SUPERMARKET MODEL ON GRAPHS

• 成果类型: Article
• 署名作者: Budhiraja, Amarjit; Mukherjee, Debankur; Wu, Ruoyu
• 署名单位: University of North Carolina; University of North Carolina Chapel Hill; Brown University; University of Michigan System; University of Michigan
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/18-AAP1437
• 发表日期: 2019
• 页码: 1740-1777
• 关键词: shortest; Join

摘要

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 Nth graph given as an Erdos-Renyi 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) ->infinity, and under a stronger condition c(N)/ln N ->infinity, con- vergence (in probability) is shown for almost every realization of the random graph.