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

Praise for the Third Edition: "This is one of the best books available. Its excellent organizational structure allows quick reference to specific models and its clear presentation . . . solidifies the understanding of the concepts being presented."IIE Transactions on Operations EngineeringThoroughly revised and expanded to reflect the latest developments in the field, Fundamentals of Queueing Theory, Fourth Edition continues to present the basic statistical principles that are necessary to analyze the probabilistic nature of queues. Rather than presenting a narrow focus on the subject, this update illustrates the wide-reaching, fundamental concepts in queueing theory and its applications to diverse areas such as computer science, engineering, business, and operations research.This update takes a numerical approach to understanding and making probable estimations relating to queues, with a comprehensive outline of simple and more advanced queueing models. Newly featured topics of the Fourth Edition include:Retrial queuesApproximations for queueing networksNumerical inversion of transformsDetermining the appropriate number of servers to balance quality and cost of serviceEach chapter provides a self-contained presentation of key concepts and formulae, allowing readers to work with each section independently, while a summary table at the end of the book outlines the types of queues that have been discussed and their results. In addition, two new appendices have been added, discussing transforms and generating functions as well as the fundamentals of differential and difference equations. New examples are now included along with problems that incorporate QtsPlus software, which is freely available via the book's related Web site.With its accessible style and wealth of real-world examples, Fundamentals of Queueing Theory, Fourth Edition is an ideal book for courses on queueing theory at the upper-undergraduate and graduate levels. It is also a valuable resource for researchers and practitioners who analyze congestion in the fields of telecommunications, transportation, aviation, and management science.

Fundamentals of Queueing Theory

To be agile and cost effective, data centers should allow dynamic resource allocation across large server pools. In particular, the data center network should enable any server to be assigned to any service. To meet these goals, we present VL2, a practical network architecture that scales to support huge data centers with uniform high capacity between servers, performance isolation between services, and Ethernet layer-2 semantics. VL2 uses (1) flat addressing to allow service instances to be placed anywhere in the network, (2) Valiant Load Balancing to spread traffic uniformly across network paths, and (3) end-system based address resolution to scale to large server pools, without introducing complexity to the network control plane. VL2's design is driven by detailed measurements of traffic and fault data from a large operational cloud service provider. VL2's implementation leverages proven network technologies, already available at low cost in high-speed hardware implementations, to build a scalable and reliable network architecture. As a result, VL2 networks can be deployed today, and we have built a working prototype. We evaluate the merits of the VL2 design using measurement, analysis, and experiments. Our VL2 prototype shuffles 2.7 TB of data among 75 servers in 395 seconds - sustaining a rate that is 94% of the maximum possible.

/pdf/vl2-a-scalable-and-flexible-data-center-network-dkbtxdift1.pdf

VL2: a scalable and flexible data center network

We analyze 20 large sets of actual variable-bit-rate (VBR) video data, generated by a variety of different codecs and representing a wide range of different scenes. Performing extensive statistical and graphical tests, our main conclusion is that long-range dependence is an inherent feature of VBR video traffic, i.e., a feature that is independent of scene (e.g., video phone, video conference, motion picture video) and codec. In particular, we show that the long-range dependence property allows us to clearly distinguish between our measured data and traffic generated by VBR source models currently used in the literature. These findings give rise to novel and challenging problems in traffic engineering for high-speed networks and open up new areas of research in queueing and performance analysis involving long-range dependent traffic models. A small number of analytic queueing results already exist, and we discuss their implications for network design and network control strategies in the presence of long-range dependent traffic. >

Long-range dependence in variable-bit-rate video traffic

To be agile and cost effective, data centers must allow dynamic resource allocation across large server pools. In particular, the data center network should provide a simple flat abstraction: it should be able to take any set of servers anywhere in the data center and give them the illusion that they are plugged into a physically separate, noninterfering Ethernet switch with as many ports as the service needs. To meet this goal, we present VL2, a practical network architecture that scales to support huge data centers with uniform high capacity between servers, performance isolation between services, and Ethernet layer-2 semantics. VL2 uses (1) flat addressing to allow service instances to be placed anywhere in the network, (2) Valiant Load Balancing to spread traffic uniformly across network paths, and (3) end system--based address resolution to scale to large server pools without introducing complexity to the network control plane. VL2's design is driven by detailed measurements of traffic and fault data from a large operational cloud service provider. VL2's implementation leverages proven network technologies, already available at low cost in high-speed hardware implementations, to build a scalable and reliable network architecture. As a result, VL2 networks can be deployed today, and we have built a working prototype. We evaluate the merits of the VL2 design using measurement, analysis, and experiments. Our VL2 prototype shuffles 2.7 TB of data among 75 servers in 395 s---sustaining a rate that is 94% of the maximum possible.

