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25
MaxWeight Scheduling in a Generalized Switch: State Space Collapse and Equivalent Workload Minimization under
 Complete Resource Pooling. Bell Labarotories Technical Report
, 2001
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Pathwise optimality of the exponential scheduling rule for wireless channels
 Advances in Applied Probability
, 2004
"... We consider the problem of scheduling transmissions of multiple data users (flows) sharing the same wireless channel (server). The unique feature of this problem is the fact that the capacity (service rate) of the channel varies randomly with time and asynchronously for different users. We study a s ..."
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Cited by 49 (13 self)
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We consider the problem of scheduling transmissions of multiple data users (flows) sharing the same wireless channel (server). The unique feature of this problem is the fact that the capacity (service rate) of the channel varies randomly with time and asynchronously for different users. We study a scheduling policy called Exponential scheduling rule, which was introduced in an earlier paper. Given a system with N users, and any set of positive numbers {an},n = 1,2,...,N, we show that in a heavytraffic limit, under a nonrestrictive complete resource pooling condition, this algorithm has the property that, for each time t, it (asymptotically) minimizes maxn an˜qn(t), where ˜qn(t) is user n queue length in the heavy traffic regime.
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 ..."
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Cited by 48 (8 self)
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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.
Performance Evaluation and Policy Selection in Multiclass Networks
, 2002
"... This paper concerns modelling and policy synthesis for regulation of multiclass queueing networks. A 2parameter network model is introduced to allow independent modelling of variability and mean processingrates, while maintaining simplicity of the model. Policy synthesis is based on consideration ..."
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Cited by 24 (18 self)
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This paper concerns modelling and policy synthesis for regulation of multiclass queueing networks. A 2parameter network model is introduced to allow independent modelling of variability and mean processingrates, while maintaining simplicity of the model. Policy synthesis is based on consideration of more tractable workload models, and then translating a policy from this abstraction to the discrete network of interest. Translation is made possible through the use of safetystocks that maintain feasibility of workload trajectories. This is a wellknown approach in the queueing theory literature, and may be viewed as a generic approach to avoid deadlock in a discreteevent dynamical system. Simulation is used to evaluate a given policy, and to tune safetystock levels. These simulations are accelerated through a variance reduction technique that incorporates stochastic approximation to tune the variance reduction. The search for appropriate safetystock levels is coordinated through a cutting plane algorithm. Both the policy synthesis and the simulation acceleration rely heavily on the development of approximations to the value function through fluid model considerations.
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 ..."
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Cited by 17 (0 self)
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We consider a class of open stochastic processing networks, with feedback routing and overlapping server capabilities, in heavy traffic. The networks
A multiclass queue in heavy traffic with throughput time constraints: Asymptotically optimal dynamic controls. Queueing Syst
 Theory Appl
, 2001
"... Abstract. Consider a singleserver queueing system with K job classes, each having its own renewal input process and its own general service time distribution. Further suppose the queue is in heavy traffic, meaning that its traffic intensity parameter is near the critical value of one. A system mana ..."
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Cited by 8 (3 self)
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Abstract. Consider a singleserver queueing system with K job classes, each having its own renewal input process and its own general service time distribution. Further suppose the queue is in heavy traffic, meaning that its traffic intensity parameter is near the critical value of one. A system manager must decide whether or not to accept new jobs as they arrive, and also the order in which to serve jobs that are accepted. The goal is to minimize penalties associated with rejected jobs, subject to upper bound constraints on the throughput times for accepted jobs; both the penalty for rejecting a job and the bound on the throughput time may depend on job class. This problem formulation does not make sense in a conventional queueing model, because throughput times are random variables, but we show that the formulation is meaningful in an asymptotic sense, as one approaches the heavy traffic limit under diffusion scaling. Moreover, using a method developed recently by Bramson and Williams, we prove that a relatively simple dynamic control policy is asymptotically optimal in this framework. Our proposed policy rejects jobs from one particular class when the server’s nominal workload is above a threshold value, accepting all other arrivals; and the sequencing rule for accepted jobs is one that maintains near equality of the relative backlogs for different classes, defined in a natural sense. 1. Introduction and
Existence condition for the diffusion approximations of multiclass priority queueing networks
 Faculty of Commerce and Business Administration, UBC
, 2001
"... In this paper, we extend the work of Chen and Zhang (2000b) and establish a new sufficient condition for the existence of the (conventional) diffusion approximation for multiclass queueing networks under priority service disciplines. This sufficient condition relates to the weak stability of the flu ..."
