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Sequencing and routing in multiclass queueing networks part I: Feedback regulation
 SIAM J. Control Optim
"... Abstract. Part II continues the development of policy synthesis techniques for multiclass queueing networks based upon a linear fluid model. The following are shown: (i) A relaxation of the fluid model based on workload leads to an optimization problem of lower dimension. An analogous workloadrelax ..."
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Cited by 55 (12 self)
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Abstract. Part II continues the development of policy synthesis techniques for multiclass queueing networks based upon a linear fluid model. The following are shown: (i) A relaxation of the fluid model based on workload leads to an optimization problem of lower dimension. An analogous workloadrelaxation is introduced for the stochastic model. These relaxed control problems admit pointwise optimal solutions in many instances. (ii) A translation to the original fluid model is almost optimal, with vanishing relative error as the networkload ρ approaches one. It is pointwise optimal after a short transient period, provided a pointwise optimal solution exists for the relaxed control problem. (iii) A translation of the optimal policy for the fluid model provides a policy for the stochastic networkmodel that is almost optimal in heavy traffic, over all solutions to the relaxed stochastic model, again with vanishing relative error. The regret is of order  log(1 − ρ).
A simplex based algorithm to solve separated continuous linear programs
 Mathematical Programming
, 2008
"... We consider the separated continuous linear programming problem with linear data. We characterize the form of its optimal solution, and present an algorithm which solves it in a finite number of steps, using simplex pivot iterations. 1 ..."
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Cited by 30 (5 self)
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We consider the separated continuous linear programming problem with linear data. We characterize the form of its optimal solution, and present an algorithm which solves it in a finite number of steps, using simplex pivot iterations. 1
From fluid relaxations to practical algorithms for job shop scheduling: the makespan objective
 Mathematical Programming
, 2002
"... We design an algorithm for the highmultiplicity jobshop scheduling problem with the objective of minimizing the total holding cost by appropriately rounding an optimal solution to a fluid relaxation in which we replace discrete jobs with the flow of a continuous fluid. The algorithm solves the flu ..."
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Cited by 23 (4 self)
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We design an algorithm for the highmultiplicity jobshop scheduling problem with the objective of minimizing the total holding cost by appropriately rounding an optimal solution to a fluid relaxation in which we replace discrete jobs with the flow of a continuous fluid. The algorithm solves the fluid relaxation optimally and then aims to keep the schedule in the discrete network close to the schedule given by the fluid relaxation. If the number of jobs from each type grow linearly with N,then the algorithm is within an additive factor O�N � from the optimal (which scales as O�N 2�); thus,it is asymptotically optimal. We report computational results on benchmark instances chosen from the OR library comparing the performance of the proposed algorithm and several commonly used heuristic methods. These results suggest that for problems of moderate to high multiplicity,the proposed algorithm outperforms these methods,and for very high multiplicity the overperformance is dramatic. For problems of low to moderate multiplicity,however,the relative errors of the heuristic methods are comparable to those of the proposed algorithm,and the best of these methods performs better overall than the proposed method. Received December 1999; revisions received July 2000,September 2001; accepted September 2002. Subject classifications: Production/scheduling,deterministic: approximation algorithms for deterministic job shops. Queues,optimization: asymptotically optimal solutions to queueing networks. Area of review: Manufacturing,Service,and Supply Chain Operations. 1.
Multiproduct systems with both setup times and costs: Fluid bounds and schedules
 Operations Research
, 2004
"... This paper considers a multiproduct, singleserver production system where both setup times and costs are incurred whenever the server changes product. The system is maketoorder with a per unit backlogging cost. The objective is to minimize the longrun average cost per unit time. Using a fluid m ..."
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Cited by 18 (0 self)
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This paper considers a multiproduct, singleserver production system where both setup times and costs are incurred whenever the server changes product. The system is maketoorder with a per unit backlogging cost. The objective is to minimize the longrun average cost per unit time. Using a fluid model, we provide a closedform lower bound on system performance. This bound is also shown to provide a lower bound for stochastic systems when scheduling is static, but is only an approximation when scheduling is dynamic. Heavytraffic analysis yields a refined bound that includes secondmoment terms. The fluid bound suggests both dynamic and static scheduling In this paper we consider a production environment where a number of different products are produced on a single machine and setup activities are necessary when switches of product type are made. These setup activities require both time and cost that depend on the specific product type. Throughout the paper we assume that the setups do not depend on the previous product produced
Scheduling and control of manufacturing systems — a fluid approach
 Proceedings of the 37 Allerton Conference
, 1999
"... We suggest a fluid framework for solving scheduling and control problems of manufacturing systems. We formulate the fluid approximation, show how to solve it, give an important graphical display of the fluid solution, construct a schedule from the fluid solution, and provide probabilistic bound of i ..."
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Cited by 17 (10 self)
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We suggest a fluid framework for solving scheduling and control problems of manufacturing systems. We formulate the fluid approximation, show how to solve it, give an important graphical display of the fluid solution, construct a schedule from the fluid solution, and provide probabilistic bound of its suboptimality. 1
Workload Models for Stochastic Networks: Value Functions and Performance Evaluation
, 2005
"... This paper concerns control and performance evaluation for stochastic network models. Structural properties of value functions are developed for controlled Brownian motion (CBM) and deterministic (fluid) workloadmodels, leading to the following conclusions: Outside of a nullset of network paramete ..."
