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13
Analysis of a decentralized productioninventory system. Working paper, Sloan School of Management,
, 1999
"... W e model an isolated portion of a competitive supply chain as a M/M/1 maketostock queue. The retailer carries finished goods inventory to service a Poisson demand process, and specifies a policy for replenishing his inventory from an upstream supplier. The supplier chooses the service rate, i.e., ..."
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W e model an isolated portion of a competitive supply chain as a M/M/1 maketostock queue. The retailer carries finished goods inventory to service a Poisson demand process, and specifies a policy for replenishing his inventory from an upstream supplier. The supplier chooses the service rate, i.e., the capacity of his manufacturing facility, which behaves as a singleserver queue with exponential service times. Demand is backlogged and both agents share the backorder cost. In addition, a linear inventory holding cost is charged to the retailer, and a linear cost for building production capacity is incurred by the supplier. The inventory level, demand rate, and cost parameters are common knowledge to both agents. Under the continuousstate approximation where the M/M/1 queue has an exponential rather than geometric steadystate distribution, we characterize the optimal centralized and Nash solutions, and show that a contract with linear transfer payments replicates a costsharing agreement and coordinates the system. We also compare the total system costs, the agents' decision variables, and the customer service levels of the centralized versus Nash versus Stackelberg solutions. (MaketoStock Queue; Game Theory )
Multiproduct systems with both setup times and costs: Fluid bounds and schedules.
 Operations Research,
, 2006
"... Abstract 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 ..."
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Abstract 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 local or static, but is only an approximation when scheduling is global or dynamic. The fluid bound suggests both local and global scheduling heuristics, which are tested for the stochastic system via a simulation study.
Applications of polling systems.
 Surveys in Operations Research and Management Science,
, 2011
"... Abstract Since the first paper on polling systems, written by Mack in 1957, a huge number of papers on this topic has been written. A typical polling system consists of a number of queues, attended by a single server. In several surveys, the most notable ones written by Takagi, detailed and compreh ..."
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Abstract Since the first paper on polling systems, written by Mack in 1957, a huge number of papers on this topic has been written. A typical polling system consists of a number of queues, attended by a single server. In several surveys, the most notable ones written by Takagi, detailed and comprehensive descriptions of the mathematical analysis of polling systems are provided. The goal of the present survey paper is to complement these papers by putting the emphasis on applications of polling models. We discuss not only the capabilities, but also the limitations of polling models in representing various applications. Moreover, since the publication of the previous survey papers around 10 till 15 years ago, a large number of papers on polling models has been published. The present survey is directed at both academicians and practitioners.
Mean value analysis for polling systems in heavy traffic
 In Proceedings of ValueTools, Pisa Article
, 2006
"... In this paper we present a new approach to derive heavytraffic asymptotics for polling models. We consider the classical cyclic polling model with exhaustive service at each queue, and with general servicetime and switchover time distributions, and study its behavior when the load tends to one. F ..."
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In this paper we present a new approach to derive heavytraffic asymptotics for polling models. We consider the classical cyclic polling model with exhaustive service at each queue, and with general servicetime and switchover time distributions, and study its behavior when the load tends to one. For this model, we explore the recently proposed mean value analysis (MVA), which takes a new view on the dynamics of the system, and use this view to provide an alternative way to derive closedfrom expressions for the expected asymptotic delay; the expressions were derived earlier in [32], but in a different way. Moreover, the MVAbased approach enables us to derive closedform expressions for the heavytraffic limits of the covariances between the successive visit periods, which are key performance metrics in many application areas. These results, which have not been obtained before, reveal a number of insensitivity properties of the covariances with respect to the system parameters under heavytraffic assumptions, and moreover, lead to simple approximations for the covariances between the successive visit times for stable systems. Numerical examples demonstrate that the approximations are accurate when the load is close enough to one.
Heavy Traffic Analysis of the Dynamic Stochastic InventoryRouting Problem
"... We analyze three queueing control problems that model a dynamic stochastic distribution system, where a single capacitated vehicle serves a finite number of retailers in a maketostock fashion. The objective in each of these vehicle routing and inventory problems is to minimize the long run average ..."
