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

Baris Ata, J. Michael Harrison, Nian Si
2024 arXiv   pre-print

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2024 pre-print
arXiv:2312.11823v2 fatcat:m3e4pveukrc45jbmi4f5zndkre
Motivated by applications in queueing theory, we consider a class of singular stochastic control problems whose state space is the d-dimensional positive orthant.  ...  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  ...  described as singular control of Brownian motion, or of geometric Brownian motion in some financial applications, or of reflected Brownian motion in some queueing applications.  ... 
arXiv:2312.11823v3 fatcat:yardggdosfe5jbkrzvom2zfw7a
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Baris Ata, J. Michael Harrison, Nian Si. "Singular Control of (Reflected) Brownian Motion: A Computational Method Suitable for Queueing Applications." arXiv (2024)

Stochastic Networks and Reflecting Brownian Motion: The Mathematics of Ruth Williams

Ioana Dumitriu, Todd Kemp, Kavita Ramanan
2022 Notices of the American Mathematical Society  

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Constraining Brownian motion to stay in a region by pushing it in the allowed "reflection" or constraint directions at the boundary can be thought of as a stochastic control problem, with highly singular  ...  The construction in (1) yields the reflecting Brownian motion 𝑍 as a continuous function of the driving Brownian motion 𝐵.  ... 
doi:10.1090/noti2437 fatcat:sfrj4zb7ena6pa3agoxsavk6pm
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Ioana Dumitriu, Todd Kemp, Kavita Ramanan. "Stochastic Networks and Reflecting Brownian Motion: The Mathematics of Ruth Williams." Notices of the American Mathematical Society 69.03 (2022)

A guide to Brownian motion and related stochastic processes [article]

Jim Pitman, Marc Yor
2018 arXiv   pre-print

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This is a guide to the mathematical theory of Brownian motion and related stochastic processes, with indications of how this theory is related to other branches of mathematics, most notably the classical  ...  broader perspective on the theory of Brownian motion and related stochastic processes than can be found in these texts.  ...  See Harrison [165] for further treatment of reflecting Brownian motion and its applications to stochastic flow systems. See also R.  ... 
arXiv:1802.09679v1 fatcat:yyzymarn4belbnkqjsyykbwoim
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Jim Pitman, Marc Yor. "A guide to Brownian motion and related stochastic processes." arXiv (2018)

Singular control with state constraints on unbounded domain

Rami Atar, Amarjit Budhiraja
2006 Annals of Probability  

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2006 accepted
arXiv:math/0501016v2 fatcat:jizctk26hbh37gvp4de4tl4xym
2005 pre-print
arXiv:math/0501016v1 fatcat:76iyaeau6zbctcyqkhptq5v4u4
The proofs combine probabilistic arguments and viscosity solution methods. Our framework covers a wide range of diffusion control problems that arise from queueing networks in heavy traffic.  ...  Here, x∈X, B is a Brownian motion with drift b and covariance Σ, G is a fixed matrix, and Y^∘ is the Radon--Nikodym derivative dY/d|Y|. Let L=-(1/2)trace(Σ D^2)-b· D where D denotes the gradient.  ...  Part of the work on this paper was done when the second author was visiting the Technion. Hospitality of the EE Department at the Technion is gratefully acknowledged.  ... 
doi:10.1214/009117906000000359 fatcat:7q77gtnfizhxvnlki3u67dp7iq
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Rami Atar, Amarjit Budhiraja. "Singular control with state constraints on unbounded domain." Annals of Probability 34.5 (2006) 1864-1909

Dynamic scheduling in manufacturing systems using Brownian approximations

K Ravikumar, Y Narahari
1994 Sadhana (Bangalore)  

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It then provides a review of relevant weak convergence concepts, followed by a description of the limiting behaviour of queueing systems under heavy traffic.  ...  Recently, Brownian networks have emerged as an effective stochastic model to approximate multicIass queueing networks with dynamic scheduling capability, under conditions of balanced heavy loading.  ...  We would also like to acknowledge the encouragement and comments of Professor N. Viswanadham and several critical comments of the reviewers of this paper.  ... 
doi:10.1007/bf02743936 fatcat:ldawphsoebgsvf7v5vc5waswb4
Citation

K Ravikumar, Y Narahari. "Dynamic scheduling in manufacturing systems using Brownian approximations." Sadhana (Bangalore) 19.6 (1994) 891-939

