Supermarket Model on Graphs
Amarjit Budhiraja, Debankur Mukherjee, and Ruoyu Wu

TL;DR
This paper extends the classical supermarket model to graphs beyond cliques, showing that under certain degree conditions, the queue-length distribution converges to the same deterministic limit as in the well-known model, even on sparse or random graphs.
Contribution
It demonstrates that the asymptotic behavior of the supermarket model's occupancy process is robust to various network topologies, including sparse and Erdős-Rényi random graphs, under specific degree conditions.
Findings
Occupancy process converges to the same ODE system as the classical model for dense graphs.
Convergence holds for sparse graphs if minimum degree tends to infinity.
For Erdős-Rényi graphs, annealed convergence occurs if average degree grows unboundedly.
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 , and each task is irrevocably assigned to the shortest queue among the one it first appears and its 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- 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…
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