Diffusion approximations for load balancing mechanisms in cloud storage systems

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.

Scalable Load Balancing in Networked Systems: Universality Properties and Stochastic Coupling Methods

“…It is noteworthy that the scaled occupancy process loses its diffusive behavior for fixed d. It is further shown in [11] that with high probability the steady-state fraction of queues with length at least log d (N/η(N )) − ω(1) tasks approaches unity, which in turn implies that with high probability the steadystate delay is at least log d (N/η(N )) − O(1) as N → ∞. The diffusion approximation of the JSQ(d) policy in the Halfin-Whitt regime (2.1), starting from a different initial scaling, has been studied by Budhiraja & Friedlander [8]. Recently, Ying [47] introduced a broad framework involving Stein's method to analyze the rate of convergence of the scaled steady-state occupancy process of the JSQ(2) policy when η(N ) = N α with α > 0.8.…”

A Functional Central Limit Theorem for the Becker–Döring Model

“…This feature has implications on the choice of the Hilbert space L 2 (w) chosen to formulate the SDE (10) and, additionally, several estimates have to be derived to control the stochastic fluctuations of the first coordinate. This situation is different from the examples of the Ginzburg-Landau model in [7,28,34], since each site only have interactions with a finite number of sites, or in the stochastic network example [6] where the interaction range is also finite. It should be also noted that our evolution equations are driven by a set of independent Poisson processes whose intensity is state dependent which is not the case in the Ginzburg-Landau models for which the diffusion coefficients are constant.…”

“…For more results on related fluctuation problems in statistical mechanics, see Spohn [28,29]. Fluctuations of an infinite dimensional Markov process associated to load balancing mechanisms in large stochastic networks have been investigated in Graham [13], Budhiraja and Friedlander [6].…”

Near Equilibrium Fluctuations for Supermarket Models with Growing Choices

“…The influential works of Mitzenmacher [23,24] and Vvedenskaya et al [29] showed by considering a fluid scaling that increasing d from 1 to 2 leads to significant improvement in performance in terms of steadystate queue length distributions in that the tails of the asymptotic steady-state distributions decay exponentially when d = 1 and super-exponentially when d = 2. Limit theorems under a diffusion scaling for the JSQ(d) system, with a fixed d, can be found in [5,7]. Although JSQ(d) for a fixed d ≥ 2 leads to significant improvements over JSQ (1), as observed in [10,11], no fixed value of d provides the optimal waiting time properties of the join-the-shortest-queue system (i.e.…”