Many-Server Asymptotics for Join-the-Shortest-Queue: Large Deviations and Rare Events

The Join-the-Shortest-Queue routing policy is studied in an asymptotic regime where the number of processors $n$ scales with the arrival rate. A large deviation principle (LDP) for the occupancy process is established, as $n\to \infty$, in a suitable infinite-dimensional path space. Model features that present technical challenges include, Markovian dynamics with discontinuous statistics, a diminishing rate property of the transition probability rates, and an infinite-dimensional state space. The difficulty is in the proof of the Laplace lower bound which requires establishing the uniqueness of solutions of certain infinite-dimensional systems of controlled ordinary differential equations. 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 long queues. In particular, denoting by $E_j^n(T)$ the event that there is at least one queue with $j$ or more jobs at some time instant over $[0,T]$, we show that, in the critical case, for large $n$ and $T$, $\mathbb{P}(E^n_j(T)) \approx \exp\left [-\frac{n (j-2)^2}{4T}\right].$