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15
BALANCED ALLOCATIONS: THE HEAVILY LOADED CASE
, 2006
"... We investigate ballsintobins processes allocating m balls into n bins based on the multiplechoice paradigm. In the classical singlechoice variant each ball is placed into a bin selected uniformly at random. In a multiplechoice process each ball can be placed into one out of d ≥ 2 randomly selec ..."
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Cited by 58 (8 self)
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We investigate ballsintobins processes allocating m balls into n bins based on the multiplechoice paradigm. In the classical singlechoice variant each ball is placed into a bin selected uniformly at random. In a multiplechoice process each ball can be placed into one out of d ≥ 2 randomly selected bins. It is known that in many scenarios having more than one choice for each ball can improve the load balance significantly. Formal analyses of this phenomenon prior to this work considered mostly the lightly loaded case, that is, when m ≈ n. In this paper we present the first tight analysis in the heavily loaded case, that is, when m ≫ n rather than m ≈ n. The best previously known results for the multiplechoice processes in the heavily loaded case were obtained using majorization by the singlechoice process. This yields an upper bound of the maximum load of bins of m/n + O ( √ m ln n/n) with high probability. We show, however, that the multiplechoice processes are fundamentally different from the singlechoice variant in that they have “short memory. ” The great consequence of this property is that the deviation of the multiplechoice processes from the optimal allocation (that is, the allocation in which each bin has either ⌊m/n ⌋ or ⌈m/n ⌉ balls) does not increase with the number of balls as in the case of the singlechoice process. In particular, we investigate the allocation obtained by two different multiplechoice allocation schemes,
Analyses of Load Stealing Models Based on Differential Equations
 In Proceedings of the 10th Annual ACM Symposium on Parallel Algorithms and Architectures
, 1998
"... In this paper we develop models for and analyze several randomized work stealing algorithms in a dynamic setting. Our models represent the limiting behavior of systems as the number of processors grows to infinity using differential equations. The advantages of this approach include the ability to m ..."
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Cited by 19 (0 self)
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In this paper we develop models for and analyze several randomized work stealing algorithms in a dynamic setting. Our models represent the limiting behavior of systems as the number of processors grows to infinity using differential equations. The advantages of this approach include the ability to model a large variety of systems and to provide accurate numerical approximations of system behavior even when the number of processors is relatively small. We show how this approach can yield significant intuition about the behavior of work stealing algorithms in realistic settings.
A Generic Mean Field Convergence Result for Systems of Interacting Objects
"... We consider a model for interacting objects, where the evolution of each object is given by a finite state Markov chain, whose transition matrix depends on the present and the past of the distribution of states of all objects. This is a general model of wide applicability; we mention as examples: TC ..."
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Cited by 17 (4 self)
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We consider a model for interacting objects, where the evolution of each object is given by a finite state Markov chain, whose transition matrix depends on the present and the past of the distribution of states of all objects. This is a general model of wide applicability; we mention as examples: TCP connections, HTTP flows, robot swarms, reputation systems. We show that when the number of objects is large, the occupancy measure of the system converges to a deterministic dynamical system (the “mean field”) with dimension the number of states of an individual object. We also prove a fast simulation result, which allows to simulate the evolution of a few particular objects imbedded in a large system. We illustrate how this can be used to model the determination of reputation in large populations, with various liar strategies. 1
Fast Jackson Networks
 Annals of Applied Probability
, 1998
"... this paper, we consider networks of Jackson type whose nodes are stations of this kind, and show that flexibility of routing leads to improvement in network performance in a similar way. ..."
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Cited by 15 (2 self)
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this paper, we consider networks of Jackson type whose nodes are stations of this kind, and show that flexibility of routing leads to improvement in network performance in a similar way.
Meanfield analysis for the evaluation of gossip protocols
 SIGMETRICS Perform. Eval. Rev
, 2008
"... Abstract—Gossip protocols are designed to operate in very large, decentralised networks. A node in such a network bases its decision to interact (gossip) with another node on its partial view of the global system. Because of the size of these networks, analysis of gossip protocols is mostly done usi ..."
