Convergence of Probability Measures. By P. Billingsley. Chichester, Sussex, Wiley, 1968. xii, 253 p. 9 1/4“. 117s.

Convergence of Probability Measures

This paper analyses the stability and fairness of two classes of rate control algorithm for communication networks. The algorithms provide natural generalisations to large-scale networks of simple additive increase/multiplicative decrease schemes, and are shown to be stable about a system optimum characterised by a proportional fairness criterion. Stability is established by showing that, with an appropriate formulation of the overall optimisation problem, the network's implicit objective function provides a Lyapunov function for the dynamical system defined by the rate control algorithm. The network's optimisation problem may be cast in primal or dual form: this leads naturally to two classes of algorithm, which may be interpreted in terms of either congestion indication feedback signals or explicit rates based on shadow prices. Both classes of algorithm may be generalised to include routing control, and provide natural implementations of proportionally fair pricing.

https://link.springer.com/content/pdf/10.1057%2Fpalgrave.jors.2600523.pdf

Rate control for communication networks: shadow prices, proportional fairness and stability

A survey of current continuous nonlinear multi-objective optimization (MOO) concepts and methods is presented. It consolidates and relates seemingly different terminology and methods. The methods are divided into three major categories: methods with a priori articulation of preferences, methods with a posteriori articulation of preferences, and methods with no articulation of preferences. Genetic algorithms are surveyed as well. Commentary is provided on three fronts, concerning the advantages and pitfalls of individual methods, the different classes of methods, and the field of MOO as a whole. The Characteristics of the most significant methods are summarized. Conclusions are drawn that reflect often-neglected ideas and applicability to engineering problems. It is found that no single approach is superior. Rather, the selection of a specific method depends on the type of information that is provided in the problem, the user’s preferences, the solution requirements, and the availability of software.

Survey of multi-objective optimization methods for engineering

The author's preface gives an outline: "This book is about weakconvergence methods in metric spaces, with applications sufficient to show their power and utility. The Introduction motivates the definitions and indicates how the theory will yield solutions to problems arising outside it. Chapter 1 sets out the basic general theorems, which are then specialized in Chapter 2 to the space C[0, l ] of continuous functions on the unit interval and in Chapter 3 to the space D [0, 1 ] of functions with discontinuities of the first kind. The results of the first three chapters are used in Chapter 4 to derive a variety of limit theorems for dependent sequences of random variables. " The book develops and expands on Donsker's 1951 and 1952 papers on the invariance principle and empirical distributions. The basic random variables remain real-valued although, of course, measures on C[0, l ] and D[0, l ] are vitally used. Within this framework, there are various possibilities for a different and apparently better treatment of the material. More of the general theory of weak convergence of probabilities on separable metric spaces would be useful. Metrizability of the convergence is not brought up until late in the Appendix. The close relation of the Prokhorov metric and a metric for convergence in probability is (hence) not mentioned (see V. Strassen, Ann. Math. Statist. 36 (1965), 423-439; the reviewer, ibid. 39 (1968), 1563-1572). This relation would illuminate and organize such results as Theorems 4.1, 4.2 and 4.4 which give isolated, ad hoc connections between weak convergence of measures and nearness in probability. In the middle of p. 16, it should be noted that C*(S) consists of signed measures which need only be finitely additive if 5 is not compact. On p. 239, where the author twice speaks of separable subsets having nonmeasurable cardinal, he means "discrete" rather than "separable." Theorem 1.4 is Ulam's theorem that a Borel probability on a complete separable metric space is tight. Theorem 1 of Appendix 3 weakens completeness to topological completeness. After mentioning that probabilities on the rationals are tight, the author says it is an

Convergence of probability measures

It is shown that, particularly for terrestrial cellular telephony, the interference-suppression feature of CDMA (code division multiple access) can result in a many-fold increase in capacity over analog and even over competing digital techniques. A single-cell system, such as a hubbed satellite network, is addressed, and the basic expression for capacity is developed. The corresponding expressions for a multiple-cell system are derived. and the distribution on the number of users supportable per cell is determined. It is concluded that properly augmented and power-controlled multiple-cell CDMA promises a quantum increase in current cellular capacity. >

