Large deviations of max-weight scheduling policies on convex rate regions

Abstract—We consider a single server discrete-time system with K users where the server picks operating points from a compact, convex and co-ordinate convex set in ℜ K +. For this system we analyse the performance of a stablising policy that at any given time picks operating points from the allowed rate region that maximise a weighted sum of rate, where the weights depend upon the workloads of the users. Assuming a Large Deviations Principle (LDP) for the arrival processes in the Skorohod space of functions that are right-continuous with left-hand limits we establish an LDP for the workload process using a generalised version of the contraction principle to derive the corresponding rate function. With the LDP result available we then analyse the tail probabilities of the workloads under different buffering scenarios. I

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

In general, obtaining the exact steady-state distribution of queue lengths is not feasible. Therefore, we establish bounds for the tail probabilities of queue 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 how far the load is from the maximum allowed load. Our bounds are not limited to asymptotic cases and are applicable even for finite values of $\epsilon$, and they get sharper as $\epsilon \to 0$. Consequently, we derive non-asymptotic convergence rates for the tail probabilities. Unlike other approaches such as moment bounds based on drift arguments and bounds on Wasserstein distance using Stein's method, our method yields sharper tail bounds. Furthermore, our results offer 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 continuous time Join-the-shortest queue (JSQ) load balancing system, (ii) a discrete time single-server queue and (iii) an $M/M/n$ queue. We not only bridge the gap between classical-HT and LD regimes but also explore the large system HT regimes for JSQ and $M/M/n$ systems. In these regimes, both the system size and the system load increase simultaneously. Our results also close a gap in the existing literature on the limiting distribution of JSQ in the super-NDS (a.k.a. super slowdown) regime. This contribution is of an independent interest. Here, a key ingredient is a more refined characterization of state space collapse for JSQ system, achieved by using an exponential Lyapunov function designed to approximate the $\ell_{\infty}$ norm.Comment: 37 pages, 1 figur