systems and models of large equity markets


Abstract: We introduce and study stable multidimensional diffusions interacting through their ranks. These interactions give rise to invariant measures which are in broad agreement with stability properties observed in large equity markets over long time-periods. The models we present assign growth rates and variances that depend on both the name (identity) and the rank (according to capitalization) of each individual asset. Such models are able realistically to capture certain critical features of the observed stability in capital distribution over the past century, all the while being simple enough to allow for rather detailed analytical study.

The methodologies used in this study touch upon the question of triple points for systems of competing diffusive particles; in particular, some choices of parameters may permit triple (or higher-order) collisions to occur. We show, however, that such multiple collisions have no effect on any of the stability properties of the resulting system. This is accomplished through a detailed analysis of collision local times.

The models have connections with the analysis of Queueing Networks in heavy traffic, with multi-dimensional Diffusions reflected on the faces of the positive orthant, and with competing particle systems in Statistical Mechanics (e.g., Sherrington- Kirkpatrick model for spin-glasses). Their hydrodynamic-limit behavior is governed by generalized porous medium equations with convection, and the fluctuations around these limits by appropriate linear stochastic partial differential equations of parabolic type with additive noise. Whereas, limits of a different kind display phase transitions and are governed by Poisson-Dirichlet distributions. We survey progress on some of these fronts, and suggest open problems for further study.

Title: Sensitivity analysis of reflected Brownian motion


Abstract: Reflected Brownian motion (RBM) arises in a variety of applications ranging from stochastic networks to mathematical finance and under general stability conditions, it has a unique stationary distribution. In such applications, it is useful to characterize the sensitivity of stationary performance measures with respect to the parameters that define the RBM. In this talk, we characterize these sensitivities in terms of the long time behavior of the RBM and its so-called derivative process, which characterizes pathwise derivatives of the RBM on finite time intervals. We show that this joint process is positive recurrent, has a unique stationary distribution and parametric sensitivities of the stationary distribution of an RBM can be expressed in terms of the stationary distribution of the joint process. Our proofs require analysis of path properties of coupled RBMs and contraction properties related to the geometry of the domain and directions of reflection. This work is joint with Kavita Ramanan.

Title: Lévy driven queues: the workload correlation function is positive, decreasing and convex


Abstract: In this talk I will consider Lévy-driven queues, i.e., reflected Lévy processes, generalising the class of reflected Brownian motions, with a focus on structural properties of the workload correlation function. After having introduced the objects studied, I’ll proceed by stating the conjecture that has been around for quite a while, namely that the workload correlation is a positive, decreasing and convex function of time. As a historic account, I’ll briefly discuss the seminal contribution by Ott on the special case of the M/G/1 queue, based on exploiting properties of complete monotone functions. The same methodology has been used in the extension (by Es-Saghouani and me) to queues with spectrally positive Lévy input, whereas later (in a paper by Glynn and me) the spectrally negative case was dealt with. For a long time, there was little hope to prove the conjecture for general Lévy input (and, for that matter, for reflected random walks in discrete-time). In a recent paper (that I wrote with Berkelmans and Cichocka), we provide an elementary proof, only relying on basic properties of Lévy processes and their reflected version. Importantly, the argumentation extends to double reflection, and also covers reflected random walks. Time permitting, I also discuss various ramifications due to Kella and me, and I comment on the question whether the structural properties carry over to the Markov modulated case.

Title: Parameter estimation for skew/sticky BM


Abstract: In this talk we consider some one-dimensional diffusions whose behavior is perturbed by a barrier-point. The perturbation nature of partial-reflection (skew BM) or sticky is encoded by a parameter.
Our goal is to discuss parameter estimation from high-frequency observations of a trajectory.
In particular, in the case of skew BM we discuss the main properties of the Maximum Likelihood Estimator (MLE) such as asymptotic mixed normality with a non standard rate of 1/4 and the behavior for all possible values of the parameter. Moreover, thanks to a recent asymptotic inverse function theorem, we give an infinite series expansion of the MLE.
This talk is partially based on a joint works with A. Anagnostakis (LJK Grenoble) and A. Lejay (IECL/Inria Nancy)

Title: Palm problems arising in BAR approach

Abstract: We consider a Markov process with time homogeneous transitions which is jointly stationary with multiple point processes. Assuming that these point processes have finite intensities, we define Palm distributions concerning them. Motivated by a BAR (Basic adjoint relationship) approach, which extensively uses the Palm distributions, we are interested in two problems; (a) when this stationary Markov process inherits the same Markov structure under the Palm distributions, and (b) how the state changes at counting instants of the point processes can be handled to derive stationary equations when there are simultaneous counts and each of them influences the state changes. We call (a) and (b) Palm problems.

For (a), we show that, if the Markov process is strong Markov, then it is again a strong Markov process under the Palm distribution, and its transition operator is unchanged. However, the jump transition of the Markov process at a counting instant of each point process may require the predictability of those instants. For (b), we make it clear under what framework it can be well handled. We also discuss how those results can be applied in deriving BAR’s for queueing networks.

