Results 1  10
of
16
MaxWeight scheduling in a generalized switch: state space collapse and workload minimization in heavy traffic
 Annals of Applied Probability
, 2004
"... We consider a generalized switch model, which includes as special cases the model of multiuser data scheduling over a wireless medium, the inputqueued crossbar switch model and a discrete time version of a parallel server queueing system. Input flows n = 1,...,N are served in discrete time by a swi ..."
Abstract

Cited by 113 (9 self)
 Add to MetaCart
We consider a generalized switch model, which includes as special cases the model of multiuser data scheduling over a wireless medium, the inputqueued crossbar switch model and a discrete time version of a parallel server queueing system. Input flows n = 1,...,N are served in discrete time by a switch. The switch state follows a finite state, discrete time Markov chain. In each state m, the switch chooses a scheduling decision k from a finite set K(m), which has the associated service rate vector (µ m 1 (k),..., µm N (k)). We consider a heavy traffic regime, and assume a Resource Pooling (RP) condition. Associated with this condition is a notion of workload X = � n ζnQn, whereζ = (ζ1,...,ζN) is some fixed nonzero vector with nonnegative components, and Q1,...,QN are the queue lengths. We study the MaxWeight discipline which always chooses a decision k maximizing n γn[Qn] β µ m n (k),thatis, k ∈ arg max γn[Qn] i n β µ m n (i), where β>0, γ1> 0,...,γN> 0 are arbitrary parameters. We prove that under MaxWeight scheduling and the RP condition, in the heavy traffic limit, the queue length process has the following properties: (a) The vector (γ1Q β 1,...,γNQ β N) is always proportional to ζ (this is “State Space Collapse”), (b) the workload process converges to a Reflected Brownian Motion, (c) MaxWeight minimizes the workload among all disciplines. As a corollary of these properties, MaxWeight asymptotically minimizes the holding cost rate n γnQ β+1
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 48 (8 self)
 Add to MetaCart
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.
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 ..."
Abstract

Cited by 48 (12 self)
 Add to MetaCart
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.
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 ..."
Abstract

Cited by 24 (18 self)
 Add to MetaCart
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 ..."
Abstract

Cited by 16 (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
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 ..."
Abstract

Cited by 5 (1 self)
 Add to MetaCart
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.
HEAVY TRAFFIC LIMIT FOR A PROCESSOR SHARING QUEUE WITH SOFT DEADLINES
, 707
"... This paper considers a GI/GI/1 processor sharing queue in which jobs have soft deadlines. At each point in time, the collection of residual service times and deadlines is modeled using a random counting measure on the right halfplane. The limit of this measure valued process is obtained under diffu ..."
Abstract

Cited by 3 (0 self)
 Add to MetaCart
This paper considers a GI/GI/1 processor sharing queue in which jobs have soft deadlines. At each point in time, the collection of residual service times and deadlines is modeled using a random counting measure on the right halfplane. The limit of this measure valued process is obtained under diffusion scaling and heavy traffic conditions and is characterized as a deterministic function of the limiting queue length process. As special cases, one obtains diffusion approximations for the lead time profile and the profile of times in queue. One also obtains a snapshot principle for sojourn times. 1. Introduction. Congestion
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 ..."
Abstract

Cited by 3 (0 self)
 Add to MetaCart
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.
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 ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
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
Brownian Approximations of Multiclass Open Queueing Networks
 Operations Research
, 2001
"... We study a multiclass open queueing network with a set of singleserver stations that operate under a combination of FIFO (firstinfirstout) and priority service disciplines, and are subject to random breakdowns. Assuming that the primitive processes  in particular, external arrivals, service ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
We study a multiclass open queueing network with a set of singleserver stations that operate under a combination of FIFO (firstinfirstout) and priority service disciplines, and are subject to random breakdowns. Assuming that the primitive processes  in particular, external arrivals, service requirements, service capacities (up and down times) and the routing mechanism  follow twomoment approximations (based on functional central limit theorems), we develop a semimartingale reflected Brownian motion (SRBM) approx imation for the performance processes such as workload, queue lengths and sojourn times.