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.

Load balanced Birkhoff-von Neumann switches, part II: multi-stage buffering

Motivated by the need of a simple and high performance switch architecture that scales up with the speed of fiber optics, we propose a switch architecture with two-stage switching fabrics and one-stage buffering. The first stage performs load balancing, while the second stage is a Birkhoff-von Neumann input-buffered switch that performs switching for load balanced traffic. Such a switch is called the load balanced Birkhoff-von Neumann switch in this paper. The on-line complexity of the switch is O(1). It is shown that under a mild technical condition on the input traffic, the load balanced Birkhoff-von Neumann switch achieves 100% throughput as an output-buffered switch for both unicast and multicast traffic with fan-out splitting. When input traffic is bursty, we show that load balancing is very effective in reducing delay, and the average delay of the load balanced Birkhoff-von Neumann switch converges to that of an output-buffered switch under heavy load. Also, by simulations, we demonstrate that load balancing is more effective than the conflict resolution algorithm, i-SLIP, in heavy loads. When both the load balanced Birkhoff-von Neumann switch and the corresponding output-buffered switch are allocated with the same finite amount of buffer at each port, we also show that the packet loss probability in the load balanced Birkhoff-von Neumann switch is much smaller than that in an output-buffered switch when the buffer is large.

Load balanced Birkhoff-von Neumann switches

Traditionally, conflict resolution in an input-buffered switch is solved by finding a matching between inputs and outputs per time slot. To do this, a switch not only needs to gather the information of the virtual output queues at the inputs, hut also uses the gathered information to compute a matching. As such, both the communication overhead and the computation overhead make it difficult to scale. Recent works on the two-stage switch architecture in (6|, [7], [12], (8| showed that conflict resolution can be easily solved over time and space without communication and computation overhead. However, the main problem of such a two-stage switch architecture is that packets might be out of sequence. The main objective of this paper is to propose a scalable solution, called the mailbox switch, that solves the out-of-sequence problem in the two-stage switch architecture. The key idea of the mailbox switch is to use a set of symmetric connection patterns to create a feedback path for packet departure times. With the information of packet departure times, the mailbox switch can schedule packets so that they depart in the order of their arrivals. Despite the simplicity of the mailbox switch, we show via both the theoretical models and simulations that the throughput of the mailbox switch can be as high as 75%. With limited resequencing delay, a modified version of the mailbox switch achieves 95% throughput. We also propose a recursive way to construct the switch fabrics for the set of symmetric connection patterns. If the number of inputs, N, is a power of 2, we show that the switch fabric for the mailbox switch can be built with N/2 log2 N 2times2 switches

/pdf/mailbox-switch-a-scalable-two-stage-switch-architecture-for-4zttyzfg1k.pdf

Mailbox switch: a scalable two-stage switch architecture for conflict resolution of ordered packets

This paper is about load-balancing packets across multiple paths inside a switch, or across a network. It is motivated by the recent interest in load-balanced switches. Load-balanced switches provide an appealing alternative to crossbars with centralized schedulers. A load-balanced switch has no scheduler, is particularly amenable to optics, and - most relevant here -guarantees 100% throughput. A uniform mesh is used to load-balance packets uniformly across all 2-hop paths in the switch. In this paper we explore whether this particular method of load-balancing is optimal in the sense that it achieves the highest throughput for a given capacity of interconnect. The method we use allows the load-balanced switch to be compared with ring, torus and hypercube interconnects, too. We prove that for a given interconnect capacity, the load-balancing mesh has the maximum throughput. Perhaps surprisingly, we find that the best mesh is slightly non-uniform, or biased, and has a throughput of N/(2N - 1), where N is the number of nodes.

/pdf/optimal-load-balancing-3ooxjw0td6.pdf

Optimal load-balancing

Traditionally, conflict resolution in an input- buffered switch is solved by finding a matching between inputs and outputs per time slot, which incurs unscalable computation and communication overheads. The main objective of this paper is to propose a scalable solution, called the mailbox switch, that solves the out-of-sequence problem in the two-stage switch architecture. The key idea of the mailbox switch is to use a set of symmetric connection patterns to create a feedback path for packet departure times. With the information of packet departure times, the mailbox switch can schedule packets so that they depart in the order of their arrivals. Despite the simplicity of the mailbox switch, we show via both the theoretical models and simulations that the throughput of the mailbox switch can be as high as 75%. With limited resequencing delay, a modified version of the mailbox switch achieves 95% throughput. We also propose a recursive way to construct the switch fabrics for the set of symmetric connection patterns. If the number of inputs, N, is a power of 2, we show that the switch fabric for the mailbox switch can be built with y log2 N 2 x 2 switches.

Duan-Shin Lee

Papers

Load balanced Birkhoff-von Neumann switches, part II: multi-stage buffering

Load balanced Birkhoff-von Neumann switches

Mailbox switch: a scalable two-stage switch architecture for conflict resolution of ordered packets

Optimal load-balancing

Mailbox switch: a scalable two-stage switch architecture for conflict resolution of ordered packets