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15
State space collapse and diffusion approximation for a network operating under a fair bandwidthsharing policy, in preparation
, 2004
"... We consider a connectionlevel model of Internet congestion control, introduced by Massoulié and Roberts [36], that represents the randomly varying number of flows present in a network. Here bandwidth is shared fairly amongst elastic document transfers according to a weighted αfair bandwidth sharin ..."
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Cited by 46 (8 self)
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We consider a connectionlevel model of Internet congestion control, introduced by Massoulié and Roberts [36], that represents the randomly varying number of flows present in a network. Here bandwidth is shared fairly amongst elastic document transfers according to a weighted αfair bandwidth sharing policy introduced by Mo and Walrand [37] (α ∈ (0,∞)). Assuming Poisson arrivals and exponentially distributed document sizes, we focus on the heavy traffic regime in which the average load placed on each resource is approximately equal to its capacity. A fluid model (or functional law of large numbers approximation) for this stochastic model was derived and analyzed in a prior work [29] by two of the authors. Here we use the long time behavior of the solutions of this fluid model established in [29] to derive a property called multiplicative state space collapse, which loosely speaking shows that in diffusion scale the flow count process for the stochastic model can be approximately recovered as a continuous lifting of the workload process. Under weighted proportional fair sharing of bandwidth (α = 1) and a mild
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 36 (10 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
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 20 (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.
The impact of reneging in processor sharing queues
 ACMSigmetrics (Saint Malo), ACM/IFIP WG
, 2006
"... We investigate an overloaded processor sharing queue with renewal arrivals and generally distributed service times. Impatient customers may abandon the queue, or renege, before completing service. The random time representing a customer’s patience has a general distribution and may be dependent on ..."
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Cited by 10 (3 self)
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We investigate an overloaded processor sharing queue with renewal arrivals and generally distributed service times. Impatient customers may abandon the queue, or renege, before completing service. The random time representing a customer’s patience has a general distribution and may be dependent on his initial service time requirement. We propose a scaling procedure that gives rise to a fluid model, with nontrivial yet tractable steady state behavior. This fluid model captures many essential features of the underlying stochastic model, and we use it to analyze the impact of impatience in processor sharing queues. We show that this impact can be substantial compared with FCFS, and we propose a simple admission control policy to overcome these negative impacts.
On the Flowlevel Dynamics of a Packetswitched Network
"... Abstract: The packet is the fundamental unit of transportation in modern communication networks such as the Internet. Physical layer scheduling decisions are made at the level of packets, and packetlevel models with exogenous arrival processes have long been employed to study network performance, a ..."
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Cited by 5 (0 self)
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Abstract: The packet is the fundamental unit of transportation in modern communication networks such as the Internet. Physical layer scheduling decisions are made at the level of packets, and packetlevel models with exogenous arrival processes have long been employed to study network performance, as well as design scheduling policies that more efficiently utilize network resources. On the other hand, a user of the network is more concerned with endtoend bandwidth, which is allocated through congestion control policies such as TCP. Utilitybased flowlevel models have played an important role in understanding congestion control protocols. In summary, these two classes of models have provided separate insights for flowlevel and packetlevel dynamics of a network. In this paper, we wish to study these two dynamics together. We propose a joint flowlevel and packetlevel stochastic model for the dynamics of a network, and an associated policy for congestion control and packet scheduling that is based on αweighted policies from the literature. We provide a fluid analysis for the model that establishes the throughput optimality of the proposed policy, thus validating prior insights based on separate packetlevel and flowlevel models. By analyzing a critically scaled fluid model under the proposed policy, we provide constant factor performance bounds on the delay performance and characterize the invariant states of the system. Keywords and phrases: Flowlevel model, Packetlevel model, Congestion control, Scheduling, Utility maximization, Backpressure maximum weight.
