Results 1  10
of
35
Fluid Model for a Network Operating under a Fair BandwidthSharing Policy
 Annals of Applied Probability
, 2004
"... We consider a model of Internet congestion control, that represents the randomly varying number of ows present in a network where bandwidth is shared fairly between document transfers. We study critical uid models, obtained as formal limits under law of large numbers scalings when the average lo ..."
Abstract

Cited by 71 (7 self)
 Add to MetaCart
(Show Context)
We consider a model of Internet congestion control, that represents the randomly varying number of ows present in a network where bandwidth is shared fairly between document transfers. We study critical uid models, obtained as formal limits under law of large numbers scalings when the average load on at least one resource is equal to its capacity. We establish convergence to equilibria for uid models, and identify the invariant manifold. The form of the invariant manifold gives insight into the phenomenon of entrainment, whereby congestion at some resources may prevent other resources from working at their full capacity.
Maximum pressure policies in stochastic processing networks
, 2005
"... Complex systems like semiconductor wafer fabrication facilities (fabs), networks of data switches, and largescale call centers all demand efficient resource allocation. Deterministic models like linear programs (LP) have been used for capacity planning at both the design and expansion stages of s ..."
Abstract

Cited by 69 (6 self)
 Add to MetaCart
Complex systems like semiconductor wafer fabrication facilities (fabs), networks of data switches, and largescale call centers all demand efficient resource allocation. Deterministic models like linear programs (LP) have been used for capacity planning at both the design and expansion stages of such systems. LPbased planning is critical in setting a medium range or longterm goal for many systems, but it does not translate into a daytoday operational policy that must deal with discreteness of jobs and the randomness of the processing environment. A stochastic processing network, advanced by J. Michael Harrison (2000, 2002, 2003), is a system that takes inputs of materials of various kinds and uses various processing resources to produce outputs of materials of various kinds. Such a network provides a powerful abstraction of a wide range of realworld systems. It provides highfidelity stochastic models in diverse economic sectors including manufacturing, service, and information technology. We propose a family of maximum pressure service policies for dynamically allocating service capacities in a stochastic processing network. Under a mild assumption on network structure, we prove that a network operating under a maximum pressure policy achieves maximum throughput predicted by LPs. These policies are semilocal in the sense that each
On Dynamic Scheduling of a Parallel Server System with Complete Resource Pooling
 In Analysis of Communication Networks: Call Centres, Traffic and Performance
, 2000
"... scientific noncommercial use only for individuals, with permission from the authors. We consider a parallel server queueing system consisting of a bank of buffers for holding incoming jobs and a bank of flexible servers for processing these jobs. Incoming jobs are classified into one of several dif ..."
Abstract

Cited by 66 (5 self)
 Add to MetaCart
scientific noncommercial use only for individuals, with permission from the authors. We consider a parallel server queueing system consisting of a bank of buffers for holding incoming jobs and a bank of flexible servers for processing these jobs. Incoming jobs are classified into one of several different classes (or buffers). Jobs within a class are processed on a firstinfirstout basis, where the processing of a given job may be performed by any server from a given (classdependent) subset of the bank of servers. The random service time of a job may depend on both its class and the server providing the service. Each job departs the system after receiving service from one server. The system manager seeks to minimize holding costs by dynamically scheduling waiting jobs to available servers. We consider a parameter regime in which the system satisfies both a heavy traffic and a complete resource pooling condition. Our cost function is an expected cumulative discounted cost of holding jobs in the system, where the (undiscounted) cost per unit time is a linear function of normalized (with heavy traffic scaling) queue length. In a prior work [40], the second author proposed a continuous review threshold control policy for use in such a parallel server system. This policy was advanced as an “interpretation ” of the analytic solution to an associated Brownian control problem (formal heavy
Newsvendor Networks: Inventory Management and Capacity Investment with Discretionary Activities
, 2002
"... We introduce a class of models, called newsvendor networks, that allow for multiple products and multiple processing and storage points and investigate how their singleperiod properties extend to dynamic settings. Such models provide a parsimonious framework to study various problems of stochastic ..."
Abstract

Cited by 43 (7 self)
 Add to MetaCart
We introduce a class of models, called newsvendor networks, that allow for multiple products and multiple processing and storage points and investigate how their singleperiod properties extend to dynamic settings. Such models provide a parsimonious framework to study various problems of stochastic capacity investment and inventory management, including assembly, commonality, distribution, flexibility, substitution and transshipment. Newsvendor networks are stochastic models with recourse that are characterized by linear revenue and cost structures and a linear inputoutput transformation. While capacity and inventory decisions are locked in before uncertainty is resolved, some managerial discretion remains via expost inputoutput activity decisions. Expost decisions involve both the choice of activities and their levels and can result in subtle benefits. This discretion in choice is captured through alternate or "nonbasic" activities that can redeploy inputs and resources to best respond to resolved uncertain events. Nonbasic activities are never used in a deterministic environment; their value stems from discretionary flexibility to meet stochastic demand deviations from the operating point. The optimal capacity and inventory decisions balance overages with underages. Continuing the classic newsvendor analogy, the optimal balancing conditions can be interpreted as
Asymptotic optimality of maximum pressure policies in stochastic processing networks
 Annals of Applied Probability
, 2008
"... We consider a class of stochastic processing networks. Assume that the networks satisfy a complete resource pooling condition. We prove that each maximum pressure policy asymptotically minimizes the workload process in a stochastic processing network in heavy traffic. We also show that, under each q ..."
Abstract

