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We consider a queueing system with $n$ parallel queues operating according tothe so-called "supermarket model" in which arriving customers join the shortestof $d$ randomly selected queues. Assuming rate $n\lambda_{n}$ Poisson arrivalsand rate $1$ exponentially distributed service times, we consider this model inthe heavy traffic regime, described by $\lambda_{n}\uparrow 1$ as $n\to\infty$.We give a simple expectation argument establishing that majority of queues havesteady state length at least $\log_d(1-\lambda_{n})^{-1} - O(1)$ withprobability approaching one as $n\rightarrow\infty$, implying the same for thesteady state delay of a typical customer.  Our main result concerns the detailed behavior of queues with length smallerthan $\log_d(1-\lambda_{n})^{-1}-O(1)$. Assuming $\lambda_{n}$ converges to $1$at rate at most $\sqrt{n}$, we show that the dynamics of such queues does notfollow a diffusion process, as is typical for queueing systems in heavytraffic, but is described instead by a deterministic infinite system of lineardifferential equations, after an appropriate rescaling. The unique fixed pointsolution of this system is shown explicitly to be of the form$\pi_{1}(d^{i}-1)/(d-1), i\ge 1$, which we conjecture describes the steadystate behavior of the queue lengths after the same rescaling. Our result isobtained by combination of several technical ideas including establishing theexistence and uniqueness of an associated infinite dimensional system ofnon-linear integral equations and adopting an appropriate stopped process as anintermediate step.

Supermarket Queueing System in the Heavy Traffic Regime. Short Queue Dynamics