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

Classical Gaussian white noise in communications and signal processing is viewed as the limit of zero mean second order Gaussian processes with a compactly supported flat spectral density as the support goes to infinity. The difficulty of developing a theory to deal with nonlinear transformations of white noise has been to interpret the corresponding limits. In this paper we show that a renormalization and centering of powers of band-limited Gaussian processes is Gaussian white noise and as a consequence, homogeneous polynomials under suitable renormalization remain white noises.

On Powers of Gaussian White Noise

We show in this paper that if a stationary traffic source is regulated by a leaky bucket with leak rate p and bucket size a, then the amount of information generated in successive time intervals is dominated, in the increasing convex ordering sense, by that of a Poisson arrival process with rate p/a, with each arrival bringing an amount of information equal to σ. By exploiting this property, we then show that the mean value in the stationary regime of the content of a buffer drained at constant rate and fed with the superposition of regulated flows is less than the mean value of the same buffer fed with an adequate Poisson process, whose characteristics depend upon the regulated input flows.

/pdf/a-stochastic-ordering-property-for-leaky-bucket-regulated-3mfkgg81if.pdf

A Stochastic Ordering Property for Leaky Bucket Regulated Flows in Packet Networks

In this paper we consider a generalization of the so called M/G/∞ model where M types of sessions enter a buffer. The instantaneous rates of the sessions are functions of the occupancy of an M/G/∞ system with Weibullian G distributions. In particular we assume that a session of type i transmits ri cells per unit time and lasts for a random time τ with a Weibull distribution given by Pr (τ>x)∼e−γixαi, where 0 0. We show that the complementary buffer occupancy distribution for large buffer size is Weibullian whose parameters can be determined as the solution of a deterministic nonlinear knapsack problem. For αi<0.5, upper and lower bound factors are determined. When specialized to the homogeneous case, i.e., when all the sessions are identical, the result coincides with a lower bound reported in the literature.

Large Buffer Asymptotics for Fluid Queues with Heterogeneous M / G /∞ Weibullian Inputs

In this paper, we study a joint power allocation and base-station assignment problem for the downlink in multi-class CDMA networks. In a previous paper (J.W. Lee et al., 2002), we developed a power allocation scheme based on a utility and pricing framework considering a single cell in the system. In this paper, we consider a multi-cell system in which each cell uses the power allocation algorithm developed in (J.W. Lee et al., 2002). Based on the power allocation algorithm, we propose a pricing based base-station assignment algorithm, which results in a joint power allocation and base-station assignment scheme. In this algorithm, a base-station is assigned to a mobile taking into account the congestion level of the cell as well as the transmission environment between the base-station and the mobile. We compare the performance of the proposed algorithm with that of the traditional SIR based base-station assignment algorithm. The result shows that our proposed algorithm outperforms the SIR based algorithm.

Joint power allocation and base-station assignment based on pricing for the downlink in multi-class CDMA networks

Let M be a nonlinear transformation on a separable Hilbert space with range in H. Let mu denote a standard Gauss measure on H. It is shown that, under suitable conditions on M, there exists an exponential transformation L (completely characterized by M) on H such that d eta =Ld mu defines a finitely additive or cylindrical probability measure on H under which (I-M)(.) is white noise. This is the white noise version of the Girsanov theorem. The nonlinear filtering model is considered, and the Radon-Nikodym derivative of the cylindrical measure induced by the observation process on H is interpreted, showing that it has a pathwise characterization in terms of the nonlinear filter map. It is then shown that if the signal process is the solution to a nonlinear differential equation with a white noise input, then the innovation process is white noise under the cylindrical measure induced by the observation process and the innovations process is related to the observation process by a continuous, causally invertible map. >

Ravi R. Mazumdar

Papers

On Powers of Gaussian White Noise

A Stochastic Ordering Property for Leaky Bucket Regulated Flows in Packet Networks

Large Buffer Asymptotics for Fluid Queues with Heterogeneous M / G /∞ Weibullian Inputs

Joint power allocation and base-station assignment based on pricing for the downlink in multi-class CDMA networks

A white noise version of the Girsanov theorem