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

Ein Fernbedienungssystem umfasst ein erstes Netzwerk, das zur Kommunikation uber ein erstes Format konfiguriert ist. Das erste Netzwerk umfasst einen Router, der dazu konfiguriert ist, paketierte Fernbedienungscodes zu empfangen, die zum Router im ersten Format gesendet werden. Das Fernbedienungssystem umfasst ferner ein zweites Netzwerk, das zur Kommunikation uber einen Satz von zweiten Formaten konfiguriert ist. Das zweite Netzwerk umfasst einen Umsetzer, der mit dem Router uber eine Netzwerkverbindung kommunikativ gekoppelt ist. Der Umsetzer ist dazu konfiguriert, die paketierten Fernbedienungscodes vom Router im ersten Format zu empfangen und die Fernbedienungscodes vom ersten Format in einen Satz von zweiten Formaten umzusetzen. Das zweite Netzwerk umfasst ferner einen Verteiler, der mit dem Umsetzer kommunikativ gekoppelt ist. Der Verteiler ist dazu konfiguriert, die paketierten Fernbedienungscodes im Satz von zweiten Formaten zu empfangen und die paketierten Fernbedienungscodes im Satz von zweiten Formaten zu einem Satz von Verbraucherelektronikvorrichtungen zu senden.

Vorrichtung und Betriebsverfahren für ein Fernbedienungssystem

We study problems arising in real-time auction markets, common in e-commerce and computational advertising, where bidders face the problem of calculating optimal bids. We focus upon a contract management problem where a demand aggregator is subject to multiple contractual obli-gations requiring them to acquire items of heterogeneous types at a speciﬁed rate, which they will seek to fulﬁll at minimum cost. Our main results show that, through a transformation of variables, this problem can be formulated as a convex optimization problem, for both ﬁrst and second price auctions. Convexity results in efﬁcient algorithms for solving instances of this problem, and the resulting duality theory admits rich structure and interpretations. Additionally, we show that the transformation of variables used to formulate this problem as a convex program can also be used to guarantee the convexity of optimal bidding problems studied by other authors (who did not leverage convexity). Finally, we show how the expected cost of bidding in second price auctions is formally identical to certain transaction costs when submitting market orders in limit order book markets. This fact is used to analyze a Markowitz portfolio problem which accounts for these transaction costs, establishing an interesting connection between ﬁnance and optimal bidding.

Convexity and Duality in Optimum Real-time Bidding and Related Problems

The approximation of a stationary time-series by finite orde r autoregressive (AR) and moving averages (MA) is a problem that occurs in many applications. In this paper we study asymptotic behavior of the spectral density of finite order approximations of wide sens e stationary time series. It is shown that when the on the spectral density is non-vanishing in [−π,π] and the covariance is summable, the spectral density of the approximating autoregressive sequence converges at the origin. Under additional mild conditions on the coefficients of the Wold decomposition it is also shown th at the spectral densities of both moving average and autoregressive approximations converge in L2 as the order of approximation increases.

On the Convergence of Finite Order Approximations of Stationary Time Series

In this paper we study charging schemes for bandwidth or server usage under the processor sharing discipline. Specifically, we analyze post-payment and pre-payment (or payment on arrival) schemes in three charging frameworks: fixed-rate charging, Vickrey---Clarke---Groves based charging, and congestion based charging for users with logarithmic utilities. We show that in the absence of QoS constraints, the network operator can earn unbounded profits and thus there is a need to devise schemes where users are only charged if they are given a minimum rate. We obtain explicit characterizations for mean user payments and the operator's mean revenue for these frameworks. We also analyze charge volatility via the second moments of the above implementations of arrival-based payments and post-payments. The volatility reflects the confidence in mean revenue for the operator and expected charges for a user. We present conditions under which a pre-payment mechanism is preferable over a post-payment mechanism. We also show that the same analysis can be applied to a scenario with admission control where each entering user is guaranteed a minimum service rate.

Pricing schemes in processor sharing systems

In this survey we present an overview of the current state-of-the-art and the future issues to be resolved for the deployment of integrated communications networks based on the Asynchronous Transfer Mode (ATM) Our presentation is from the perspective of traffic and congestion control and management issues which hold the key to the successful achievement of this deployment since it is mainly in these aspects that ATM networks differ from conventional communication networks.

/pdf/traffic-engineering-in-atm-networks-current-trends-and-3mo86syskt.pdf

Ravi R. Mazumdar

Papers

Vorrichtung und Betriebsverfahren für ein Fernbedienungssystem

Convexity and Duality in Optimum Real-time Bidding and Related Problems

On the Convergence of Finite Order Approximations of Stationary Time Series

Pricing schemes in processor sharing systems

Traffic engineering in ATM networks: Current trends and future issues