Factorization in the Cloud: Integer Factorization Using F# and Windows Azure

Implementations are presented of two common algorithms for integer factorization, Pollard’s “p – 1” method and the SQUFOF method. The algorithms are implemented in the F# language, a functional programming language developed by Microsoft and officially released for the first time in 2010. The algorithms are thoroughly tested on a set of large integers (up to 64 bits in size), running both on a physical machine and a Windows Azure machine instance. Analysis of the relative performance between the two environments indicates comparable performance when taking into account the difference in computing power. Further analysis reveals that the relative performance of the Azure implementation tends to improve as the magnitudes of the integers increase, indicating that such an approach may be suitable for larger, more complex factorization tasks. Finally, several questions are presented for future research, including the performance of F# and related languages for more efficient, parallelizable algorithms, and the relative cost and performance of factorization algorithms in various environments, including physical hardware and commercial cloud computing offerings from the various vendors in the industry.

Library Technology Services at Boston University: an un-roadmap

A working paper for discussion

A Two-Tiered On-Line Server-Side Bandwidth Reservation Framework for the Real-Time Delivery of Multiple Video Streams

The advent of virtualization and cloud computing technologies necessitates the development of effective mechanisms for the estimation and reservation of resources needed by content providers to deliver large numbers of video-on-demand (VOD) streams through the cloud. Unfortunately, capacity planning for the QoS-constrained delivery of a large number of VOD streams is inherently difficult as VBR encoding schemes exhibit significant bandwidth variability. In this paper, we present a novel resource management scheme to make such allocation decisions using a mixture of per-stream reservations and an aggregate reservation, shared across all streams to accommodate peak demands. The shared reservation provides capacity slack that enables statistical multiplexing of peak rates, while assuring analytically bounded frame-drop probabilities, which can be adjusted by trading off buffer space (and consequently delay) and bandwidth. Our two-tiered bandwidth allocation scheme enables the delivery of any set of streams with less bandwidth (or equivalently with higher link utilization) than state-of-the-art deterministic smoothing approaches. The algorithm underlying our proposed frame-work uses three per-stream parameters and is linear in the number of servers, making it particularly well suited for use in an on-line setting. We present results from extensive trace-driven simulations, which confirm the efficiency of our scheme especially for small buffer sizes and delay bounds, and which underscore the significant realizable bandwidth savings, typically yielding losses that are an order of magnitude or more below our analytically derived bounds.

Collocation Games and Their Application to Distributed Resource Management

We introduce Collocation Games as the basis of a general framework for modeling, analyzing, and facilitating the interactions between the various stakeholders in distributed systems in general, and in cloud computing environments in particular. Cloud computing enables fixed-capacity (processing, communication, and storage) resources to be offered by infrastructure providers as commodities for sale at a fixed cost in an open marketplace to independent, rational parties (players) interested in setting up their own applications over the Internet. Virtualization technologies enable the partitioning of such fixed-capacity resources so as to allow each player to dynamically acquire appropriate fractions of the resources for unencumbered use. In such a paradigm, the resource management problem reduces to that of partitioning the entire set of applications (players) into subsets, each of which is assigned to fixed-capacity cloud resources. If the infrastructure and the various applications are under a single administrative domain, this partitioning reduces to an optimization problem whose objective is to minimize the overall deployment cost. In a marketplace, in which the infrastructure provider is interested in maximizing its own profit, and in which each player is interested in minimizing its own cost, it should be evident that a global optimization is precisely the wrong framework. Rather, in this paper we use a game-theoretic framework in which the assignment of players to fixed-capacity resources is the outcome of a strategic "Collocation Game". Although we show that determining the existence of an equilibrium for collocation games in general is NP-hard, we present a number of simplified, practically-motivated variants of the collocation game for which we establish convergence to a Nash Equilibrium, and for which we derive convergence and price of anarchy bounds. In addition to these analytical results, we present an experimental evaluation of implementations of some of these variants for cloud infrastructures consisting of a collection of multidimensional resources of homogeneous or heterogeneous capacities. Experimental results using trace-driven simulations and synthetically generated datasets corroborate our analytical results and also illustrate how collocation games offer a feasible distributed resource management alternative for autonomic/self-organizing systems, in which the adoption of a global optimization approach (centralized or distributed) would be neither practical nor justifiable.