Abstract. When studying convergence of measures, an important issue is the choice of probability metric. We provide a summary and some new results concerning bounds among some important probability metrics/distances that are used by statisticians and probabilists. Knowledge of other metrics can provide a means of deriving bounds for another one in an applied problem. Considering other metrics can also provide alternate insights. We also give examples that show that rates of convergence can strongly depend on the metric chosen. Careful consideration is necessary when choosing a metric. Abrégé. Le choix de métrique de probabilité est une décision très importante lorsqu’on étudie la convergence des mesures. Nous vous fournissons avec un sommaire de plusieurs métriques/distances de probabilité couramment utilisées par des statisticiens(nes) at par des probabilistes, ainsi que certains nouveaux résultats qui se rapportent à leurs bornes. Avoir connaissance d’autres métriques peut vous fournir avec un moyen de dériver des bornes pour une autre métrique dans un problème appliqué. Le fait de prendre en considération plusieurs métriques vous permettra d’approcher des problèmes d’une manière différente. Ainsi, nous vous démontrons que les taux de convergence peuvent dépendre de façon importante sur votre choix de métrique. Il est donc important de tout considérer lorsqu’on doit choisir une métrique. 1.

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This paper introduces a new aspect of queueing theory, the study of systems that service customers with specic timing requirements (e.g. due dates or deadlines). Unlike standard queueing theory in which common performance measures are customer delay, queue length and server utilization, real-time queueing theory focuses on the ability of a queue discipline to meet customer timing requirements, e.g., the fraction of customers who meet their requirements and the distribution of customer lateness. It also focuses on queue control policies to reduce or minimize lateness, although these control aspects are not explicitly addressed in this paper. To study these measures, one must keep track of the lead-times (deadline minus current time) of each customer, hence the system state is of unbounded dimension. A heavy trac analysis is presented for the earliest deadline rst (EDF) scheduling policy. This analysis decomposes the behavior of the real-time queue into two parts: the number in the sys...

This paper presents a heavy traffic analysis of the behavior of multi-class acyclic queueing networks in which the customers have deadlines. We assume the queueing system consists of J stations, and there are K different customer classes. Customers from each class arrive to the network according to independent renewal processes. The customers from each class are assigned a random deadline drawn from a deadline distribution associated with that class and they move from station to station according to a fixed acyclic route. The customers at a given node are processed according to the earliest-deadline-first (EDF) queue discipline. At any time, the customers of each type at each node have a lead time, the time until their deadline lapses. We model these lead times as a random counting measure on the real line. Under heavy traffic conditions and suitable scaling, it is proved that the measure-valued lead-time process converges to a deterministic function of the workload process. A two-station example is worked out in details, and simulation results are presented to illustrate the predictive value of the theory. This work is a generalization of Doytchinov, Lehoczky and Shreve [5], which developed these results for the single queue case.

A single queueing station which serves K input streams is considered. Each stream is an independent renewal process, with customers having random leadtimes. Customers are served by processor sharing across streams. Within each stream, two disciplines are considered { earliest-deadline-rst and rstin -rst-out. The set of current lead times of the K streams is modeled as a K-dimensional vector of random counting measures on R , and the limit of this vector of measure-valued processes is obtained under heavy trac conditions. Short title: Multiple-Input Queues Keywords: Due dates, heavy trac, queueing, diusion limits, random measures AMS subject classication: Primary 60K25; Secondary 60G57, 60J65 1 Supported by the Center for Nonlinear Analysis (NSF Grant No. DMS-98-03791). 2 Supported by NSF Grant No. DMS-98-02464. 1 Introduction Over the last 10 years, communication technology has become dramatically more sophisticated. There are now many dierent types of communication serv...

In a recent paper the author obtained optimal bounds for the strong Gaussian approximation of sums of independent R d-valued random vectors with finite exponential moments. The results may be considered as generalizations of well-known results of Komlós–Major–Tusnády and Sakhanenko. The dependence of constants on the dimension d and on distributions of summands is given explicitly. Some related problems are discussed.

This paper presents a second-order heavy traffic analysis of a single server queue that processes customers having deadlines using the earliest-deadline-first scheduling policy. For such systems, referred to as real-time queueing systems, performance is measured by the fraction of customers who meet their deadline, rather than more traditional performance measures, such as customer delay, queue length or server utilization. To model such systems, one must keep track of customer lead times (the time remaining until a customer deadline elapses) or equivalent information. This paper reviews the earlier heavy traffic analysis of such systems that provided approximations to the system’s behavior. The main result of this paper is the development of a second-order analysis that gives the accuracy of the approximations and the rate of convergence of the sequence of realtime queueing systems to its heavy traffic limit. 1. Introduction.

Mathematical models for climate studies are treated as a coupling link between physical and computational models. These models are characterized by the fact that small-scale phenomena influence the large-scale properties of the modelling system, yet the former cannot be extracted from the latter using available hardware and computational procedures. Climate systems belong to the class of systems whose dynamics are only observable in transient states. As a result, the sensitivity of models to coupling procedures requires an examination of the schemes responsible for transporting data between components. It is proposed to perform such an examination, based on the connection between error growth and the degree of coupling of model components, using adaptive error control. Key words: mathematical climate system models, coupling and decoupling procedures, hydrodynamic stability. 1 1 Introduction Elements of the mathematical modelling of climate can be traced back to Aristotle's Meteorol...

To assess the quality of solutions in stochastic inverse problems, a proper measure for the distance of random variables is essential. The aim of this note is the comparison of the metrics of Ky Fan and Prokhorov with other concepts such as expected values, probability estimates and almost sure convergence. In ill-posed problems one aims to find an appropriate solution x † to an equation of the form F (x) = y, when the operator F is not continuously invertible. Therefore, the problems of interest are unstable; when only noisy data y δ are available, special techniques (so called regularization methods) must be applied to obtain regularized solutions x δ α that are reasonable approximations to x †. To assess the quality of different regularization methods, in the theory of ill-posed problems convergence rate results, i. e., results of the form ∥x † − x δ ∥ α = O ( f ( ∥ ∥y − y δ∥∥)) are an accepted quality criterion (see [12] for an introduction into this topic). So in a nutshell, in the deterministic theory of inverse problems the aim is to bound the distance between desired and regularized solution, in terms of the distance between exact and noisy data.

Abstract — A MIMO downlink system in which data is transmitted to two users over a common wireless channel is considered. The channel is assumed to be fixed for all transmissions over the period of interest and the ratio of anticipated average arrival rates for the two users, also known as the relative traffic rate, is the system design parameter. A packet-based traffic model is considered where data for each user is queued at the transmit end. A queueing analogue for this system leads to a coupled queueing system for which a simple policy is known to be throughputoptimal under Markovian assumptions. Since an exact expression for the performance is not available, as a measure of performance (in heavy traffic), a diffusion approximation is established. This diffusion process is a two-dimensional semimartingale reflecting Brownian motion living in the positive quadrant of two-dimensional space. Index Terms — Coupled queueing systems, diffusion approximation, heavy traffic, multi-input multi-output (MIMO), semimartingale reflecting Brownian motion (SRBM). I.