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Cited by 5 (1 self)
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In this paper, we extend the work of Chen and Zhang (2000b) and establish a new sufficient condition for the existence of the (conventional) diffusion approximation for multiclass queueing networks under priority service disciplines. This sufficient condition relates to the weak stability of the fluid networks and the stability of the high priority classes of the fluid networks that correspond to the queueing networks under consideration. Using this sufficient condition, we prove the existence of the diffusion approximation for the lastbufferfirstserved reentrant lines. We also study a threestation network example, and observe that the diffusion approximation may not exist, even if the “proposed” limiting semimartingale reflected Brownian motion (SRBM) exists.
Modeling a Healthcare System as a Queueing Network: The Case of a Belgian Hospital
"... The performance of healthcare systems in terms of patient flow times and utilization of critical resources can be assessed through queueing and simulation models. We model the orthopaedic department of the Middelheim hospital (Antwerpen, Belgium) focusing on the impact of outages (preemptive and non ..."
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Cited by 4 (0 self)
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The performance of healthcare systems in terms of patient flow times and utilization of critical resources can be assessed through queueing and simulation models. We model the orthopaedic department of the Middelheim hospital (Antwerpen, Belgium) focusing on the impact of outages (preemptive and nonpreemptive outages) on the effective utilization of resources and on the flow time of patients. Several queueing network solution procedures are developed such as the decomposition and Brownian motion approaches. Simulation is used as a validation tool. We present new approaches to model outages. The model offers a valuable tool to study the tradeoff between the capacity structure, sources of variability and patient flow times.
Heavy traffic limit for a processor sharing queue with soft deadlines
 Ann. Appl. Probab
, 2007
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Directional derivatives of oblique reflection maps
, 2008
"... Given an oblique reflection map Γ and functions ψ, χ ∈ Dlim (the space of IR Kvalued functions that have left and right limits at every point), the directional derivative ∇χΓ(ψ) of Γ along χ, evaluated at ψ, is defined to be the pointwise limit (as ε ↓ 0) of the family of functions ∇ε χΓ(ψ). = ε−1 ..."
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Cited by 2 (0 self)
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Given an oblique reflection map Γ and functions ψ, χ ∈ Dlim (the space of IR Kvalued functions that have left and right limits at every point), the directional derivative ∇χΓ(ψ) of Γ along χ, evaluated at ψ, is defined to be the pointwise limit (as ε ↓ 0) of the family of functions ∇ε χΓ(ψ). = ε−1 [Γ(ψ + εχ) − Γ(ψ)]. Directional derivatives are shown to exist and lie in Dlim for oblique reflection maps associated with reflection matrices of the socalled HarrisonReiman class. When ψ and χ are continuous, the convergence of ∇ε χΓ(ψ) to ∇χΓ(ψ) is shown to be uniform on compact subsets of continuity points of the limit ∇χΓ(ψ) and the derivative ∇χΓ(ψ) is shown to have an autonomous characterization as the unique fixed point of an associated map. Motivation for the study of directional derivatives stems from the fact that they arise as functional central limit approximations to timeinhomogeneous queueing networks as well as transient timehomogeneous queueing networks. This work also shows how the various types of discontinuities of the derivative ∇χΓ(ψ) are related to the reflection matrix and properties of the function Γ(ψ). In the queueing network context, this describes the influence of the topology of the network and the states (of underloading, overloading or