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Diffusion approximations for load balancing mechanisms in cloud storage systems

Part of: Limit theorems Special processes Operations research and management science Stochastic analysis Probability theory on algebraic and topological structures Markov processes

Published online by Cambridge University Press:  22 July 2019

Amarjit Budhiraja  and
Eric Friedlander
Amarjit Budhiraja*
Affiliation:
University of North Carolina at Chapel Hill
Eric Friedlander*
Affiliation:
University of North Carolina at Chapel Hill
*
*Postal address: Department of Statistics and Operations Research, University of North Carolina at Chapel Hill, NC 27599, USA.
*Postal address: Department of Statistics and Operations Research, University of North Carolina at Chapel Hill, NC 27599, USA.
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Abstract

In large storage systems, files are often coded across several servers to improve reliability and retrieval speed. We study load balancing under the batch sampling routeing scheme for a network of n servers storing a set of files using the maximum distance separable (MDS) code (cf. Li (2016)). Specifically, each file is stored in equally sized pieces across L servers such that any k pieces can reconstruct the original file. When a request for a file is received, the dispatcher routes the job into the k-shortest queues among the L for which the corresponding server contains a piece of the file being requested. We establish a law of large numbers and a central limit theorem as the system becomes large (i.e. n → ∞), for the setting where all interarrival and service times are exponentially distributed. For the central limit theorem, the limit process take values in ℓ2, the space of square summable sequences. Due to the large size of such systems, a direct analysis of the n-server system is frequently intractable. The law of large numbers and diffusion approximations established in this work provide practical tools with which to perform such analysis. The power-of-d routeing scheme, also known as the supermarket model, is a special case of the model considered here.

Keywords

Mean-field approximationdiffusion approximationstochastic networkcylindrical Brownian motionpropagation of chaoscloud storage systemsupermarket modelMDS codingpower-of-d

MSC classification

Primary: 60K25: Queueing theory 60K35: Interacting random processes; statistical mechanics type models; percolation theory 60F05: Central limit and other weak theorems 60H30: Applications of stochastic analysis (to PDE, etc.)
Secondary: 90B22: Queues and service 60J28: Applications of continuous-time Markov processes on discrete state spaces 60J70: Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.) 60B12: Limit theorems for vector-valued random variables (infinite-dimensional case)

Information

Type
Original Article
Information
Advances in Applied Probability , Volume 51 , Issue 1 , March 2019 , pp. 41 - 86
Copyright
© Applied Probability Trust 2019 

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