arXiv subjects

J. Michael Harrison

Publications and source records attributed to J. Michael Harrison.

5 recordsLinked to original sources

arXiv2026

Diffusion-Based Policies for Dynamic Control of Stochastic Processing Networks

We consider a processing network model with $m$ job classes or buffers, exogenous input flows into some classes, $n$ processing activities, and $p$ servers. Each activity is either a specified server processing jobs of a specified class, or a fictional input server delivering jobs of a specified class; jobs change class in Markovian fashion after completing service. A standard multiclass queueing network, with its one-to-one correspondence between job classes and activities, is a special case, but our general model allows two or more ways to process a given class, and some or all input flows may be turned away at the system manager's discretion. Costs are linear: a holding cost per time unit for each class $i$ job in the system, and a rejection penalty for each class $i$ arrival denied access $(i=1,\ldots,m)$. The system manager makes input control, job routing, and order-of-service decisions to minimize expected discounted costs over an infinite horizon. We formulate an approximating Brownian control problem (BCP) whose state space is the $m$-dimensional nonnegative orthant; control is a drift vector chosen from a bounded polyhedral set, based on dynamic state observations. Using recently developed computational methods, the BCP can be solved numerically in dimensions up to at least $m=50$, and we explain how the numerical solution is translated into an implementable control policy for the queueing system of original interest. Previous work on heavy traffic diffusion approximations suggests that this policy is nearly optimal in the heavy traffic parameter regime, and numerical examples support that conjecture. We also discuss its advantage over an alternative approach, featured in our previous work, where the BCP is replaced by a lower-dimensional "equivalent workload formulation" that is computationally efficient but difficult to interpret in the network of original interest.

math.OC↗

arXiv2024

Drift Control of High-Dimensional RBM: A Computational Method Based on Neural Networks

Motivated by applications in queueing theory, we consider a stochastic control problem whose state space is the $d$-dimensional positive 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 system manager chooses a drift vector $θ(t)$ at each time $t$ based on the history of $Z$, and the cost rate at time $t$ depends on both $Z(t)$ and $θ(t)$. 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. For test problems studied thus far, our method is accurate to within a fraction of one percent, and is computationally feasible in dimensions up to at least $d=30$.

eess.SY↗

arXiv2024

Singular Control of (Reflected) Brownian Motion: A Computational Method Suitable for Queueing Applications

Motivated by applications in queueing theory, we consider a class of singular stochastic control problems whose state space is the d-dimensional positive orthant. The original problem is approximated by a drift control problem, to which we apply a recently developed computational method that is feasible for dimensions up to d=30 or more. To show that nearly optimal solutions are obtainable using this method, we present computational results for a variety of examples, including queueing network examples that have appeared previously in the literature.

eess.SY↗

arXiv2022

Advertising Media and Target Audience Optimization via High-dimensional Bandits

We present a data-driven algorithm that advertisers can use to automate their digital ad-campaigns at online publishers. The algorithm enables the advertiser to search across available target audiences and ad-media to find the best possible combination for its campaign via online experimentation. The problem of finding the best audience-ad combination is complicated by a number of distinctive challenges, including (a) a need for active exploration to resolve prior uncertainty and to speed the search for profitable combinations, (b) many combinations to choose from, giving rise to high-dimensional search formulations, and (c) very low success probabilities, typically just a fraction of one percent. Our algorithm (designated LRDL, an acronym for Logistic Regression with Debiased Lasso) addresses these challenges by combining four elements: a multiarmed bandit framework for active exploration; a Lasso penalty function to handle high dimensionality; an inbuilt debiasing kernel that handles the regularization bias induced by the Lasso; and a semi-parametric regression model for outcomes that promotes cross-learning across arms. The algorithm is implemented as a Thompson Sampler, and to the best of our knowledge, it is the first that can practically address all of the challenges above. Simulations with real and synthetic data show the method is effective and document its superior performance against several benchmarks from the recent high-dimensional bandit literature.

cs.LG↗

arXiv2006

Correction. Brownian models of open processing networks: canonical representation of workload

Due to a printing error the above mentioned article [Annals of Applied Probability 10 (2000) 75--103, doi:10.1214/aoap/1019737665] had numerous equations appearing incorrectly in the print version of this paper. The entire article follows as it should have appeared. IMS apologizes to the author and the readers for this error. A recent paper by Harrison and Van Mieghem explained in general mathematical terms how one forms an ``equivalent workload formulation'' of a Brownian network model. Denoting by $Z(t)$ the state vector of the original Brownian network, one has a lower dimensional state descriptor $W(t)=MZ(t)$ in the equivalent workload formulation, where $M$ can be chosen as any basis matrix for a particular linear space. This paper considers Brownian models for a very general class of open processing networks, and in that context develops a more extensive interpretation of the equivalent workload formulation, thus extending earlier work by Laws on alternate routing problems. A linear program called the static planning problem is introduced to articulate the notion of ``heavy traffic'' for a general open network, and the dual of that linear program is used to define a canonical choice of the basis matrix $M$. To be specific, rows of the canonical $M$ are alternative basic optimal solutions of the dual linear program. If the network data satisfy a natural monotonicity condition, the canonical matrix $M$ is shown to be nonnegative, and another natural condition is identified which ensures that $M$ admits a factorization related to the notion of resource pooling.

math.PR↗