On the Size Distribution of Autonomous Systems

This paper explores reasons for the high degree of variability in the sizes of ASes that have recently been observed, and the processes by which this variable distribution develops. AS size distribution is important for a number of reasons. First, when modeling network topologies, an AS size distribution assists in labeling routers with an associated AS. Second, AS size has been found to be positively correlated with the degree of the AS (number of peering links), so understanding the distribution of AS sizes has implications for AS connectivity properties. Our model accounts for AS births, growth, and mergers. We analyze two models: one incorporates only the growth of hosts and ASes, and a second extends that model to include mergers of ASes. We show analytically that, given reasonable assumptions about the nature of mergers, the resulting size distribution exhibits a power law tail with the exponent independent of the details of the merging process. We estimate parameters of the models from measurements obtained from Internet registries and from BGP tables. We then compare the models solutions to empirical AS size distribution taken from Mercator and Skitter datasets, and find that the simple growth-based model yields general agreement with empirical data. Our analysis of the model in which mergers occur in a manner independent of the size of the merging ASes suggests that more detailed analysis of merger processes is needed.

Geometric Generalizations of the Power of Two Choices

A well-known paradigm for load balancing in distributed systems is the``power of two choices,''whereby an item is stored at the less loaded of two (or more) random alternative servers. We investigate the power of two choices in natural settings for distributed computing where items and servers reside in a geometric space and each item is associated with the server that is its nearest neighbor. This is in fact the backdrop for distributed hash tables such as Chord, where the geometric space is determined by clockwise distance on a one-dimensional ring. Theoretically, we consider the following load balancing problem. Suppose that servers are initially hashed uniformly at random to points in the space. Sequentially, each item then considers d candidate insertion points also chosen uniformly at random from the space,and selects the insertion point whose associated server has the least load. For the one-dimensional ring, and for Euclidean distance on the two-dimensional torus, we demonstrate that when n data items are hashed to n servers,the maximum load at any server is log log n / log d + O(1) with high probability. While our results match the well-known bounds in the standard setting in which each server is selected equiprobably, our applications do not have this feature, since the sizes of the nearest-neighbor regions around servers are non-uniform. Therefore, the novelty in our methods lies in developing appropriate tail bounds on the distribution of nearest-neighbor region sizes and in adapting previous arguments to this more general setting. In addition, we provide simulation results demonstrating the load balance that results as the system size scales into the millions.

On the Emergence of Highly Variable Distributions in the Autonomous System Topology

Recent studies have noted that vertex degree in the autonomous system (AS) graph exhibits a highly variable distribution [15, 22]. The most prominent explanatory model for this phenomenon is the Barabási-Albert (B-A) model [5, 2]. A central feature of the B-A model is preferential connectivity—meaning that the likelihood a new node in a growing graph will connect to an existing node is proportional to the existing node’s degree. In this paper we ask whether a more general explanation than the B-A model, and absent the assumption of preferential connectivity, is consistent with empirical data. We are motivated by two observations: first, AS degree and AS size are highly correlated [11]; and second, highly variable AS size can arise simply through exponential growth. We construct a model incorporating exponential growth in the size of the Internet, and in the number of ASes. We then show via analysis that such a model yields a size distribution exhibiting a power-law tail. In such a model, if an AS’s link formation is roughly proportional to its size, then AS degree will also show high variability. We instantiate such a model with empirically derived estimates of growth rates and show that the resulting degree distribution is in good agreement with that of real AS graphs.

Measuring the Behavior of a World-Wide Web Server

Server performance has become a crucial issue for improving the overall performance of the World-Wide Web. This paper describes Webmonitor, a tool for evaluating and understanding server performance, and presents new results for a realistic workload. Webmonitor measures activity and resource consumption, both within the kernel and in HTTP processes running in user space. Webmonitor is implemented using an efficient combination of sampling and event-driven techniques that exhibit low overhead. Our initial implementation is for the Apache World-Wide Web server running on the Linux operating system. We demonstrate the utility of Webmonitor by measuring and understanding the performance of a Pentium-based PC acting as a dedicated WWW server. Our workload uses a file size distribution with a heavy tail. This captures the fact that Web servers must concurrently handle some requests for large audio and video files, and a large number of requests for small documents, containing text or images. Our results show that in a Web server saturated by client requests, over 90% of the time spent handling HTTP requests is spent in the kernel. Furthermore, keeping TCP connections open, as required by TCP, causes a factor of 2-9 increase in the elapsed time required to service an HTTP request. Data gathered from Webmonitor provide insight into the causes of this performance penalty. Specifically, we observe a significant increase in resource consumption along three dimensions: the number of HTTP processes running at the same time, CPU utilization, and memory utilization. These results emphasize the important role of operating system and network protocol implementation in determining Web server performance.