On the capacity of a cellular CDMA system

In this paper we derive a multidimensional version of the rate conservation law (RCL) for cadlag processes of bounded variation. These results are then used to analyze queueing models which have a natural multidimensional characterization, such as priority queues. In particular the RCL is used to establish certain conservation laws between the idle probabilities for such queues. We use the relations to provide a detailed analysis of preemptive resume priority queues with M/G inputs. Special attention is paid to the validity of the so-called reduced service rate approximation.

Rate Conservation Laws for Multidimensional Processes of Bounded Variation with Applications to Priority Queueing Systems

In this paper, we study a large multi-server loss model under the SQ(d) routeing scheme when the service time distributions are general with finite mean. Previous works have addressed the exponential service time case when the number of servers goes to infinity, giving rise to a mean field model. The fixed point of the limiting mean field equations (MFEs) was seen to be insensitive to the service time distribution in simulations, but no proof was available. While insensitivity is well known for loss systems, the models, even with state-dependent inputs, belong to the class of linear Markov models. In the context of SQ(d) routeing, the resulting model belongs to the class of nonlinear Markov processes (processes whose generator itself depends on the distribution) for which traditional arguments do not directly apply. Showing insensitivity to the general service time distributions has thus remained an open problem. Obtaining the MFEs in this case poses a challenge due to the resulting Markov description of the system being in positive orthant as opposed to a finite chain in the exponential case. In this paper, we first obtain the MFEs and then show that the MFEs have a unique fixed point that coincides with the fixed point in the exponential case, thus establishing insensitivity. The approach is via a measure-valued Markov process representation and the martingale problem to establish the mean field limit.

/pdf/insensitivity-of-the-mean-field-limit-of-loss-systems-under-2v8zrozj9l.pdf

Insensitivity of the mean field limit of loss systems under SQ(d) routeing

http://www.lk.cs.ucla.edu/data/files/Rosenberg/On%20the%20Analysis%20of%20Exponential%20Queuing%20Systems%20with.pdf

On the analysis of exponential queuing systems with randomly changing arrival rates: stability conditions and finite buffer scheme with a resume level

Classical Gaussian white noise in communications and signal processing is viewed as the limit of zero-mean second-order Gaussian processes with a compactly supported flat spectral density as the support goes to infinity. The difficulty of developing a theory to deal with nonlinear transformations of white noise has been to interpret the corresponding limits. In this paper, we show that a renormalization and centering of powers of band-limited Gaussian processes is Gaussian white noise, and, as a consequence, homogeneous polynomials under suitable renormalization remain white noises.

On Powers of Gaussian White Noise

In this paper, we study large multi-server loss models under power-of-$d$ routing scheme when service time distributions are general with finite mean. Previous works have addressed the exponential service time case when the number of servers goes to infinity giving rise to a mean field model. The fixed point of limiting mean field equations (MFE) was shown to be insensitive to the service time distribution through simulation. Showing insensitivity to general service time distributions has remained an open problem. Obtaining the MFE in this case poses a challenge due to the resulting Markov description of the system being in positive orthant as opposed to a finite chain in the exponential case. In this paper, we first obtain the MFE and then show that the MFE has a unique fixed point that coincides with the fixed point in the exponential case thus establishing insensitivity. The approach is via a measure-valued Markov process representation and the martingale problem to establish the mean-field limit. The techniques can be applied to other queueing models.

Ravi R. Mazumdar

Papers

Rate Conservation Laws for Multidimensional Processes of Bounded Variation with Applications to Priority Queueing Systems

Insensitivity of the mean field limit of loss systems under SQ(d) routeing

On the analysis of exponential queuing systems with randomly changing arrival rates: stability conditions and finite buffer scheme with a resume level

On Powers of Gaussian White Noise

Insensitivity of the mean-field Limit of Loss Systems Under Power-of-d Routing