Title: Asymptotics of Random Walks in the Quarter Plane

Abstract: The main question posed in this presentation will be about the behaviour of the number of lattice paths starting from a fixed point and ending at (possibly another) fixed point, where we let the number of steps go to infinity. As the scaling limit of a random walk is a Brownian motion, it is not all that surprising that these asymptotics closely resemble those of the equivalent continuous problem; something that has been shown for the first order terms in 2011 by Denisov and Wachtel.
In this presentation it will be shown that for finite, orbit-summable models a generalisation of their result holds, which allows us to write an asymptotic expansion as a series of polyharmonic functions with polynomial weights.

Title: Scaling Limits for Shortest Remaining Processing Time Queues


Abstract: We develop a heavy traffic limit theorem for the measure valued state descriptors for a sequence of shortest remaining processing time (SRPT) queues using nonstandard scaling.  In an SRPT queue, the server preemptively serves the job with the shortest remaining processing time first.  SRPT is inherently of interest due its optimality property as a queue length minimizer.  A natural state descriptor for an SRPT queue is a measure valued process which at a given time is the finite nonnegative Borel measure on the nonnegative real line that puts a unit atom at the remaining processing time of each job in system.  Gromoll, Kruk, and Puha (2011) proved that for processing time distributions with unbounded support, SRPT minimizes the queue length so efficiently that the heavy traffic diffusion limit for a sequence of SRPT queues satisfying natural asymptotic assumptions is the process that is identically equal to the zero measure, i.e., the limiting queue length process is identically zero.  In this work, we utilize an alternative scaling to obtain a nontrivial limit that involves both an adjustment to the spatial scaling, as well as a relocation mass to account for the growth of the remaining processing times.  For processing time distributions with finite second moments and regularly varying tails, we show that the (nonstandardly) scaled measure valued processes converge in distribution (in the space of paths of measures) to a measure valued limit process described in terms of a nonnegative valued random field determined from a single Brownian motion.  Puha (2015) previously considered processing time distributions with rapidly varying tails showing that the limiting measure is an atom at one, of randomly varying total mass.  In contrast, the remaining processing times for regularly varying tails spread out, as described by the random field.

Joint with Sayan Banerjee (UNC) and Amarjit Budhiraja (UNC)

Title: Pathwise differentiability of reflected diffusions


Abstract:  Differentiability of flows and sensitivity analysis are classical topics in dynamical systems. In the stochastic context, stochastic flows have been well studied by Kunita and others.  However, the analysis of these properties for constrained or reflected processes, which arise in a variety of applications, is challenging due to the discontinuous dynamics at the boundary of the domain, and is further complicated when the boundary is non-smooth. We show that the study of both differentiability of flows and sensitivities of constrained processes in convex polyhedral domains can be largely reduced to the study of directional derivatives of an associated map, called the Skorokhod map, and we introduce an axiomatic framework to characterize these directional derivatives. In addition, we establish pathwise differentiability of a large class of reflected diffusions in convex polyhedral domains and show that they can be described in terms of certain constrained stochastic differential equations with time-varying domains and directions of reflection.  This is based on joint works with David Lipshutz.

Title: RBM with Drift in a Wedge


Abstract: We study reflecting Brownian motion with drift constrained to a wedge in the plane. Our first set of results provide necessary and sufficient conditions for existence and uniqueness of a solution to the corresponding submartingale problem with drift. Next, we study a version of the problem with absorption at the vertex of the wedge. In this case, we provide a condition for existence and uniqueness of a solution to the problem and some results on the probability of the vertex being reached. We also prove that in the case of 1 < alpha < 2, RBM in a wedge is a Dirichlet process. Specifically, its unique Doob-Meyer type decomposition is given by Z=X+Y, where X is a two-dimensional Brownian motion and Y is a continuous process of zero energy. Furthermore, we show that for p > alpha , the strong p-variation of the sample paths of Y is finite on compact intervals, and, for 0 < p <= alpha,  the strong p-variation of Y is infinite on [0,T] whenever Z has been started from the origin. We also show that on excursion intervals of Z away from the origin, (Z,Y) satisfies the standard Skorokhod problem for X. However, on the entire time horizon (Z,Y) does not satisfy the standard Skorokhod problem for X, but nevertheless we show that it satisfies the extended Skorkohod problem.

Title: Parallel Server Systems under an Extended Heavy Traffic Condition

Abstract: Parallel server systems (PSS) are queueing control problems with a relatively simple structure. There are I classes of customers and K servers, and the service rate depends on the class-server pair. The controls involve both deciding which server serves each customer, as well as the order in which they are served. The objective is to minimize the expected infinite horizon total discounted cost, where the cost rate is a linear function of the queue length vector. A linear program (LP) known as the static allocation problem was introduced in order to define heavy traffic for these systems. Previous work on PSS made the key assumption that the static allocation LP has a unique solution. We relax this assumption, introducing the extended heavy traffic condition, which allows multiple solutions.
The existence of multiple solutions to the LP introduces complications in the treatment of this problem. In particular, there is now freedom to choose among the LP solutions, which determine the fluid scale allocation of effort by the servers. This non-deterministic fluid limit leads to technical problems along with a more complicated limiting workload control problem (WCP). Under the additional assumption of complete resource pooling, we provide an asymptotic lower bound on the achievable cost in terms of the solution of the WCP. In addition, for the simple special case of I=K=2, where the WCP is solvable, we translate the optimal control into a control policy for the original PSS and prove that it is asymptotically optimal.
(Based on joint work with Rami Atar and Eyal Castiel.)