HeavyTraffic Approximations for Linear Networks Operating under AlphaFair BandwidthSharing Policies
 In Proceedings of VALUETOOLS (2006
"... We consider the flowlevel performance of a linear network supporting elastic traffic, where the service capacity is shared among the various classes of users according to a weighted alphafair policy. Assuming Poisson arrivals and exponentially distributed service requirements for each class, the ..."
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Cited by 4 (3 self)
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We consider the flowlevel performance of a linear network supporting elastic traffic, where the service capacity is shared among the various classes of users according to a weighted alphafair policy. Assuming Poisson arrivals and exponentially distributed service requirements for each class, the dynamics of the user population may be described by a Markov process. While valuable stability results have been established for the family of alphafair policies, the distribution of the number of active users has remained intractable in all but a few special cases. In order to gain further insight in the flowlevel performance in more general scenarios, we develop approximations for the mean number of users based on the assumption that one or two of the nodes experience heavytraffic conditions. In case of just a single ‘bottleneck ’ node, we exploit the fact that this node approximately behaves as a twoclass Discriminatory ProcessorSharing model. In the case that there are two nodes critically loaded, we rely on the observation that the joint workload process at these nodes is asymptotically independent of the fairness coefficient alpha, provided all classes have equal weights. In particular, the distribution of the joint workload process is roughly equal to that for an unweighted Proportional Fair policy, which is exactly known. In both cases, the numbers of users at nonbottleneck nodes can be approximated by that in an M/M/1 queue with reduced service capacity. Extensive numerical experiments indicate that the resulting approximations tend to be reasonably accurate across a wide range of parameters, ∗This research has been funded by the Dutch
Monotonicity properties for multiclass queueing systems. Discrete Event Dynamic Systems DOI
, 2009
"... We study multidimensional stochastic processes that arise in queueing models used in the performance evaluation of wired and wireless networks. The evolution of the stochastic process is determined by the scheduling policy used in the associated queueing network. For general arrival and service pro ..."
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Cited by 4 (2 self)
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We study multidimensional stochastic processes that arise in queueing models used in the performance evaluation of wired and wireless networks. The evolution of the stochastic process is determined by the scheduling policy used in the associated queueing network. For general arrival and service processes, we give sufficient conditions in order to compare samplepath wise the workload and the number of users under different policies. This allows us to evaluate the performance of the system under various policies in terms of stability, the mean overall delay and the mean holding cost. We apply the general framework to linear networks, where users of one class require service from several shared resources simultaneously. For the important family of weighted αfair policies, stability results are derived and monotonicity of the mean holding cost with respect to the fairness parameter α and the relative weights is established. In order to broaden the comparison results, we investigate a heavytraffic regime and perform numerical experiments. In addition, we study a singleserver queue with two user classes, and show that under Discriminatory Processor Sharing (DPS) or Generalized Processor Sharing (GPS) the mean overall sojourn time is monotone with respect to the ratio of the weights. Finally we extend the framework to obtain comparison results that cover the singleserver queue with an arbitrary number of classes as well. 1
Comparison of bandwidthsharing policies in a linear network
 In Proceedings of ValueTools
, 2008
"... In bandwidthsharing networks, users of various classes require service from different subsets of shared resources simultaneously. These networks have been proposed to analyze the performance of wired and wireless networks. For general arrival and service processes, we give sufficient conditions in ..."
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Cited by 2 (2 self)
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In bandwidthsharing networks, users of various classes require service from different subsets of shared resources simultaneously. These networks have been proposed to analyze the performance of wired and wireless networks. For general arrival and service processes, we give sufficient conditions in order to compare samplepath wise the workload and the number of users under different policies in a linear bandwidthsharing network. This allows us to compare the performance of the system under various policies in terms of stability, the mean overall delay and the weighted mean number of users. For the important family of weighted αfair policies, we derive stability results and establish monotonicity of the weighted mean number of users with respect to the fairness parameter α and the relative weights. In order to broaden the comparison results, we investigate a heavytraffic regime and perform numerical experiments. 1.
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 ..."
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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