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June 2019 Supermarket model on graphs
Amarjit Budhiraja, Debankur Mukherjee, Ruoyu Wu
Ann. Appl. Probab. 29(3): 1740-1777 (June 2019). DOI: 10.1214/18-AAP1437

Abstract

We consider a variation of the supermarket model in which the servers can communicate with their neighbors and where the neighborhood relationships are described in terms of a suitable graph. Tasks with unit-exponential service time distributions arrive at each vertex as independent Poisson processes with rate $\lambda$, and each task is irrevocably assigned to the shortest queue among the one it first appears and its $d-1$ randomly selected neighbors. This model has been extensively studied when the underlying graph is a clique in which case it reduces to the well-known power-of-$d$ scheme. In particular, results of Mitzenmacher (1996) and Vvedenskaya et al. (1996) show that as the size of the clique gets large, the occupancy process associated with the queue-lengths at the various servers converges to a deterministic limit described by an infinite system of ordinary differential equations (ODE). In this work, we consider settings where the underlying graph need not be a clique and is allowed to be suitably sparse. We show that if the minimum degree approaches infinity (however slowly) as the number of servers $N$ approaches infinity, and the ratio between the maximum degree and the minimum degree in each connected component approaches $1$ uniformly, the occupancy process converges to the same system of ODE as the classical supermarket model. In particular, the asymptotic behavior of the occupancy process is insensitive to the precise network topology. We also study the case where the graph sequence is random, with the $N$th graph given as an Erdős–Rényi random graph on $N$ vertices with average degree $c(N)$. Annealed convergence of the occupancy process to the same deterministic limit is established under the condition $c(N)\to\infty$, and under a stronger condition $c(N)/\ln N\to\infty$, convergence (in probability) is shown for almost every realization of the random graph.

Citation

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Amarjit Budhiraja. Debankur Mukherjee. Ruoyu Wu. "Supermarket model on graphs." Ann. Appl. Probab. 29 (3) 1740 - 1777, June 2019. https://doi.org/10.1214/18-AAP1437

Information

Received: 1 December 2017; Revised: 1 September 2018; Published: June 2019
First available in Project Euclid: 19 February 2019

zbMATH: 07057465
MathSciNet: MR3914555
Digital Object Identifier: 10.1214/18-AAP1437

Subjects:
Primary: 60F17, 60K25, 68M20
Secondary: 60J27, 60K35

Rights: Copyright © 2019 Institute of Mathematical Statistics

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Vol.29 • No. 3 • June 2019
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