A Scaling Analysis of a Transient Stochastic Network

Feuillet, Mathieu; Pierpoint, Robert

doi:10.1017/s0001867800007199

Cited by 7 publications

(14 citation statements)

References 27 publications

(33 reference statements)

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“…Specifically, the fraction of class-c nodes with n ≥ 1 packets in the buffer "increases" at rate λ c and "decreases" at rate ν c π x0 (Ω −c ), where the measure π x0 (Ω −c ) represents the limiting "instantaneous measure" on the activity states that allows class-c nodes to back-off. The argument is thus based on a stochastic averaging principle which follows the same lines of ideas of [21,25].…”

Section: Overview Of the Main Resultsmentioning

confidence: 99%

Mean-field limits for multi-hop random-access networks

Cecchi

Ven

Shneer³

2018

SIGMETRICS Perform. Eval. Rev.

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We establish mean-field limits for large-scale random-access networks with buffer dynamics and arbitrary interference graphs. While saturated-buffer scenarios have been widely investigated and yield useful throughput estimates for persistent sessions, they fail to capture the fluctuations in buffer contents over time, and provide no insight in the delay performance of flows with intermittent packet arrivals. Motivated by that issue, we explore in the present paper random-access networks with buffer dynamics, where flows with empty buffers refrain from competition for the medium. The occurrence of empty buffers thus results in a complex dynamic interaction between activity states and buffer contents, which severely complicates the performance analysis. Hence we focus on a many-sources regime where the total number of nodes grows large, which not only offers mathematical tractability but is also highly relevant with the densification of wireless networks as the Internet of Things emerges. We exploit time scale separation properties to prove that the properly scaled buffer occupancy process converges to the solution of a deterministic initial-value problem, and establish the existence and uniqueness of the associated fixed point. This approach simplifies the performance analysis of networks with huge numbers of nodes to a low-dimensional fixed-point calculation. For the case of a complete interference graph, we demonstrate asymptotic stability, provide a simple closed-form expression for the fixed point, and prove interchange of the mean-field and steady-state limits. This yields asymptotically exact approximations for key performance metrics, in particular the stationary buffer content and packet delay distributions.

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Section: Overview Of the Main Resultsmentioning

confidence: 99%

Mean-field limits for multi-hop random-access networks

Cecchi

Ven

Shneer³

2018

SIGMETRICS Perform. Eval. Rev.

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“…The first result establishes the existence and uniqueness of a stochastic process satisfying the SDEs (12) and (13). For T > 0, let D T def.…”

Section: An Asymptotic Processmentioning

confidence: 98%

“…Simplified models using finite birth and death processes have been often used, see Chun et al [9], Picconi et al [24] and Ramabhadran and Pasquale [26]. In Feuillet and Robert [13] and Sun et al [30], the authors studied how the durability T (δ) scales with the number of servers N and the maximum number of copies d of each file, under simplifying assumptions on file losses and the duplication mechanism. For this later work, each copy of a file is assumed to be lost at a certain fixed rate, independently of the other copies of files.…”

Section: Models With Independent Losses Of Copies and Global Duplicationmentioning

confidence: 99%

A large scale analysis of unreliable stochastic networks

Aghajani¹,

Pierpoint²,

Sun³

2018

Ann. Appl. Probab.

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The problem of reliability of a large distributed system is analyzed via a new mathematical model. A typical framework is a system where a set of files are duplicated on several data servers. When one of these servers breaks down, all copies of files stored on it are lost. In this way, repeated failures may lead to losses of files. The efficiency of such a network is directly related to the performances of the mechanism used to duplicate files on servers. In this paper we study the evolution of the network using a natural duplication policy giving priority to the files with the least number of copies.We investigate the asymptotic behavior of the network when the number N of servers is large. The analysis is complicated by the large dimension of the state space of the empirical distribution of the state of the network. A stochastic model of the evolution of the network which has values in state space whose dimension does not depend on N is introduced. Despite this description does not have the Markov property, it turns out that it is converging in distribution, when the number of nodes goes to infinity, to a nonlinear Markov process. The rate of decay of the network, which is the key characteristic of interest of these systems, can be expressed in terms of this asymptotic process. The corresponding mean-field convergence results are established. A lower bound on the exponential decay, with respect to time, of the fraction of the number of initial files with at least one copy is obtained.

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“…Therefore, during the last decade considerable attention has been paid to developing stochastic model analysis for assessing file lifetime and data security in large-scale data center networks with file replication mechanism. Also see Picconi et al [11] and [12], Kersch and Szabo [5] and Feuillet and Robert [4] for more details.…”

Section: Introductionmentioning

confidence: 99%

“…In addition, there are still some interesting issues, such as cost analysis of recovering lost data, effect of file replication mechanism and bandwidth limitation, durability and availability of data, and how to control the file lost probability. Readers may refer to recent publications for details, among which, data storage systems by Blake and Rodrigues [2], Utard and Vernois [18], Lian et al [7], Chun [3] and Ramabhadran and Pasquale [14], [15] and [16]; DHT replication by Picconi et al [11] and [12], Kersch and Szabo [5], Pace et al [10] and Kniesburges et al [6]; failure prediction by Pinheiro et al [13]; and large-scale stochastic networks with unreliable processors by Feuillet and Robert [4], Sun et al [17] and Aghajani et al [1].…”

Section: Introductionmentioning

confidence: 99%

A Stochastic Model for File Lifetime and Security in Data Center Networks

2018

Computational Data and Social Networks

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Data center networks are an important infrastructure in various applications of modern information technologies. Note that each data center always has a finite lifetime, thus once a data center fails, then it will lose all its storage files and useful information. For this, it is necessary to replicate and copy each important file into other data centers such that this file can increase its lifetime of staying in a data center network. In this paper, we describe a large-scale data center network with a file d-threshold policy, which is to replicate each important file into at most d − 1 other data centers such that this file can maintain in the data center network under a given level of data security in the long-term. To this end, we develop three relevant Markov processes to propose two effective methods for assessing the file lifetime and data security. By using the RG-factorizations, we show that the two methods are used to be able to more effectively evaluate the file lifetime of large-scale data center networks. We hope the methodology and results given in this paper are applicable in the file lifetime study of more general data center networks with replication mechanism.

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A Scaling Analysis of a Transient Stochastic Network

Cited by 7 publications

References 27 publications

Mean-field limits for multi-hop random-access networks

Mean-field limits for multi-hop random-access networks

A large scale analysis of unreliable stochastic networks

A Stochastic Model for File Lifetime and Security in Data Center Networks

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