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Cited by 12 (5 self)
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Abstract—Gossip protocols are designed to operate in very large, decentralised networks. A node in such a network bases its decision to interact (gossip) with another node on its partial view of the global system. Because of the size of these networks, analysis of gossip protocols is mostly done using simulations, that tend to be expensive in computation time and memory consumption. We employ meanfield approximation for an analytical evaluation of gossip protocols. Nodes in the network are represented by small identical stochastic models. Joining all nodes would result in an enormous stochastic process. If the number of nodes goes to infinity, however, meanfield analysis allows us to replace this intractably large stochastic process by a small deterministic process. This process approximates the behaviour of very large gossip networks, and can be evaluated using simple matrixvector multiplications. I.
c ○ 2005 Springer Science + Business Media, Inc. Manufactured in The Netherlands. Balancing Queues by Mean Field Interaction ∗
, 2004
"... Abstract. Consider a queueing network with N nodes in which queue lengths are balanced through meanfield interaction. When N is large, we study the performance of such a network in terms of limiting results as N goes to infinity. ..."
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Abstract. Consider a queueing network with N nodes in which queue lengths are balanced through meanfield interaction. When N is large, we study the performance of such a network in terms of limiting results as N goes to infinity.
MeanField Approximation for Stochastic Transportation Network and Stability of Dynamical System
, 1999
"... A queuing system is considered, with a virtual node, N nodes, and rN servers. At each node (of these N nodes) the arrivals of particles form a Poisson flow of rate (t). For an empty node a particle leaves the system. A server at the node takes the particle and moves to a random node. Travelling time ..."
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A queuing system is considered, with a virtual node, N nodes, and rN servers. At each node (of these N nodes) the arrivals of particles form a Poisson flow of rate (t). For an empty node a particle leaves the system. A server at the node takes the particle and moves to a random node. Travelling time is exponential of mean 1. The number of servers at each node (of these N nodes) is bounded by m. We discuss the property of stability for the limiting deterministic process as N ! 1. Furthermore, we consider a particular queuing system with dynamical routing. 1 Introduction Consider a network, consisting of N nodes, a virtual node and rN servers. At each node (of these N nodes) the arrivals of particles form a Poisson flow of rate (t). If a particle arrives at an empty node then the particle leaves the system. Otherwise, if there is a server at the node then the server takes the particle and jumps to the virtual node. At this node the server waits for an exponential time of mean 1. Afte...
Scotland, UK A Generic Mean Field Convergence Result for Systems of Interacting Objects
"... We consider a model for interacting objects, where the evolution of each object is given by a finite state Markov chain, whose transition matrix depends on the present and the past of the distribution of states of all objects. This is a general model of wide applicability; we mention as examples: TC ..."
Abstract
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We consider a model for interacting objects, where the evolution of each object is given by a finite state Markov chain, whose transition matrix depends on the present and the past of the distribution of states of all objects. This is a general model of wide applicability; we mention as examples: TCP connections, HTTP flows, robot swarms, reputation systems. We show that when the number of objects is large, the occupancy measure of the system converges to a deterministic dynamical system (the “mean field”) with dimension the number of states of an individual object. We also prove a fast simulation result, which allows to simulate the evolution of a few particular objects imbedded in a large system. We illustrate how this can be used to model the determination of reputation in large populations, with various liar strategies. 1
HashBased Data Structures for Extreme Conditions
, 2008
"... This thesis is about the design and analysis of Bloom filter and multiple choice hash table variants for application settings with extreme resource requirements. We employ a very flexible methodology, combining theoretical, numerical, and empirical techniques to obtain constructions that are both an ..."
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This thesis is about the design and analysis of Bloom filter and multiple choice hash table variants for application settings with extreme resource requirements. We employ a very flexible methodology, combining theoretical, numerical, and empirical techniques to obtain constructions that are both analyzable and practical. First, we show that a wide class of Bloom filter variants can be effectively implemented using very easily computable combinations of only two fully random hash functions. From a theoretical perspective, these results show that Bloom filters and related data structures can often be substantially derandomized with essentially no loss in performance. From a practical perspective, this derandomization allows for a significant speedup in certain query intensive applications. The rest of this work focuses on designing spaceefficient, openaddressed, multiple choice hash tables for implementation in highperformance router hardware. Using multiple hash functions conserves space, but requires every hash table operation to consider multiple hash buckets, forcing a tradeoff between the slow speed of examining these buckets serially