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, a power allocation scheme is proposed to maximize the number of active terminals in fading multi-user channels in which a minimum rate must be maintained for all active users. It is assumed that receivers and transmitters have perfect channel state information. Both uplink and downlink scenarios are considered and under the assumption of independent Rayleigh fading channels for different terminals, the optimal number of active terminals is asymptotically obtained as the total number of users, n, is large enough. For broadcast channels with successive interference cancellation at receivers and multiple- access channels with joint decoding at the receiver, the maximum number of active terminals is shown to be arbitrarily close to In ( Ptotal/sigma2\ In n) /Rmin with probability approaching one, where Ptotal and Rmin denote total transmit power and the minimum rate respectively, and sigma2 represents the noise variance.

User Capacity of Fading Multi-User Channels with a Minimum Rate Constraint

In this paper we explain the line of development that led to a direct modeling of white noise in stochastic systems. This is followed by the mathematical theory of finitely additive white noise and some recent results on modeling the state process with white noise input. Likelihood ratio plays a central role throughout this paper.

Direct modeling of white noise in stochastic systems

We consider the problem of determining asymptotic bounds on the capacity of a random ad hoc network. Previous approaches assumed a link layer model in which if a transmitter-receiver pair can communicate with each other, i.e., the Signal to Interference and Noise Ratio (SINR) is above a certain threshold, then every transmitted packet is received error-free by the receiver thereby. Using this model, the per node capacity of the network was shown to be $\Theta(\frac{1}{\sqrt{n\log{n}}})$. In reality, for any finite link SINR, there is a non-zero probability of erroneous reception of the packet. We show that in a large network, as the packet travels an asymptotically large number of hops from source to destination, the cumulative impact of packet losses over intermediate links results in a per-node throughput of only $O(\frac{1}{n})$. We then propose a new scheduling scheme to counter this effect. The proposed scheme provides tight guarantees on end-to-end packet loss probability, and improves the per-node throughput to $\Omega(\frac{1}{\sqrt{n} ({\log{n}})^{\frac{\alpha{+2}}{2(\alpha-2)}}})$ where $\alpha>2$ is the path loss exponent.

The Capacity of Ad hoc Networks under Random Packet Losses

We consider the network control problem for wireless networks with flow level dynamics under the general k-hop interference model. In particular, we investigate the control problem in low load and high load regimes. In the low load regime, we show that the network can be stabilized by a regulated maximal scheduling policy considering flow level dynamics if the offered load satisfies a constraining bound condition. Because maximal matching is a general scheduling rule whose implementation is not specified, we propose a constant-time and distributed scheduling algorithm for a general k-hop interference model which can approximate the maximal scheduling policy within an arbitrarily small error. Under the stability condition, we show how to calculate transmission rates for different user classes such that the long-term (time average) network utility is maximized. Our results imply that congestion control is unnecessary when the offered load is low and optimal user rates can be determined to maximize users' long-term satisfaction. In the high load regime where the network can be unstable under the regulated maximal scheduling policy, we propose the cross-layer congestion control and scheduling algorithm which can stabilize the network under arbitrary network load. Through numerical analysis for some typical networks, we show that the proposed scheduling algorithm has much lower overhead than other existing queue-length-based constant-time scheduling schemes in the literature, and it achieves performance much better than the guaranteed bound. In addition, using congestion control in the low load condition results in much lower average utility compared to that due to the optimal transmission rate derived in the paper.

/pdf/appropriate-control-of-wireless-networks-with-flow-level-1s3bx4oxq6.pdf

Appropriate control of wireless networks with flow level dynamics

We study binary opinion dynamics in a fully connected network of interacting agents. The agents are assumed to interact according to one of the following rules: (1) Voter rule: An updating agent simply copies the opinion of another randomly sampled agent; (2) Majority rule: An updating agent samples multiple agents and adopts the majority opinion in the selected group. We focus on the scenario where the agents are biased towards one of the opinions called the {\em preferred opinion}. Using suitably constructed branching processes, we show that under both rules the mean time to reach consensus is $\Theta(\log N)$, where $N$ is the number of agents in the network. Furthermore, under the majority rule model, we show that consensus can be achieved on the preferred opinion with high probability even if it is initially the opinion of the minority. We also study the majority rule model when stubborn agents with fixed opinions are present. We find that the stationary distribution of opinions in the network in the large system limit using mean field techniques.

Ravi R. Mazumdar

Papers

User Capacity of Fading Multi-User Channels with a Minimum Rate Constraint

Direct modeling of white noise in stochastic systems

The Capacity of Ad hoc Networks under Random Packet Losses

Appropriate control of wireless networks with flow level dynamics

Opinion Dynamics under Voter and Majority Rule Models with Biased and Stubborn Agents.