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In general, obtaining the exact steady-state distribution of queue lengths isnot feasible. Therefore, we establish bounds for the tail probabilities ofqueue lengths. Specifically, we examine queueing systems under Heavy-Traffic(HT) conditions and provide exponentially decaying bounds for the probability$\mathbb P(\epsilon q > x)$, where $\epsilon$ is the HT parameter denoting howfar the load is from the maximum allowed load. Our bounds are not limited toasymptotic cases and are applicable even for finite values of $\epsilon$, andthey get sharper as $\epsilon \to 0$. Consequently, we derive non-asymptoticconvergence rates for the tail probabilities. Unlike other approaches such asmoment bounds based on drift arguments and bounds on Wasserstein distance usingStein's method, our method yields sharper tail bounds. Furthermore, our resultsoffer bounds on the exponential rate of decay of the tail, given by$-\frac{1}{x} \log \mathbb P(\epsilon q > x)$ for any finite value of $x$.These can be interpreted as non-asymptotic versions of Large Deviation (LD)results.  We demonstrate our approach by presenting tail bounds for: (i) a continuoustime Join-the-shortest queue (JSQ) load balancing system, (ii) a discrete timesingle-server queue and (iii) an $M/M/n$ queue. We not only bridge the gapbetween classical-HT and LD regimes but also explore the large system HTregimes for JSQ and $M/M/n$ systems. In these regimes, both the system size andthe system load increase simultaneously. Our results also close a gap in theexisting literature on the limiting distribution of JSQ in the super-NDS(a.k.a. super slowdown) regime. This contribution is of an independentinterest. Here, a key ingredient is a more refined characterization of statespace collapse for JSQ system, achieved by using an exponential Lyapunovfunction designed to approximate the $\ell_{\infty}$ norm.

Exponential Tail Bounds on Queues: A Confluence of Non-Asymptotic Heavy Traffic and Large Deviations