Title: CLT for reflecting Brownian motion in generalized parabolic domains


Abstract: We consider an obliquely reflected Brownian motion in a 1+d dimensional domain given by the points (x,y) such that |y|_d is smaller than a function which asymptotically grows as a (possibly negative) power of x. Depending on that power and on the reflection field, the resulting process has a different behaviour at the first order: the x component can either explode in finite time, or be diffusive, or diverge toward infinity as any power of t greater than a half. In this last intermediate case, we will explain this first order behaviour as well as the central limit theorem that comes after.

This talk is based on a joint work with A. Mijatović and A. Wade.

Title: Universal order statistics for random walks


Abstract: I will study the statistics of the gaps $\Delta_{k,n}$ between the $k^\text{th}$ and $(k+1)^\text{th}$ maximum of the set of positions $\{x_1,\ldots,x_n\}$ of a random walk after $n$ steps, starting from the origin. I will discuss the case of continuous jump distributions $f(\eta)$, with a corresponding Lévy index $0 < \mu \leq 2$. I will show that one can obtain an exact analytical expression for the probability distribution $P_{k,n}(\Delta)$ valid for any $k$ and $n$, and jump distribution $f(\eta)$, which is amenable to an asymptotic large $n$ analysis. In this limit, one finds that the gap distribution becomes stationary, i.e., $\lim_{n\to \infty} P_{k,n}(\Delta)=P_k(\Delta)$. Furthermore, for $\mu>1$, I will show that in the limit of $k\to\infty$ the stationary distribution exhibits a universal scaling form $P_k(\Delta) \sim k^{1-1/\mu} \mathcal{P}_\mu(k^{1-1/\mu}\Delta)$ which depends only on the Lévy index $\mu$, but not on the details of the jump distribution. This limiting scaling function $\mathcal{P}_\mu(x)$ can be computed explicitly in terms of Mittag-Leffler functions.

Title: Biochemical reaction networks and reflecting diffusion processes


Abstract: In 2019, Leite and Williams proposed certain reflected diffusion processes as approximations to continuous time Markov chain models frequently used to model biochemical reaction networks. These diffusions live in the positive orthant of a d-dimensional space and are confined there by a smoothly varying oblique reflection field on the boundary. Leite and Williams showed that, under mild conditions, these diffusions can be obtained as weak limits of certain jump-diffusion extensions of the traditional Langevin approximations, and therefore called these constrained Langevin approximations. In this talk, we will review this approximation and describe some progress on proving error estimates for strong versions of this approximation and also describe some remaining open problems. Part of this work is joint with Felipe Campos.

Title: Routing in a Queueing Loss Model under Imperfect Classifications


Abstract: Motivated by reducing the recidivism (reoffense) rate in criminal justice we study a multiclass loss model (multiclass G/M/s/s+M) to help determine admission decisions to incarceration diversion programs that have limited capacities, which serves as an alternative to incarceration and could lower the likelihood of recidivism but cannot prevent reoffense while in the program. Service time in the model corresponds to time in the diversion program, and reneging corresponds to an offense being committed while in service. A (potentially imperfect) classification algorithm predicts which individuals may benefit the most from the programs and which individuals have the most risk to recidivate. The admission decisions aim to minimize the long-run average total cost, which consists of recidivism cost, diversion program cost, and incarceration cost. The major trade-off is to whether admit the current arriving individual or reserve the spot for upcoming individuals who could potentially benefit more.
We first formulate a fluid control problem and show that the solution to that fluid control problem motivates using a simple prioritization scheme to determine admission decisions, assuming either no prediction error or existence of a stochastic ordering property. Next, moving to diffusion control, under a heavy-traffic regime, we approximate the system state (program population for different classes) by a multi-dimensional reflected Brownian motion in a rectangle with lower left-hand corner at the origin, and dimension equal to the number of classes. For a single class setting, the one-dimensional diffusion problem can be analytically solved. However, for the multiclass setting, the multidimensional diffusion control problem is analytically very challenging, and we are investigating potential decomposition heuristics leveraging the priority schedule from the fluid control to solve approximately.


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Title: Brownian motion with asymptotically normal reflection in unbounded domains: from transience to stability


Title: Wasserstein bounds in the CLT of the MLE for the drift coefficient of a stochastic partial differential equation


Title: Reflected Brownian Motion in a wedge :sum-of-exponential absorption densities


Title: Uniform Moment Bounds for Generalized Jackson Networks in Multi-scale Heavy Traffic


Title: Reflected Brownian motion, transient case and asymptotics of Green functions


Title: Routing in a Queueing Loss Model under Imperfect Classifications