We will review some of the major results in random graphs and some of the more challenging open problems. We will cover algorithmic and structural questions. We will touch on newer models, including those related to the WWW.

Random graphs

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Convex Analysisの二,三の進展について

Convergence of Probability Measures. By P. Billingsley. Chichester, Sussex, Wiley, 1968. xii, 253 p. 9 1/4“. 117s.

Convergence of Probability Measures

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

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Introduction To Multivariate Statistical Analysis

Statistical evidence suggests that the autocorrelation function p(k) (k=0,1,...) of a compressed-video sequence is better captured by p(k)=e/sup -/spl beta//spl radic/k/ than by p(k)=k/sup -/spl beta//=e/sup -/spl beta/logk/ (long-range dependence) or p(k)=e/sup -/spl beta/k/ (Markovian). A video model with such a correlation structure is introduced based on the so-called M/G//spl infin/ input processes. In essence, the M/G//spl infin/ process is a stationary version of the busy-server process of a discrete-time M/G//spl infin/ queue. By varying G, many forms of time dependence can be displayed, which makes the class of M/G//spl infin/ input models a good candidate for modeling many types of correlated traffic in computer networks. For video traffic, we derive the appropriate G that gives the desired correlation function p(k)=e/sup -/spl beta//spl radic/k/. Though not Markovian, this model is shown to exhibit short-range dependence. Poisson variates of the M/G//spl infin/ model are appropriately transformed to capture the marginal distribution of a video sequence. Using the performance of a real video stream as a reference, we study via simulations the queueing performance under three video models: our M/G//spl infin/ model, the fractional ARIMA model (which exhibits LRD), and the DAR(1) model (which exhibits a Markovian structure). Our results indicate that only the M/G//spl infin/ model is capable of consistently providing acceptable predictions of the actual queueing performance. Furthermore, only O(n) computations are required to generate an M/G//spl infin/ trace of length n, compared to O(n/sup 2/) for an F-ARIMA trace.

Modeling video traffic using M/G//spl infin/ input processes: a compromise between Markovian and LRD models

A simple queueing system, known as the fork-join queue, is considered with basic performance measure defined as the delay between the fork and join dates. Simple lower and upper bounds are derived for some of the statistics of this quantity. They are obtained, in both transient and steady-state regimes, by stochastically comparing the original system to other queueing systems with a structure simpler than the original system, yet with identical stability characteristics. In steady-state, under renewal assumptions, the computation reduces to standard GI/GI /1 calculations and the bounds constitute a first sizing-up of system performance. These bounds can also be used to show that for homogeneous fork-join queue system under assumptions, the moments of the system response time grow logarithmically in the number of parallel processors provided the service time distribution has rational Laplace–Stieltjes transform. The bounding arguments combine ideas from the theory of stochastic ordering with the notion of associated random variables, and are of independent interest to study various other queueing systems with synchronization constraints. The paper is an abridged version of a more complete report on the matter [6].

https://www.show.docjava.com/courses/cr346/data/papers/1427640.pdf

The fork-join queue and related systems with synchronization constraints: stochastic ordering and computable bounds

The authors consider queueing that occur naturally in the study of a class of resource-sharing problems under synchronization constraints such as resequencing and fork-join primitives These queueing models are amenable to a representation in terms of a state recursion The proposed methods of analysis are complementary and draw on classical ideas of queuing theory as well as on mathematical tools from the theory of stochastic ordering and ergodic theory The state recursion is at the center of all aspects of the analysis, be it for developing the exact solutions, obtaining bounds on system performance or establishing the stability conditions The ideas are illustrated on simple models of resequencing and fork-join synchronization, which emphasis put on deriving computable bounds on the performance measures >

Queueing models for systems with synchronization constraints

The random key graph is a random graph naturally associated with the random key predistribution scheme introduced by Eschenauer and Gligor in the context of wireless sensor networks (WSNs). For this class of random graphs, we establish a new version of a conjectured zero-one law for graph connectivity as the number of nodes becomes unboundedly large. The results reported here complement and strengthen recent work on this conjecture by Blackburn and Gerke. In particular, the results are given under conditions which are more realistic for applications to WSNs.

http://users.ece.cmu.edu/~oyagan/Journals/YaganMakowski_Connectivity_Revised.pdf

Armand M. Makowski

Papers

Modeling video traffic using M/G//spl infin/ input processes: a compromise between Markovian and LRD models

The fork-join queue and related systems with synchronization constraints: stochastic ordering and computable bounds

Queueing models for systems with synchronization constraints

Zero–One Laws for Connectivity in Random Key Graphs

K competing queues with geometric service requirements and linear costs: The μc-rule is always optimal☆