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We analyze three queueing control problems that model a dynamic stochastic distribution system, where a single capacitated vehicle serves a finite number of retailers in a maketostock fashion. The objective in each of these vehicle routing and inventory problems is to minimize the long run average inventory (holding and backordering) and transportation cost. In all three problems, the controller dynamically specifies whether a vehicle at the warehouse should idle or embark with a full load. In the first problem, the vehicle must travel along a prespecified (TSP) tour of all retailers, and the controller dynamically decides how many units to deliver to each retailer. In the second problem, the vehicle delivers an entire load to one retailer (direct shipping) and the controller decides which retailer to visit next. The third problem allows the additional dynamic choice between the TSP and direct shipping options. Motivated by existing heavy traffic limit theorems, we make a time scale decomposition assumption that allows us to approximate these queueing control problems by diffusion control problems, which are explicitly solved in the fixed route problems, and numerically solved in the dynamic routing case. Simulation experiments confirm that the heavy traffic approximations are quite accurate over a broad range of problem parameters. Our results lead to some new observations about the behavior of this complex system. A prototypical example of the inventoryrouting problem (IRP) is the challenge faced by a large oil company as it distributes gasoline to its various gas stations: several warehouses hold inventory of a particular item (gasoline) and serve a set of retailers (stations) in a maketostock fashion; arriving customers (automobiles) consume the product at these retail sites, and a fleet of finite capacity vehicles (tanker trucks) is used to transport the product from the warehouse to the various retailers. The management decisions involved in the design and operation of such a system are manyfold and complex. Traditionally, a hierarchical decomposition of the problem is used to allow for a solvable model
Simulation Optimization for the Stochastic Economic Lot Scheduling Problem with SequenceDependent Setup Times
"... We consider the stochastic economic lot scheduling problem (SELSP) with lost sales and random demand, where switching between products is subject to sequencedependent setup times. We propose a solution based on simulation optimization using an iterative twostep procedure which combines global poli ..."
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We consider the stochastic economic lot scheduling problem (SELSP) with lost sales and random demand, where switching between products is subject to sequencedependent setup times. We propose a solution based on simulation optimization using an iterative twostep procedure which combines global policy search with local search heuristics for the traveling salesman sequencing subproblem. To optimize the production cycle, we compare two criteria: minimizing total setup times and evenly distributing setups to obtain a more regular production cycle. Based on a numerical study, we find that a policy with a balanced production cycle outperforms other policies with unbalanced cycles.
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"... Towards a unifying theory on branchingtype polling systems in heavy traffic R.D. van der Mei ..."
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Towards a unifying theory on branchingtype polling systems in heavy traffic R.D. van der Mei
Heavy Traffic Limit Theorems for Two Tandem Polling Stations
"... We consider two tandem singleserver stations in heavy traffic. There are two job types and they visit both stations in order. The first station processes jobs in an exhaustive service or gated service fashion and the second station uses an arbitrary nonidling service discipline. The stations have z ..."
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We consider two tandem singleserver stations in heavy traffic. There are two job types and they visit both stations in order. The first station processes jobs in an exhaustive service or gated service fashion and the second station uses an arbitrary nonidling service discipline. The stations have zero switchover times. We prove heavy traffic limit theorems for the twodimensional total workload processes, one under exhaustive service and one under gated service. The limiting processes are twodimensional reflected Brownian motions (RBM) and the limit under exhaustive service is equal in distribution to the limiting process when the first station performs one of two buffer priority policies.
Polling Systems with Periodic Server Routing in HeavyTraffic: Renewal Arrivals
"... This paper considers heavytraffic limit theorems for polling models with periodic server routing with general renewal arrivals and with mixtures of gated and exhaustive service policies, in a general parameter setting. We provide a strong conjecture for the limiting waitingtime distribution in a g ..."
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This paper considers heavytraffic limit theorems for polling models with periodic server routing with general renewal arrivals and with mixtures of gated and exhaustive service policies, in a general parameter setting. We provide a strong conjecture for the limiting waitingtime distribution in a general parameter setting when the load tends to 1, under proper heavytraffic scalings. The results in this paper, which generalize those in [14] for the special case of Poisson arrivals, are obtained by following the nonPoisson transient systems approach developed in [6]. Numerical results with simulations support the correctness our results, and moreover, show that the expressions for the waitingtime distributions in the limiting case lead to sharp approximations for the waitingtime distributions for stable polling systems with medium and heavy load. Keywords: polling system, polling table, periodic routing, delay, waiting time, heavytraffic, This paper considers heavytraffic limit theorems for polling models with periodic server routing under renewal arrivals. As such, we generalize results in Olsen and Van der Mei [14], which were proven for these systems under Poisson arrivals in steadystate. In such systems the order in which
The Stochastic Economic Lot Scheduling Problem: A Survey
, 2005
"... We consider the production of multiple standardized products on a single machine with limited capacity and setup times under random demands and random production times, i.e., the socalled stochastic economic lot scheduling problem (SELSP). The main task for the production manager in this setting i ..."
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We consider the production of multiple standardized products on a single machine with limited capacity and setup times under random demands and random production times, i.e., the socalled stochastic economic lot scheduling problem (SELSP). The main task for the production manager in this setting is the construction of a production plan for the machine that minimizes the total costs, i.e., the sum of holding, backlogging and setup costs. Based on the critical elements of such a production plan, we give a classification and extensive overview of the research on the SELSP together with an indication of open research areas.