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Stationary Distribution Convergence for Generalized Jackson Networks in Heavy Traffic
 Mathematics of Operations Research
"... In a recent paper [5] it was shown that under suitable conditions stationary distributions of the (scaled) queue lengths process for a generalized Jackson network converge to the stationary distribution of the associated reflected Brownian motion in the heavy traffic limit. The proof relied on certa ..."
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Cited by 18 (3 self)
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In a recent paper [5] it was shown that under suitable conditions stationary distributions of the (scaled) queue lengths process for a generalized Jackson network converge to the stationary distribution of the associated reflected Brownian motion in the heavy traffic limit. The proof relied on certain exponential integrability assumptions on the primitives of the network. In this note we show that the above result holds under much weaker integrability conditions. We provide an alternative proof of this result making (in addition to natural heavy traffic and stability assumptions) only standard independence and square integrability assumptions on the network primitives that are commonly used in heavy traffic analysis. Furthermore, under additional integrability conditions we establish convergence of moments of stationary distributions.
Convergent numerical scheme for singular stochastic control with state constraints in a portfolio selection problem
 SIAM Journal on Control and Optimization
"... Abstract. We consider a singular stochastic control problem with state constraints that arises in problems of optimal consumption and investment under transaction costs. Numerical approximations for the value function using the Markov chain approximation method of Kushner and Dupuis are studied. The ..."
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Cited by 7 (1 self)
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Abstract. We consider a singular stochastic control problem with state constraints that arises in problems of optimal consumption and investment under transaction costs. Numerical approximations for the value function using the Markov chain approximation method of Kushner and Dupuis are studied. The main result of the paper shows that the value function of the Markov decision problem (MDP) corresponding to the approximating controlled Markov chain converges to that of the original stochastic control problem as various parameters in the approximation approach suitable limits. All our convergence arguments are probabilistic; the main assumption that we make is that the value function be finite and continuous. In particular, uniqueness of the solutions of the associated HJB equations is neither needed nor available (in the generality under which the problem is considered). Specific features of the problem that make the convergence analysis nontrivial include unboundedness of the state and control space and the cost function; degeneracies in the dynamics; mixed boundary (Dirichlet–Neumann) conditions; and presence of both singular and absolutely continuous controls in the dynamics. Finally, schemes for computing the value function and optimal control policies for the MDP are presented and illustrated with a numerical study.
Existence of optimal controls for singular control problems with state constraints
 Ann. Appl. Prob
, 2006
"... We establish the existence of an optimal control for a general class of singular control problems with state constraints. The proof uses weak convergence arguments and a time rescaling technique. The existence of optimal controls for Brownian control problems [14], associated with a broad family of ..."
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We establish the existence of an optimal control for a general class of singular control problems with state constraints. The proof uses weak convergence arguments and a time rescaling technique. The existence of optimal controls for Brownian control problems [14], associated with a broad family of stochastic networks, follows as a consequence.
Optimal control of a stochastic network driven by a fractional Brownian motion input
, 2008
"... We consider a stochastic control model driven by a fractional Brownian motion. This model is a formal approximation to a queueing network with an ONOFF input process. We study stochastic control problems associated with the longrun average cost, the infinite horizon discounted cost, and the finite ..."
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Cited by 1 (1 self)
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We consider a stochastic control model driven by a fractional Brownian motion. This model is a formal approximation to a queueing network with an ONOFF input process. We study stochastic control problems associated with the longrun average cost, the infinite horizon discounted cost, and the finite horizon cost. In addition, we find a solution to a constrained minimization problem as an application of our solution to the longrun average cost problem. We also establish Abelian limit relationships among the value functions of the above control problems.
Controlled Stochastic Networks in Heavy Traffic: Convergence of Value Functions.
, 2010
"... Abstract: Scheduling control problems for a family of unitary networks under heavy traffic with general interarrival and service times, probabilistic routing and an infinite horizon discounted linear holding cost are studied. Diffusion control problems, that have been proposed as approximate models ..."
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Abstract: Scheduling control problems for a family of unitary networks under heavy traffic with general interarrival and service times, probabilistic routing and an infinite horizon discounted linear holding cost are studied. Diffusion control problems, that have been proposed as approximate models for the study of these critically loaded controlled stochastic networks, can be regarded as formal scaling limits of such stochastic systems. However, to date, a rigorous limit theory that justifies the use of such approximations for a general family of controlled networks has been lacking. It is shown that, under broad conditions, the value function of the suitably scaled network control problem converges to that of the associated diffusion control problem. This scaling limit result, in addition to giving a precise mathematical basis for the above approximation approach, suggests a general strategy for constructing near optimal controls for the physical stochastic networks by solving the associated diffusion control problem.
Critical Branching Processes, (ii)Large Deviation Properties of Weakly Interacting Processes, (iii)Multiscale Diffusion Approximations for Stochastic Networks in Heavy Traffic, (iv) Adaptive Ergodic Control of Markov Chains, (v)Controlled Stochastic Netwo
, 2010
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The public reporting burden for this collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources, gathering and maintaining the data needed, and completing and reviewing the collection of information. Send comments regarding this burden estimate or any other aspect of this collection of information, including suggesstions for reducing this burden, to Washington
STATIONARY DISTRIBUTION CONVERGENCE FOR A MULTICLASS SINGLESERVER QUEUE IN HEAVY TRAFFIC
"... Abstract. In the author's recent paper In this work, relaxing the assumption of moment generating function on the primitives in that paper to the moment condition of secondorder or higherorder, we obtain the corresponding approximation result in a multiclass singleserver queue. The key to ..."
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Abstract. In the author's recent paper In this work, relaxing the assumption of moment generating function on the primitives in that paper to the moment condition of secondorder or higherorder, we obtain the corresponding approximation result in a multiclass singleserver queue. The key to our analysis is to use the framework of Budhiraja and Lee Introduction. In the author's recent paper
DIFFUSION MODELS FOR DOUBLEENDED QUEUES WITH RENEWAL ARRIVAL PROCESSES
"... We study a doubleended queue where buyers and sellers arrive to conduct trades. When there is a pair of buyer and seller in the system, they immediately transact a trade and leave. Thus there cannot be a nonzero number of buyers and sellers simultaneously in the system. We assume that sellers and ..."
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We study a doubleended queue where buyers and sellers arrive to conduct trades. When there is a pair of buyer and seller in the system, they immediately transact a trade and leave. Thus there cannot be a nonzero number of buyers and sellers simultaneously in the system. We assume that sellers and buyers arrive at the system according to independent renewal processes, and they would leave the system after independent exponential patience times. We establish fluid and diffusion approximations for the queue length process under a suitable asymptotic regime. The fluid limit is the solution of an ordinary differential equation, and the diffusion limit is a timeinhomogeneous asymmetric OrnsteinUhlenbeck process (OU process). A heavy traffic analysis is also developed, and the diffusion limit in the stronger heavy traffic regime is a timehomogeneous asymmetric OU process. The limiting distributions of both diffusion limits are obtained. We also show the interchange of the heavy traffic and steady state limits. 1. Introduction. Consider
with impatient customers in heavy traffic.
, 2010
"... buffer size and dynamic rate control for a queueing system ..."
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