Workload reduction of a generalized Brownian network

J. M. Harrison, R. J. Williams
2005 The Annals of Applied Probability  

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2006 pre-print
arXiv:math/0602495v1 fatcat:mecfmoc4erehlghgnoanhwtiwa
The RBCP is a singular stochastic control problem, in which both the controls and the cumulative cost process are locally of bounded variation.  ...  We show that the BCP is equivalent, in a certain sense, to a reduced Brownian control problem (RBCP) of lower dimension.  ...  For Brownian network data arising from a certain class of stochastic processing network models, a method for reducing the choices for M and G to a finite set was described in [9] .  ... 
doi:10.1214/105051605000000458 fatcat:aiijvfeoqravxb7gfpum377uty
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J. M. Harrison, R. J. Williams. "Workload reduction of a generalized Brownian network." The Annals of Applied Probability 15.4 (2005) 2255-2295

Light tail asymptotics in multidimensional reflecting processes for queueing networks

Masakiyo Miyazawa
2011 TOP - An Official Journal of the Spanish Society of Statistics and Operations Research  

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For this, we consider tail asymptotics of the stationary distributions. The purpose of this paper is twofolds. We first overview the current approaches to attack the problem from a unified viewpoint.  ...  We mainly consider the tail asymptotics for two-dimensional reflecting processes, but also discuss how we can approach the case of more than two dimensions.  ...  In particular, I am grateful to Jim Dai, Robert Foley, David McDonald, Tomasz Rolski and Yiqiang A part of this paper was prepared during my stay at Cambridge in 2010 for attending the research program  ... 
doi:10.1007/s11750-011-0179-7 fatcat:qdhzaccyyzfnjcglvcbrdprcki
Citation

Masakiyo Miyazawa. "Light tail asymptotics in multidimensional reflecting processes for queueing networks." TOP - An Official Journal of the Spanish Society of Statistics and Operations Research 19.2 (2011) 233-299

Scheduling Networks of Queues: Heavy Traffic Analysis of a Multistation Network with Controllable Inputs

Lawrence M. Wein
1992 Operations Research  

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Linear programming is used to reduce the workload formulation of this control problem to a constrained singular control problem for a multidimensional Brownian motion.  ...  control problem for a one-dimensional Brownian motion, and (3) finding a closed-form solution to the constrained singular control problem.  ...  Acknowledgements This research wets partially supported by a Junior Faculty Grant from the Leaders for Manufacturing Program at MIT.  ... 
doi:10.1287/opre.40.3.s312 fatcat:ii6325v4zbdbbi5mlqznjps7jq
Citation

Lawrence M. Wein. "Scheduling Networks of Queues: Heavy Traffic Analysis of a Multistation Network with Controllable Inputs." Operations Research 40.3-supplement-2 (1992) S312-S334

Iterated foldings of discrete spaces and their limits: candidates for the role of Brownian map in higher dimensions [article]

Luca Lionni, Jean-François Marckert
2020 arXiv   pre-print

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2019 pre-print
arXiv:1908.02259v1 fatcat:tobglgoo65cftmprye7lfa2xqq
The construction relies on some objects that we name iterated Brownian snakes, which are branching analogues of iterated Brownian motions, and which are moreover limits of iterated discrete snakes.  ...  In this paper, we present a sequence of random objects that we call Dth-random feuilletages (denoted by r[D]), indexed by a parameter D≥ 0 and which are candidate to play the role of the Brownian map in  ...  tree of a branching Brownian motion.  ... 
arXiv:1908.02259v2 fatcat:sqfbdluomjfxrmu25ehuwwa5ua
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Luca Lionni, Jean-François Marckert. "Iterated foldings of discrete spaces and their limits: candidates for the role of Brownian map in higher dimensions." arXiv (2020)

Twentieth conference on stochastic processes and their applications

1992 Stochastic Processes and their Applications  

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Reflected Brownian motions in polyhedral domains In this talk, the current state of knowledge and recent research on reflected Brownian motions in polyhedral domains (RBM's) will be described.  ...  The optimal control for this problem has a simple description in terms of the following indexfunctions: l;(q)= " I q,:'(u) du. 0 Namely, an optimal control runs, at each point in time, any Brownian motion  ...  It is the case for instance for the Poisson process and the family of Poisson distributions on -1; or for the standard Brownian motion and the family of Gaussian distributions with mean 0.  ... 
doi:10.1016/0304-4149(92)90130-i fatcat:jrcpx42brbezdhyxzbdpicmlda
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"Twentieth conference on stochastic processes and their applications." Stochastic Processes and their Applications 41.2 (1992) 325-363