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Cited by 16 (9 self)
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This paper concerns control and performance evaluation for stochastic network models. Structural properties of value functions are developed for controlled Brownian motion (CBM) and deterministic (fluid) workloadmodels, leading to the following conclusions: Outside of a nullset of network parameters, (i) The fluid valuefunction is a smooth function of the initial state. Under further minor conditions, the fluid valuefunction satisfies the derivative boundary conditions that are required to ensure it is in the domain of the extended generator for the CBM model. Exponential ergodicity of the CBM model is demonstrated as one consequence. (ii) The fluid valuefunction provides a shadow function for use in simulation variance reduction for the stochastic model. The resulting simulator satisfies an exact large deviation principle, while a standard simulation algorithm does not satisfy any such bound. (iii) The fluid valuefunction provides upper and lower bounds on performance for the CBM model. This follows from an extension of recent linear programming approaches to performance evaluation.
Large deviation asymptotics and control variates for simulating large functions
, 2005
"... Consider the normalized partial sums of a realvalued function F of a Markov chain, φn: = n −1 n−1 F(Φ(k)), n ≥ 1. k=0 The chain {Φ(k) : k ≥ 0} takes values in a general state space X, with transition kernel P, and it is assumed that the Lyapunov drift condition holds: PV ≤ V −W +bIC where V: X → (0 ..."
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Cited by 16 (6 self)
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Consider the normalized partial sums of a realvalued function F of a Markov chain, φn: = n −1 n−1 F(Φ(k)), n ≥ 1. k=0 The chain {Φ(k) : k ≥ 0} takes values in a general state space X, with transition kernel P, and it is assumed that the Lyapunov drift condition holds: PV ≤ V −W +bIC where V: X → (0, ∞), W: X → [1, ∞), the set C is small, and W dominates F. Under these assumptions, the following conclusions are obtained: (i) It is known that this drift condition is equivalent to the existence of a unique invariant distribution π satisfying π(W) < ∞, and the Law of Large Numbers holds for any function F dominated by W: φn → φ: = π(F), a.s., n → ∞. (ii) The lower error probability defined by P{φn ≤ c}, for c < φ, n ≥ 1, satisfies a large deviation limit theorem when the function F satisfies a monotonicity condition. Under additional minor conditions an exact large deviations expansion is obtained. (iii) If W is nearmonotone then controlvariates are constructed based on the Lyapunov function V, providing a pair of estimators that together satisfy nontrivial large asymptotics for the lower and upper error probabilities. In an application to simulation of queues it is shown that exact large deviation asymptotics are possible even when the estimator does not satisfy a Central Limit Theorem.
A push pull queueing system
 Operations Research Letters
, 2002
"... We consider a two node multiclass queueing network given by two machines each with two classes. There are two streams of jobs: One stream originates in machine 1, which feeds it for further processing to machine 2, and the other stream moves in the opposite direction. We describe a policy for this s ..."
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Cited by 13 (8 self)
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We consider a two node multiclass queueing network given by two machines each with two classes. There are two streams of jobs: One stream originates in machine 1, which feeds it for further processing to machine 2, and the other stream moves in the opposite direction. We describe a policy for this system which is stable and which keeps both machines busy at all times. We obtain explicit expressions for its steady state behavior under M/M/ · and under M/G/ · assumptions. We also describe a similar M machine system.
Management of DemandDriven Production Systems
 IEEE TRANS. AUTOMAT. CONTROL
, 2004
"... Controlsynthesis techniques are developed for demand driven production systems. The resulting policies are timeoptimal for a deterministic model, and approximately timeoptimal for a stochastic model. Moreover, they are easily adapted to take into account a range of issues that arise in a realisti ..."
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Cited by 10 (6 self)
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Controlsynthesis techniques are developed for demand driven production systems. The resulting policies are timeoptimal for a deterministic model, and approximately timeoptimal for a stochastic model. Moreover, they are easily adapted to take into account a range of issues that arise in a realistic, dynamic environment. In particular, control synthesis techniques are developed for models in which resources are temporarily unavailable. This may be due to failure, maintenance, or an unanticipated change in demand. These conclusions are based upon the following development...
Efficient Algorithms for Separated Continuous Linear Programs: The Multicommodity Flow Problem with Holding Costs and Extensions
, 2005
"... We give an approximation scheme for separated continuous linear programming problems. Such problems arise as fluid relaxations of multiclass queueing networks and are used to find approximate solutions to complex job shop scheduling problems. In a network with linear flow costs and linear, perunit ..."
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Cited by 6 (0 self)
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We give an approximation scheme for separated continuous linear programming problems. Such problems arise as fluid relaxations of multiclass queueing networks and are used to find approximate solutions to complex job shop scheduling problems. In a network with linear flow costs and linear, perunittime holding costs, our algorithm finds a drainage of the network that, for given constants �>0 and �>0, has total cost �1 + ��OPT + �, where OPT is the cost of the minimum cost drainage. The complexity of our algorithm is polynomial in the size of the input network, 1/�, and log�1/��. The fluid relaxation is a continuous problem. While the problem is known to have a piecewise constant solution, it is not known to have a polynomially sized solution. We introduce a natural discretization of polynomial size and prove that this discretization produces a solution with low cost. This is the first polynomial time algorithm with a provable approximation guarantee for fluid relaxations.