John Hasenbein
Fluid Models: A Memoir
In this retrospective talk, I will discuss the impact and scope of Jim's fluid model results through the lens of our time working together. In addition, I will survey some of my favorite "fluid model" results over the years that illustrate the power and beauty of this approach. Much of this talk will be accessible to a general OR audience.
Peter Glynn
The Limit Interchange Problem for Markov Processes
Suppose that a sequence of Markov processes converges to a limiting Markov process over finite time intervals. When these processes are positive recurrent, it is natural to ask when their stationary distributions converge. In the setting of diffusion approximations for queueing networks, this question was first explored by Gamarnik and Zeevi (2006), and has subsequently been studied by many others, including Jim Dai and his collaborators. In this talk, we will discuss various aspects of this problem, including a stronger “diagonal” version of this interchange property which allows the limiting stationary distribution to be used as an approximation to the large-time transient distributions of the pre-limit systems. We will describe a framework for establishing this stronger interchange property, and discuss connections between the limit interchange problem and several related computational methods.
This talk is based on joint work with Jose Blanchet, Yanlin Qu, and Zeyu Zheng.
Ruth J. Williams
What a difference a little reflection can make!
Brownian motions with oblique reflection at the boundary of the positive orthant arise as scaling limits of queueing networks and more general stochastic processing networks. In these applications, typically the reflection field is constant on each boundary face. Such reflected diffusion processes have provided measures of performance and led to elegant solutions of some dynamic scheduling problems for stochastic processing networks. Indeed, in some problems, oblique reflection at the boundary has translated to improved performance through (rarely used) non-basic activities. In a different application context, reflected diffusion processes with a smoothly varying reflection field on the boundary of the positive orthant have been proposed as approximations for continuous time Markov chains used in modeling biochemical reaction networks. In this talk, we will review these diffusion approximations, describe some recent progress on proving error estimates for strong versions of these approximations, and explore the role of reflection (even when small in magnitude) in these approximations.
Part of this work is joint with Felipe Campos and Zhaolong Han.
Barış Ata
A Computational Method for Solving the Stochastic Joint Replenishment Problem in High Dimensions
We consider a discrete-time formulation for a class of high-dimensional stochastic joint replenishment problems. First, we approximate the problem by a continuous-time impulse control problem. Exploiting connections among the impulse control problem, backward stochastic differential equations (BSDEs) with jumps, and the stochastic target problem, we develop a novel, simulation-based computational method that relies on deep neural networks to solve the impulse control problem. Based on that solution, we propose an implementable inventory control policy for the original (discrete-time) stochastic joint replenishment problem, and test it against the best available benchmarks in a series of test problems. For the problems studied thus far, our method matches or beats the best benchmark we could find, and it is computationally feasible up to at least 50 dimensions - that is, 50 stock-keeping units (SKUs). If time permits, I will also mention follow up work.
Joint work with Wouter J.E.C. van Ekelen and Yuan Zhong
J. Michael Harrison
Three Kinds of Value Derived from Stochastic Models
Stochastic models can yield value in any or all of three distinct modes: providing the core logic for online system control (operational models); as tools for quantitative offline decision support (planning models); and providing qualitative insights into the mechanisms that underlie system performance (conceptual models). Examples of each value mode will be presented, with emphasis on the last one, and some milestone developments in the field will be reviewed.
Based on joint work with Peter Glynn
Onno Boxma
Threshold models in queueing/risk/storage with an alternating L´evy input
We first study a queueing process for which the dynamics are changed once the workload in the queue exceeds a predefined threshold, and these new dynamics stay in force until the queue is emptied, at which point the previous dynamics are again reinstalled. For general spectrally positive L´evy processes in each case, we derive expressions for the workload at an independent exponentially distributed random time horizon, and study in detail properties of the workload in stationarity. We work out explicit formulas as well as expressions for the optimal changing threshold for special cases of the underlying process dynamics and a chosen set of involved switching, holding and service cost functions.
Joint work with Hansjoerg Albrecher, Offer Kella and Michel Mandjes.
If time permits, there will also be a brief discussion of ongoing work with Michel Mandjes, Daniel Rutgers and Werner Scheinhardt on the transient behavior of a Cram´er-Lundberg insurance risk model (and its queueing counterpart), with the added feature that the process changes dynamics above a predefined threshold, and changes back to the old regime as soon as that threshold is again crossed from above.
Assaf Zeevi
Playing Games and Sharp Prophet Inequalities
We will discuss a well studied class of finite horizon optimal stopping problems: a player observes a sequence of iid random variables, and the objective is to design stopping rules that attempt to select the random variable with the highest value in the sequence. The performance of any stopping rule may be benchmarked relative to the selection of a “prophet" that has perfect foreknowledge of the largest value. Such comparisons are typically stated in the form of “prophet inequalities.” There is a fairly large literature that focuses on such problems and establishes various bounds on the relative loss of the “mortal” vs the “prophet.” These results tend to be almost exclusively asymptotic (in the horizon) and case-specific insofar as the problem primitives. In this talk we develop a more principled approach that allows one to compute in an efficient manner sharp non asymptotic prophet inequalities along with optimal stopping rules.
Based on Joint work with A. Goldenshluger and Y. Malinovsky.
Amy Ward
Data-Driven Matching for Impatient and Heterogeneous Demand and Supply
This paper develops a framework that integrates finite-sample statistical learning with queueing asymptotic analysis to design matching policies. The stochastic matching model we consider assumes heterogeneous demand (customers) and heterogeneous supply (workers) arrive randomly over time, each with a randomly sampled patience time, and are lost (renege) if forced to wait longer than that time to be matched. Since the inter-arrival and patience time distributions are unknown, matching decisions must be made based on historical (offline) data. We leverage asymptotic analysis to formulate a deterministic, data-driven (fluid) matching problem (DDMP) that approximates the original stochastic matching problem. We establish finite-sample statistical guarantees on the objective value gap between the DDMP solution and the ground-truth matching problem solution, which requires a novel uniform error bound involving the patience time quantile function. We show that a discrete-review, estimate-then-match-type policy is epsilon-asymptotically optimal with high probability as arrival rates grow large.
Joint work with Weiliang Liu and Xun Zhang