Coupling, local times, immersions

Wilfrid S. Kendall
2015 Bernoulli  

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2012 pre-print
arXiv:1212.1670v1 fatcat:4l7bet4ysndbfdfmduy62coroi
2012 pre-print
arXiv:1212.1670v2 fatcat:p77b6c77u5dfjizuweegppfehu
2015 accepted
arXiv:1212.1670v4 fatcat:l7qabc4s65cjjozrc5bonlxmgq
2014 pre-print
arXiv:1212.1670v3 fatcat:oahqadowejflpntcsceq6ez7rm
This survey is followed by a detailed case-study of the simpler but potentially thematic problem of coupling Brownian motion together with its local time at 0.  ...  at least when not started at pairs of initial points lying in a certain singular set.  ...  Michel Émery, whose question concerning BKR diffusions encouraged me to develop the initial idea of coupling Brownian motion together with local time, and who pointed out how to improve the last part of  ... 
doi:10.3150/14-bej596 fatcat:sykheidygfcnvjw7filddof6se
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Wilfrid S. Kendall. "Coupling, local times, immersions." Bernoulli 21.2 (2015) 1014-1046

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

Baris Ata, J. Michael Harrison, Nian Si
2024 arXiv   pre-print

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2023 pre-print
arXiv:2309.11651v1 fatcat:3jp7yoyeardsjkmgbpzd6qyoaq
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.  ...  Motivated by applications in queueing theory, we consider a stochastic control problem whose state space is the d-dimensional positive orthant.  ...  In particular, a limit theorem proved by Reiman [39] justifies the use of d-dimensional reflected Brownian motion (RBM) as an approximate model of a d-station queueing network.  ... 
arXiv:2309.11651v3 fatcat:uzfb4pskbfd2ffcbdzafioicyi
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Baris Ata, J. Michael Harrison, Nian Si. "Drift Control of High-Dimensional RBM: A Computational Method Based on Neural Networks." arXiv (2024)

A superharmonic vector for a nonnegative matrix with QBD block structure and its application to a Markov modulated two dimensional reflecting process [article]

Masakiyo Miyazawa
2015 arXiv   pre-print

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2015 pre-print
arXiv:1501.03913v2 fatcat:aptxada2efenvniey4cgs7sjyy
2015 pre-print
arXiv:1501.03913v1 fatcat:bdniccqieratnlhdvnbtekjkse
In this spirit, we modulate a two dimensional reflecting skip-free random walk in such a way that its state transitions in the boundary faces and interior of a nonnegative integer quadrant are controlled  ...  Markov modulation is versatile in generalization for making a simple stochastic model which is often analytically tractable to be more flexible in application.  ...  This work is supported by Japan Society for the Promotion of Science under grant No. 24310115. A part of this work was presented at a workshop of Sigmetrics 2014 (see its abstract [29] ).  ... 
arXiv:1501.03913v3 fatcat:azzbywcoira6hbktcr4c6i24jy
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Masakiyo Miyazawa. "A superharmonic vector for a nonnegative matrix with QBD block structure and its application to a Markov modulated two dimensional reflecting process." arXiv (2015)

Continuous-time modeling of random searches: statistical properties and inference

Reiichiro Kawai
2012 Journal of Physics A: Mathematical and Theoretical  

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Among the proposed models, the Brownian motion is most tractable in various ways while its Gaussianity and infinite variation of sample paths do not well describe the reality.  ...  To address this issue, in this paper, we propose to model the continuoustime search paths directly with a continuous-time stochastic process for which the observer makes statistical inference based on  ...  Stable Lévy Motion The stable Lévy motion has been used in several fields of application, such as statistical physics, queueing theory, mathematical finance, to mention just a few.  ... 
doi:10.1088/1751-8113/45/23/235004 fatcat:n3dayckkuvdshaoqoyyqrk7pju
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Citation

Reiichiro Kawai. "Continuous-time modeling of random searches: statistical properties and inference." Journal of Physics A: Mathematical and Theoretical 45.23 (2012) 235004

Optimum feedback strategy for access control mechanism modelled as stochastic differential equation in computer network

N. U. Ahmed, Cheng Li
2004 Mathematical Problems in Engineering  

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We consider optimum feedback control strategy for computer communication network, in particular, the access control mechanism.  ...  This is very important for optimization of high-dimensional systems serving a large number of aggregate users.  ...  Acknowledgment This work was partially supported by the National Science and Engineering Research Council of Canada under Grant A7109.  ... 
doi:10.1155/s1024123x04403081 fatcat:q2pnlzbubrhb7oluvnwwvvzd4u
DOAJ Szczepanski
Citation

N. U. Ahmed, Cheng Li. "Optimum feedback strategy for access control mechanism modelled as stochastic differential equation in computer network." Mathematical Problems in Engineering 2004.3 (2004) 263-276

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