The dynamic frame sizing algorithm is a throughput-optimal algorithm that can achieve maximum network throughput without the knowledge of arrival rates. Motivated by the need for energy-efficient communication in wireless networks, in this paper, we propose a new dynamic frame sizing algorithm, called the Greenput algorithm, that takes power allocation into account. In our Greenput algorithm, time is partitioned into frames, and the frame size of each frame is determined based on the backlogs presented at the beginning of a frame. To obtain a good delay-energy efficiency tradeoff, the key insight of our Greenput algorithm is to reduce transmit power to save energy when the backlogs are low so as not to incur too much packet delay. For this, we define a threshold parameter $T_{\max }$ (for the minimum time to empty the backlogs with maximum power allocation), and the Greenput algorithm enters the (mixed) power-saving mode when the backlogs are below the threshold. Using a large deviation bound, we prove that our Greenput algorithm is still throughput optimal. In addition to the stability result, we also perform a fluid approximation analysis for energy efficiency and average packet delay when $T_{\max }$ is very large. To show the delay-energy efficiency tradeoff, we conduct extensive computer simulations by using the Shannon formula as the channel model in a wireless network. Our simulation results show that both energy efficiency and average packet delay are quite close to their fluid approximations even when $T_{\max }$ is moderately large.

Greenput: A Power-Saving Algorithm That Achieves Maximum Throughput in Wireless Networks

In this letter, we derive the average number of times an optical packet recirculates through the optical switch and the fiber delay lines in our previous constructions of optical priority queues (see Figure 1 in Section I) under Bernoulli arrival traffic, Bernoulli control input, and uniform priority assignment. The analytical results on the average number of recirculations are further verified through simulations. Through simulations, we also find that these analytical results are useful in choosing the number of fiber delay lines in our constructions of optical priority queues when there is a limitation on the number of times an optical packet can be recirculated through the optical switch and the fiber delay lines.

/pdf/average-number-of-recirculations-in-sdl-constructions-of-3vng15ygjf.pdf

Average Number of Recirculations in SDL Constructions of Optical Priority Queues

Two-stage constructions and banyan-type networks play important roles in designing high speed switch fabrics. Switch fabrics designed from two-stage constructions and banyan-type networks cannot realize all the permutations and are known as conditionally nonblocking switches. A renowned property for conditionally nonblocking switches is the closure property, i.e., if all the switches in a two-stage construction can realize a subset of permutations that satisfy a certain property P, then the switch resulted from the two-stage construction can also realize a subset of permutations that satisfy the same property P. However, such a closure property is mostly stated for the sufficient part in the literature and the necessary part of the statement is in general either not true or unknown. Finding such a necessary and sufficient result is of fundamental importance to two-stage constructions as it can completely characterize the permutations that are realizable by two-stage constructions. In this paper, we prove a necessary and sufficient closure property for two-stage constructions. For this, we consider uniform mapping permutations and define uniform mapping switches as switches that can only realize the set of uniform mapping permutations. We show that a switch constructed by a two-stage construction is a uniform mapping switch if and only if all the switches at the first stage and the second stage are uniform mapping switches. Such a necessary and sufficient result provides a complete characterization of the permutations that can be realized by two-stage constructions. Since banyan-type networks are constructed recursively by using two-stage constructions with 2 × 2 switches, we obtain a complete characterization for any realizable permutation of any banyan-type network as a composition of its trace, a uniform mapping permutation and its guide.

/pdf/a-necessary-and-sufficient-closure-property-for-two-stage-4s3f5tvruq.pdf

A necessary and sufficient closure property for two-stage constructions of switching networks

TCP congestion control has been widely discussed on various networks under wireline or wireless environment. Many research studies have been raised to help TCP be more sensitive to the network congestion by adding explicit notifications in TCP connection or letting the intermediate nodes in the network to join TCP congestion control. In GPRS/EGPRS systems and other 3G cellular networks, the radio network controller chooses among several channel coders according to the signal to interference ratio. Hence, the channel data rate that an individual user experiences is time-dependent and variable. We propose a TCP window control mechanism that adapts to the variable channel data rates. Our simulations show that the window adaptive TCP decreases the buffer occupancy and packet transmission delay, and achieves better throughput utilization when compared with the standard TCP Tahoe version and the Reno version.

Window adaptive TCP for EGPRS networks

The network embedding problem aims to map nodes that are similar to each other to vectors in a Euclidean space that are close to each other. Like centrality analysis (ranking) and community detection, network embedding is in general considered as an ill-posed problem, and its solution may depend on a person's view on this problem. In this book chapter, we adopt the framework of sampled graphs that treat a person's view as a sampling method for a network. The modularity for a sampled graph, called the generalized modularity in the book chapter, is a similarity matrix that has a specific probabilistic interpretation. One of the main contributions of this book chapter is to propose using the generalized modularity matrix for network embedding and show that the network embedding problem can be treated as a trace maximization problem like the community detection problem. Our generalized modularity embedding approach is very general and flexible. In particular, we show that the Laplacian eigenmaps is a special case of our generalized modularity embedding approach. Also, we show that dimensionality reduction can be done by using a particular sampled graph. Various experiments are conducted on real datasets to illustrate the effectiveness of our approach.

Duan-Shin Lee

Papers

Greenput: A Power-Saving Algorithm That Achieves Maximum Throughput in Wireless Networks

Average Number of Recirculations in SDL Constructions of Optical Priority Queues

A necessary and sufficient closure property for two-stage constructions of switching networks

Window adaptive TCP for EGPRS networks

Generalized Modularity Embedding: a General Framework for Network Embedding.