Large Deviation Asymptotics for the Supermarket Model with Growing Choices

We consider the Markovian supermarket model with growing choices, where jobs arrive at rate $n\lambda_n$ and each of $n$ parallel servers processes jobs in its queue at rate $1$. Each incoming job joins the shortest among $d_n \in \{1,\dotsc,n\}$ randomly selected queues. Under the assumption $d_n \to \infty$ and $\lambda_n \to \lambda \in (0,\infty)$ as $n\to \infty$, a large deviation principle (LDP) for the occupancy process is established in a suitable infinite-dimensional path space, and it is shown that the rate function is invariant with respect to the manner in which $d_n \to \infty$. The LDP gives information on the rate of decay of probabilities of various types of rare events associated with the system. We illustrate this by establishing explicit exponential decay rates for probabilities of large total number of jobs in the system. As a corollary, we also show that probabilities of certain rare events can indeed depend on the rate of $d_n \to \infty$.