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30
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 34 (10 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).
Simulation run lengths to estimate blocking probabilities
 ACM Transactions on Modelling and Computer Simulation
, 1996
"... We derive formulas approximating the asymptotic variance of four estimators for the steadystate blocking probability in a multiserver loss system, exploiting diffusion process limits. These formulas can be used to predict simulation run lengths required to obtain desired statistical precision befor ..."
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Cited by 24 (19 self)
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We derive formulas approximating the asymptotic variance of four estimators for the steadystate blocking probability in a multiserver loss system, exploiting diffusion process limits. These formulas can be used to predict simulation run lengths required to obtain desired statistical precision before the simulation has been run, which can aid in the design of simulation experiments. They also indicate that one estimator can be much better than another, depending on the loading. An indirect estimator based on estimating the mean occupancy is significantly more (less) efficient than a direct estimator for heavy (light) loads. A major concern is the way computational effort scales with system size. For all the estimators, the asymptotic variance tends to be inversely proportional to the system size, so that the computational effort (regarded as proportional to the product of the asymptotic variance and the arrival rate) does not grow as system size increases. Indeed, holding the blocking probability fixed, the computational effort with a good estimator decreases to 0 as the system size increases. The asymptotic variance formulas also reveal the impact of the arrivalprocess and servicetime variability on the statistical precision. We validate these formulas by comparing them to exact numerical
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.
Fundamental Bounds on the Accuracy of Network Performance Measurements
 in ACM SIGMETRICS
, 2005
"... This paper considers the basic problem of "how accurate can we make Internet performance measurements". The answer is somewhat counterintuitive in that there are bounds on the accuracy of such measurements, no matter how many probes we can use in a given time interval, and thus arises a type of Hei ..."
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Cited by 19 (3 self)
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This paper considers the basic problem of "how accurate can we make Internet performance measurements". The answer is somewhat counterintuitive in that there are bounds on the accuracy of such measurements, no matter how many probes we can use in a given time interval, and thus arises a type of Heisenberg inequality describing the bounds in our knowledge of the performance of a network. The results stem from the fact that we cannot make independent measurements of a system's performance: all such measures are correlated, and these correlations reduce the efficacy of measurements. The degree of correlation is also strongly dependent on system load. The result has important practical implications that reach beyond the design of Internet measurement experiments, into the design of network protocols.
Transient Behavior of the M/G/1 Workload Process
, 1992
"... In this paper we describe the timedependent moments of the workload process in the M/G/1 queue. The k th moment as a function of time can be characterized in terms of a differential equation involving lower moment functions and the timedependent serveroccupation probability. For general initial ..."
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Cited by 17 (9 self)
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In this paper we describe the timedependent moments of the workload process in the M/G/1 queue. The k th moment as a function of time can be characterized in terms of a differential equation involving lower moment functions and the timedependent serveroccupation probability. For general initial conditions, we show that the first two moment functions can be represented as the difference of two nondecreasing functions, one of which is the moment function starting at zero. The two nondecreasing components can be regarded as probability cumulative distribution functions (cdf's) after appropriate normalization. The normalized moment functions starting empty are called moment cdf's; the other normalized components are called momentdifference cdf's. We establish relations among these cdf's using stationaryexcess relations. We apply these relations to calculate moments and derivatives at the origin of these cdf's. We also obtain results for the covariance function of the stationary workload process. It is interesting that these various timedependent characteristics can be described directly in terms of the steadystate workload distribution. Subject classification: queues, transient results: M/G/1 workload process. queues, busyperiod analysis: M/G/1 queue. In this paper, we derive some simple descriptions of the transient behavior of the classical M/G/1 queue. In particular, we focus on the workload process {W(t) : t 0} (also known as the unfinished work process and the virtual waiting time process), which is convenient to analyze because it is a Markov process. Our main results describe the timedependent probability that the server is busy, P(W(t) > 0), the timedependent moments of the workload process, E[W(t) k ], and the covariance function of the stationary ...
