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

The flow control problem in a multiclass communications environment is modelled as a multilevel problem, where one of the users or the manager of the network has a precedence in the decision process. Users control the number of packets they allow into the network to maximize their throughput under a maximum allowable average delay constraint. The original nonlinear multilevel problem is transformed to a multilevel linear program, which is subsequently solved through an equivalent linear programming formulation. The effects of delay constraints, number of allowed packets and speeds of the links on the performance of the system and on the behaviour of the individual users are studied extensively.

Multilevel flow control in telecommunication networks

The problem of estimating discrete time stochastic processes by autoregressive (AR) models is encountered in many applications The present paper explores the asymptotic behavior of the spectral density of such approximations It is shown that under certain assumptions on the spectral density and the covariance sequence of the original process, the spectral density of the approximating autoregressive sequence converges at the origin Under additional mild conditions it is also shown that the spectral density of the autoregressive approximation converges in L2 as the order of approximation increases

On the asymptotic behavior of the spectral density of autoregressive estimates

We study an optimal bidding and allocation problem that builds upon the earlier work of [4]. The model we study is of interest in online real-time advertising where the demand of a large number of advertisers is aggregated by an intermediary who bids in the advertising auction market on their behalf. This intermediary is referred to as a Demand Side Platform (DSP).

Real-time Bidding for Time Constrained Impression Contracts in First and Second Price Auctions - Theory and Algorithms

The authors consider the nonlinear filtering model with additive white noise to be the identity map on L/sub 2/[0,T] with standard Gauss measure thereon. Using a representation result for maps which are continuous in a locally convex topology generated by seminorms of Hilbert-Schmidt operators on the Hilbert space, the authors show that the filter map can be written as the composition of a continuous nonlinear map (which does not depend on the observation) with a linear Hilbert-Schmidt operator acting on the observation. In particular, this result gives a direct proof of existence of approximation of nonlinear filters in terms of Volterra polynomials.

A representation result for nonlinear filter maps in a white noise framework

We study the problem of joint congestion control and scheduling in wireless networks. We model the wireless network as a directed graph G = (V, E), where V denotes the set of nodes and E denotes the set of wireless links between the nodes. We propose a joint congestion control and scheduling scheme that achieves a fraction DI(G) of the capacity region, where dI(G) depends on certain structural properties of graph G as well as the nature of interference constraints. For specific families of graphs, which can represent a wide variety of wireless networks, dI(G) has been upper bounded by a factor independent of the number of nodes in the network for a wide range of interference models. The scheduling element of our joint congestion control and scheduling scheme is the maximal scheduling policy considered in many of the earlier works. Although, it is widely believed to be amenable to distributed implementation, no algorithms have been proposed for its implementation, except under the node-exclusive interference model which is suitable only for networks in which adjacent nodes can transmit over non-interfering channels. We propose a randomized algorithm for implementing maximal scheduling policy under a 2-hop interference model which is suitable for networks with a limited number of non-interfering channels (e.g., IEEE 802.11 DSSS networks).

/pdf/randomized-algorithms-for-cross-layer-network-control-e8rfv43vpk.pdf

Ravi R. Mazumdar

Papers

Multilevel flow control in telecommunication networks

On the asymptotic behavior of the spectral density of autoregressive estimates

Real-time Bidding for Time Constrained Impression Contracts in First and Second Price Auctions - Theory and Algorithms

A representation result for nonlinear filter maps in a white noise framework

Randomized Algorithms for Cross-Layer Network Control