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 consider the problem of assignning jobs to servers in a multi-server system consisting of N parallel processor sharing servers, categorized into M (≪ N) different types according to their proces...

/pdf/randomized-assignment-of-jobs-to-servers-in-heterogeneous-5eu8n9wauu.pdf

Randomized assignment of jobs to servers in heterogeneous clusters of shared servers for low delay

In this paper we consider a generalization of the so-called M/G//spl infin/ model where M types of long-tailed sessions enter a buffer. The instantaneous rates of the sessions are functions of the occupancy of an M/G//spl infin/ system with long-tailed G distributions. In particular we assume that a session of type i transmits r/sub i/ cells per unit time and lasts for a random time r with long-tailed distribution given by P(/spl tau/>x)/spl sim//spl alpha//sub i/x(-(1+/spl beta//sub i/)) where /spl beta//sub i/>0. We derive the mean cell loss asymptotics for large buffer size as well as the complementary distribution of the buffer occupancy exceeding a high level. When specialized to the homogeneous case we show that recent results on large buffer asymptotics, which have been shown under more restrictive assumptions, hold more generally. In the heterogeneous case, we show that the asymptotics are not necessarily governed by the sources with the smallest /spl beta//sub i/ but also depend on the rates r/sub i/ and it is the ratio of /spl beta//sub i/ to r/sub i/ which is important. Finally it is a simple observation that light-tailed (exponential tails for example) sources have no influence on the asymptotics except that they contribute to reducing the capacity available to heavy-tailed sources by their mean load.

Cell loss asymptotics in buffers fed by heterogeneous long-tailed sources

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 preferred opinion. Using suitably constructed branching processes, we show that under both rules the mean time to reach consensus is $$\varTheta (\log N)$$
 , 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.

/pdf/voter-and-majority-dynamics-with-biased-and-stubborn-agents-1rvuzt5rt1.pdf

Voter and Majority Dynamics with Biased and Stubborn Agents

This paper extends the rate conservation principle to cadlag processes whose jumps form a non-stationary point process with a time-dependent intensity. It is shown that this is a direct consequence of path integration and the strong law of large numbers for local martingales. When specialized to mean rates a non-stationary version of Miyazawa's result is obtained which is recovered in the stationary case. Some applications of the result are also given.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/216CBC88ECD25D602DBB7D3E416009B5/S0021900200042674a.pdf/div-class-title-on-rate-conservation-for-non-stationary-processes-div.pdf

On rate conservation for non-stationary processes

In this paper, we investigate the impact of random interactions between agents in a social network on the diffusion of opinions in the network. Opinion of each agent is assumed to be a binary variable and each agent is assumed to be able to interact with any other agent in the network. This models scenarios where every agent in the network has to choose from two available options and the size of the neighborhood of each agent is an increasing function of the total number agents in the network. It is assumed that each agent updates its opinion at random instants upon interacting with other randomly sampled agents. We consider two simple rules of interaction: (1) the voter rule in which the updating agent simply copies the opinion of another randomly sampled agent, (2) the majority rule, in which the updating agent samples multiple agents and adopts the majority opinion among the sampled agents and the agent itself. Under each rule, we consider two different scenarios which have not been considered in the literature thus far: (1) where the agents are 'biased' towards one of the opinions, (2) where different agents have different degrees of stubbornness. We show that the presence of biased agents reduces the consensus time for the voter rule exponentially as compared to the case where the agents are unbiased. For the majority rule model with biased agents, we show that the network reaches consensus on a particular opinion with high probability only when the initial fraction of agents having that opinion is above a certain threshold. For the majority rule model with stubborn agents, we observe metastability where the network switches back and forth between stable states spending long intervals in each state.

Ravi R. Mazumdar

Papers

Randomized assignment of jobs to servers in heterogeneous clusters of shared servers for low delay

Cell loss asymptotics in buffers fed by heterogeneous long-tailed sources

Voter and Majority Dynamics with Biased and Stubborn Agents

On rate conservation for non-stationary processes

Binary Opinion Dynamics with Biased Agents and Agents with Different Degrees of Stubbornness