TwoStage MultipleComparison Procedures for SteadyState Simulations
 Annals of Statistics
, 1999
"... this paper, the results naturally apply to (asymptotically) stationary time series. ..."
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Cited by 13 (5 self)
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this paper, the results naturally apply to (asymptotically) stationary time series.
TwoStage Stopping Procedures Based On Standardized Time Series
 Management Science
, 1994
"... this paper we will consider functions h : C[0; 1) ! ! which are typically not continuous, and we let D(h) denote the set of points x 2 C[0; 1) at which h is not continuous. Let fX ffl : ffl ? 0g be a family of random elements taking values in C[0; 1); i.e., the X ffl correspond to stochastic process ..."
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Cited by 12 (6 self)
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this paper we will consider functions h : C[0; 1) ! ! which are typically not continuous, and we let D(h) denote the set of points x 2 C[0; 1) at which h is not continuous. Let fX ffl : ffl ? 0g be a family of random elements taking values in C[0; 1); i.e., the X ffl correspond to stochastic processes with sample paths in C[0; 1). If X is a random element of C[0; 1), then the X ffl are said to converge weakly to X (written X ffl ) X as ffl ! 0) if
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 11 (7 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.
Variance reduction in simulation of loss models
 Operations Research
, 1999
"... We propose a new estimator of steadystate blocking probabilities for simulations of stochastic loss models that can be much more efficient than the natural estimator (ratio of losses to arrivals). The proposed estimator is a convex combination of the natural estimator and an indirect estimator base ..."
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Cited by 10 (8 self)
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We propose a new estimator of steadystate blocking probabilities for simulations of stochastic loss models that can be much more efficient than the natural estimator (ratio of losses to arrivals). The proposed estimator is a convex combination of the natural estimator and an indirect estimator based on the average number of customers in service, obtained from Little’s law (L = λW). It exploits the known offered load (product of the arrival rate and the mean service time). The variance reduction is dramatic when the blocking probability is high and the service times are highly variable. The advantage of the combination estimator in this regime is partly due to the indirect estimator, which itself is much more efficient than the natural estimator in this regime, and partly due to strong correlation (most often negative) between the natural and indirect estimators. In general, when the variances of two component estimators are very different, the variance reduction from the optimal convex combination is about 1 − ρ 2, where ρ is the correlation between the component estimators. For loss models, the variances of the natural and indirect estimators are very different under both light and heavy loads. The combination estimator is effective for estimating multiple blocking probabilities in loss networks with multiple traffic classes, some of which are in normal
Maximum values in queueing processes
 Prob. Engrg. and Info. Sci
, 1995
"... Motivated by extremevalue engineering in service systems, we develop and evaluate simple approximations for the distributions of maximum values of queueing processes over large time intervals. We provide approximations for several different processes, such as the waiting times of successive custome ..."
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Cited by 9 (2 self)
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Motivated by extremevalue engineering in service systems, we develop and evaluate simple approximations for the distributions of maximum values of queueing processes over large time intervals. We provide approximations for several different processes, such as the waiting times of successive customers, the remaining workload at an arbitrary time, and the queue length at an arbitrary time, in a variety of models. All our approximations are based on extremevalue limit theorems. Our first approach is to approximate the queueing process by onedimensional reflected Brownian motion (RBM). We then apply the extremevalue limit for RBM, which we derive here. Our second approach starts from exponential asymptotics for the tail of the steadystate distribution. We obtain an approximation by relating the given process to an associated sequence of i.i.d. random variables with the same asymptotic exponential tail. We use estimates of the asymptotic variance of the queueing process to determine an approximate number of variables in this associated i.i.d. sequence. Our third approach is to simplify GI/G/1 extremevalue limiting formulas in Iglehart (1972) by approximating the distribution of an idle period by the stationaryexcess distribution of an interarrival time. We use simulation to evaluate the quality of these approximations for the maximum workload. From the simulations, we obtain a rough estimate of the time when the extreme value limit theorems begin to yield good approximations.