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

We study Granger causality in the context of wide-sense stationary time series, where our focus is on the topological aspects of the underlying causality graph. We establish sufficient conditions (in particular, we develop the notion of a "strongly causal" graph topology) under which the true causality graph can be recovered via pairwise causality testing alone, and provide examples from the gene regulatory network literature suggesting that our concept of a strongly causal graph may be applicable to this field. We implement and detail finite-sample heuristics derived from our theory, and establish through simulation the efficiency gains (both statistical and computational) which can be obtained (in comparison to LASSO-type algorithms) when structural assumptions are met.

Graph Topological Aspects of Granger Causal Network Learning

We analyze the problem of how to optimally bid for ad spaces in online ad auctions. For this we consider the general case of multiple ad campaigns with overlapping targeting criteria. In our analysis we first characterize the structure of an optimal bidding strategy. In particular, we show that an optimal bidding strategies decomposes the problem into disjoint sets of campaigns and targeting groups. In addition, we show that pure bidding strategies that use only a single bid value for each campaign are not optimal when the supply curves are not continuous. For this case, we derive a lower-bound on the optimal cost of any bidding strategy, as well as mixed bidding strategies that either achieve the lower-bound, or can get arbitrarily close to it.

Optimal Bidding Strategies for Online Ad Auctions with Overlapping Targeting Criteria

We study a real-time bidding problem resulting from a set of contractual obligations stipulating that a firm win a specified number of heterogeneous impressions or ad placements over a defined duration in a real-time auction. The contracts specify item targeting criteria (which may be overlapping), and a supply requirement. Using the Pontryagin maximum principle, we show that the resulting continuous time and time inhomogeneous planning problem can be reduced into a finite dimensional convex optimization problem and solved to optimality. In addition, we provide algorithms to update the bidding plan over time via a receding horizon. Finally, we provide numerical results based on real data and show a connection to production-transportation problems.

Optimal Real-time Bidding Policies for Contract Fulfillment in Second Price Auctions.

Recently there has been a great interest in randomized load balancing schemes for large systems of parallel servers. Various policies such as SQ(d) (shortest queue amongst d randomly sampled), threshold based policies etc have been studied via a mean-field approach. All these policies are special cases of occupancy based routing decisions. In this paper we present a unified mean-field approach that holds for any routing scheme that only depends on the server occupancy in a system with a large numbers of processor sharing servers as an archetype of shared resource systems. The mean-field equations we obtain hold for general job length distributions unlike most recent works that assume exponentially distributed job lengths. We then show that the probability measure of occupancy defined on the set of non-negative integers Z_+ obtained from a fixed-point of the mean-field also satisfies the stationary mean-field equations under the assumption that the job lengths are exponential with the same average length. If the mean-field under exponential case has unique fixed-point, then the fixed point is insensitive to the job length distribution. The approach is via a measure-valued Markov process approach.

On Occupancy Based Randomized Routing Schemes in Large Systems of Shared Servers.

In this paper we provide some new insights on the stability of some Markovian stochastic network models. More precisely we show how a variation of a positive recurrence criterion for multidimensional Markov chains due to Rosberg (JAP, Vol. 17, No. 3, 1980) can be developed to generalize the necessary and sufficient conditions for stability for multiclass Markovian networks with class independent routing and any work conserving service discipline. We also discuss the stability of a coupled processor model of interest in wireless networks.

Ravi R. Mazumdar

Papers

Graph Topological Aspects of Granger Causal Network Learning

Optimal Bidding Strategies for Online Ad Auctions with Overlapping Targeting Criteria

Optimal Real-time Bidding Policies for Contract Fulfillment in Second Price Auctions.

On Occupancy Based Randomized Routing Schemes in Large Systems of Shared Servers.

Some new results on the stability of Markovian stochastic network models