Cited by 41 (4 self)
 Add to MetaCart
We consider a class of stochastic processing networks. Assume that the networks satisfy a complete resource pooling condition. We prove that each maximum pressure policy asymptotically minimizes the workload process in a stochastic processing network in heavy traffic. We also show that, under each quadratic holding cost structure, there is a maximum pressure policy that asymptotically minimizes the holding cost. A key to the optimality proofs is to prove a state space collapse result and a heavy traffic limit theorem for the network processes under a maximum pressure policy. We extend a framework of Bramson [Queueing Systems Theory Appl. 30 (1998) 89–148] and Williams [Queueing Systems Theory Appl. 30 (1998b) 5–25] from the multiclass queueing network setting to the stochastic processing network setting to prove the state space collapse result and the heavy traffic limit theorem. The extension can be adapted to other studies of stochastic processing networks.
Optimal Leadtime Differentiation via Diffusion Approximations
, 2004
"... This study illustrates how a manufacturer can use leadtime differentiation—selling the same product to different customers at different prices based on delivery leadtime—to simultaneously increase revenue and reduce capacity requirements. The manufacturer’s production facility is modeled as an expon ..."
Abstract

Cited by 24 (5 self)
 Add to MetaCart
This study illustrates how a manufacturer can use leadtime differentiation—selling the same product to different customers at different prices based on delivery leadtime—to simultaneously increase revenue and reduce capacity requirements. The manufacturer’s production facility is modeled as an exponential singleserver queue with two classes of customers that differ in price sensitivity and delay sensitivity. The manufacturer chooses the service rate and a static price for each class of customer, and then dynamically quotes leadtimes to potential customers and decides the order in which customers are processed. The arrival rate for each class decreases linearly with price and leadtime. The manufacturer’s objective is to maximize profit, subject to the constraint that each customer must be processed within the promised leadtime. Assuming that some customers will tolerate a long delivery leadtime, we show that this problem has a simple nearoptimal solution. Under our proposed policy, capacity utilization is near 100%. Impatient customers pay a premium for immediate delivery and receive priority in scheduling, whereas patient customers are quoted a leadtime proportional to the current queue length. Queue length and leadtime can be closely approximated by a reflected OrnsteinUhlenbeck diffusion process. Hence, we have a closed form expression for profit, and choose prices and capacity to optimize this. In case customers may choose either the class 1 deal or the class 2 deal, the proposed policy is made incentive compatible by quoting a leadtime for the class 2 (patient) customers that is longer than the actual queueing delay.
Heavy traffic analysis of open processing networks with complete resource pooling: asymptotic optimality of discrete review policies
 ANN. APPL. PROBAB
, 2005
"... We consider a class of open stochastic processing networks, with feedback routing and overlapping server capabilities, in heavy traffic. The networks ..."
Abstract

Cited by 23 (0 self)
 Add to MetaCart
We consider a class of open stochastic processing networks, with feedback routing and overlapping server capabilities, in heavy traffic. The networks
Fluid and Brownian Approximations for an Internet Congestion Control Model
 Proceedings of the 43rd IEEE Conference on Decision and Control, December 2004
, 2004
"... We consider a stochastic model of Internet congestion control that represents the randomly varying number of flows present in a network where bandwidth is shared fairly amongst elastic document transfers. We focus on the heavy traffic regime in which the average load placed on each resource is appr ..."
Abstract

Cited by 16 (1 self)
 Add to MetaCart
We consider a stochastic model of Internet congestion control that represents the randomly varying number of flows present in a network where bandwidth is shared fairly amongst elastic document transfers. We focus on the heavy traffic regime in which the average load placed on each resource is approximately equal to its capacity. We first describe a fluid model (or functional law of large numbers approximation) for the stochastic model. We use the long time behavior of the solutions of this fluid model to establish a property called (multiplicative) state space collapse, which shows that in diffusion scale the flow count process can be approximately recovered as a continuous lifting of the workload process. Under proportional fair sharing of bandwidth and a mild condition, we show how state space collapse can be combined with a new invariance principle to establish a Brownian model as a diffusion approximation for the workload process and hence to yield an approximation for the flow count process. The workload diffusion behaves like Brownian motion in the interior of a polyhedral cone and is confined to the cone by reflection at the boundary, where the direction of reflection is constant on any given boundary face. We illustrate this approximation result for a simple linear network. Here the diffusion lives in a wedge that is a strict subset of the positive quadrant. This geometrically illustrates the entrainment of resources, whereby congestion at one resource may prevent another resource from working at full capacity.
Near Optimal Control of Queueing Networks over a Finite Time Horizon
, 2007
"... We propose a novel approach for controlling queueing networks that minimizes weighted holding costs over a finite time horizon. Our approach approximates the discrete problem by a fluid system for which an optimization problem is formulated. This problem is a separated continuous linear program, it ..."
Abstract

Cited by 12 (10 self)
 Add to MetaCart
We propose a novel approach for controlling queueing networks that minimizes weighted holding costs over a finite time horizon. Our approach approximates the discrete problem by a fluid system for which an optimization problem is formulated. This problem is a separated continuous linear program, it is solved optimally using a simplex based algorithm of Weiss. The solution consists of piecewise constant allocations of the activities, with a finite number of breakpoints over the time horizon. Once solved, we associate a multiclass queueing network with infinite virtual queues with each interval of the fluid solution, and this measures the deviations of the original system from the fluid solution. We then track the fluid solution by using an adaptation of Dai and Lin’s maximum pressure policy that keeps these deviations rate stable. This procedure is asymptotically optimal as we scale up the number of jobs and the processing speed. We illustrate the details of the approach on a simple example composed of two servers and three queues. Simulation results confirm that the system performance is near optimal when the network is scaled up.