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 framework is developed to estimate the mean delay performance of (σ, ρ, π) regulated flows in networks with acyclic routing. We first show that the mean delay performance can be bounded by on-off type processes with exponentially distributed off periods. We then obtain per-flow bounds on the mean delay. We show that when there is no peak rate constraint, the Pollaczek-Khinchine formula for M/G/1 queues provides a tight bound thus establishing the Better-than-Poisson property for such flows. We then consider flows inside a network and show that they can be characterized by a stochastic burstiness parameter and show how the aggregate performance can be bounded from the asymptotic Better-than-Poisson property of regulated flows.

Network Calculus for Mean Delay Analysis Through a Network

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

In this note we show that if a continuous-time, nonlinear, time-invariant, finite-dimensional system evolves on a compact subset of Rn and if the Jacobian of the vector field is Hurwitz at each point of the compact set, then there is a unique equilibrium on the set and solutions exponentially converge to it. This shows that the Markus-Yamabe conjecture, which is false in general on Rn, n>2, holds on compact sets. The results of this note can be viewed as an application of Krasovskii's method for constructing Lyapunov functions and we are able to similarly construct Lyapunov-like functions valid on the given compact set. Examples are provided to illustrate the result.

Exponential Stability and the Markus-Yamabe Conjecture in Compact Spaces

Prior work has addressed the effects of multipath fading and path loss separately for broadcast and multiple-access channels. It has been shown that the number of simultaneously active users are of the order $$\Theta (\ln (\ln (n)))$$ź(ln(ln(n))) for random channel gains with exponentially-decaying tails, where n is the total number of users. Furthermore, it has been shown that assuming path loss is dominant and ignoring multipath fading, the user capacity (i.e. the maximum number of simultaneously active users in a wireless system) of multi-user channels is of the order $$\Theta (\ln (n))$$ź(ln(n)) when the users are spatially distributed on the plane with a Gaussian or Uniform distribution. In this paper, we study the shadowing effects on the number of active users for broadcast and multiple-access channels in which the users are randomly distributed on the plane. It is shown that as the total number of users in the multi-user channel goes to infinity, the number of active users is of the order $$\Theta (\sqrt{\ln (n)}).$$ź(ln(n)).

Effects of shadowing on the number of active users in random multi-user channels

In this paper we study a large system of N servers, each with capacity to process at most C simultaneous jobs; an incoming job is routed to a server if it has the lowest occupancy amongst d (out of N) randomly selected servers. A job that is routed to a server with no vacancy is assumed to be blocked and lost. Such randomized policies are referred to JSQ(d) (Join the Shortest Queue out of d) policies. Under the assumption that jobs arrive according to a Poisson process with rate where , and , we establish functional central limit theorems for the fluctuation process in both the transient and stationary regimes when service time distributions are exponential. In particular, we show that the limit is an Ornstein–Uhlenbeck process whose mean and variance depend on the mean field of the considered model. Using this, we obtain approximations to the blocking probabilities for large N, where we can precisely estimate the accuracy of first-order approximations.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/ED84156F1ADD3A8A9F67C1E2A9FF61A1/S0021900220000996a.pdf/div-class-title-sensitivity-of-mean-field-fluctuations-in-erlang-loss-models-with-randomized-routing-div.pdf

Ravi R. Mazumdar

Papers

Network Calculus for Mean Delay Analysis Through a Network

Direct modeling of white noise in stochastic systems

Exponential Stability and the Markus-Yamabe Conjecture in Compact Spaces

Effects of shadowing on the number of active users in random multi-user channels

Sensitivity of mean-field fluctuations in Erlang loss models with randomized routing