On Class-based Isolation of UDP, Short-lived and Long-lived TCP Flows

The congestion control mechanisms of TCP make it vulnerable in an environment where flows with different congestion-sensitivity compete for scarce resources. With the increasing amount of unresponsive UDP traffic in today's Internet, new mechanisms are needed to enforce fairness in the core of the network. We propose a scalable Diffserv-like architecture, where flows with different characteristics are classified into separate service queues at the routers. Such class-based isolation provides protection so that flows with different characteristics do not negatively impact one another. In this study, we examine different aspects of UDP and TCP interaction and possible gains from segregating UDP and TCP into different classes. We also investigate the utility of further segregating TCP flows into two classes, which are class of short and class of long flows. Results are obtained analytically for both Tail-drop and Random Early Drop (RED) routers. Class-based isolation have the following salient features: (1) better fairness, (2) improved predictability for all kinds of flows, (3) lower transmission delay for delay-sensitive flows, and (4) better control over Quality of Service (QoS) of a particular traffic type.

Sampling Biases in IP Topology Measurements

Considerable attention has been focused on the properties of graphs derived from Internet measurements. Router-level topologies collected via traceroute studies have led some authors to conclude that the router graph of the Internet is a scale-free graph, or more generally a power-law random graph. In such a graph, the degree distribution of nodes follows a distribution with a power-law tail. In this paper we argue that the evidence to date for this conclusion is at best insufficient. We show that graphs appearing to have power-law degree distributions can arise surprisingly easily, when sampling graphs whose true degree distribution is not at all like a power-law. For example, given a classical Erdös-Rényi sparse, random graph, the subgraph formed by a collection of shortest paths from a small set of random sources to a larger set of random destinations can easily appear to show a degree distribution remarkably like a power-law. We explore the reasons for how this effect arises, and show that in such a setting, edges are sampled in a highly biased manner. This insight allows us to distinguish measurements taken from the Erdös-Rényi graphs from those taken from power-law random graphs. When we apply this distinction to a number of well-known datasets, we find that the evidence for sampling bias in these datasets is strong.

Differentiated Predictive Fair Service for TCP Flows

The majority of the traffic (bytes) flowing over the Internet today have been attributed to the Transmission Control Protocol (TCP). This strong presence of TCP has recently spurred further investigations into its congestion avoidance mechanism and its effect on the performance of short and long data transfers. At the same time, the rising interest in enhancing Internet services while keeping the implementation cost low has led to several service-differentiation proposals. In such service-differentiation architectures, much of the complexity is placed only in access routers, which classify and mark packets from different flows. Core routers can then allocate enough resources to each class of packets so as to satisfy delivery requirements, such as predictable (consistent) and fair service. In this paper, we investigate the interaction among short and long TCP flows, and how TCP service can be improved by employing a low-cost service-differentiation scheme. Through control-theoretic arguments and extensive simulations, we show the utility of isolating TCP flows into two classes based on their lifetime/size, namely one class of short flows and another of long flows. With such class-based isolation, short and long TCP flows have separate service queues at routers. This protects each class of flows from the other as they possess different characteristics, such as burstiness of arrivals/departures and congestion/sending window dynamics. We show the benefits of isolation, in terms of better predictability and fairness, over traditional shared queueing systems with both tail-drop and Random-Early-Drop (RED) packet dropping policies. The proposed class-based isolation of TCP flows has several advantages: (1) the implementation cost is low since it only requires core routers to maintain per-class (rather than per-flow) state; (2) it promises to be an effective traffic engineering tool for improved predictability and fairness for both short and long TCP flows; and (3) stringent delay requirements of short interactive transfers can be met by increasing the amount of resources allocated to the class of short flows.