Join-the-shortest queue diffusion limit in Halfin–Whitt regime: Tail asymptotics and scaling of extrema

Abstract

Consider a system of N parallel single-server queues with unit-exponential service time distribution and a single dispatcher where tasks arrive as a Poisson process of rate λ(N). When a task arrives, the dispatcher assigns it to one of the servers according to the Join-the-Shortest Queue (JSQ) policy. Eschenfeldt and Gamarnik (2015) established that in the Halfin-Whitt regime where (N − λ(N))/ √ N → β > 0 as N → ∞, appropriately scaled occupancy measure of the system under the JSQ policy converges weakly on any finite time interval to a certain diffusion process as N → ∞. Recently, it was further established by Braverman (2018) that the convergence result extends to the steady state as well, i.e., stationary occupancy measure of the system converges weakly to the steady state of the diffusion process as N → ∞, proving the interchange of limits result.In this paper we perform a detailed analysis of the steady state of the above diffusion process. Specifically, we establish precise tail-asymptotics of the stationary distribution and scaling of extrema of the process on large time interval. Our results imply that the asymptotic steady-state scaled number of servers with queue length two or larger exhibits an Exponential tail, whereas that for the number of idle servers turns out to be Gaussian. From the methodological point of view, the diffusion process under consideration goes beyond the state-of-the-art techniques in the study of the steady state of diffusion processes. Lack of any closed form expression for the steady state and intricate interdependency of the process dynamics on its local times make the analysis significantly challenging. We develop a technique involving the theory of regenerative processes that provides a tractable form for the stationary measure, and in conjunction with several sharp hitting time estimates, acts as a key vehicle in establishing the results. The technique and the intermediate results might be of independent interest, and can possibly be used in understanding the bulk behavior of the process.

Steady-state analysis of load-balancing algorithms in the sub-Halfin–Whitt regime

“…To the best of our knowledge, there are only a few papers that deal with the steady-state analysis of many-server systems with distributed queues [1, 3, 10]. [1] and [3] analyze the steady-state distribution of JSQ in the Halfin–Whitt regime, and [10] studies Po d with . This paper complements all three, as it applies to a class of load-balancing algorithms and to any sub-Halfin–Whitt regime.…”

Join-the-Shortest Queue diffusion limit in Halfin–Whitt regime: Sensitivity on the heavy-traffic parameter

“…Remark 5. Observe that Lemma 5.6 is a major improvement over [3,Lemma 5.3] (see the statement in Lemma A.3). In Lemma A.3 a similar tail-bound is given for y β −1 log β −1 .…”

A Simple Steady-State Analysis of Load Balancing Algorithms in the Sub-Halfin-Whitt Regime

“…A recent breakthrough in this area is [6], which shows that in the Halfin-Whitt regime (α = 0.5), the diffusion-scaled process converges to a two-dimensional diffusion limit, from which it can be shown that most servers have one job in service and O( √ N) servers have two jobs (one in service and one in buffer). This seminal work has led to several significant developments: (i) [3] proved that the stationary distribution indeed converges to the stationary distribution of the two-dimensional diffusion limit based on Stein's method; and (ii) via stochastic coupling, [13] showed that the diffusion limit of Pod converges to that of JSQ in the Halfin-Whitt regime at the process level (over finite time) when d = Θ( √ N log N); and (iii) when α < 1/6, [10] proved that the waiting probability of a job is asymptotically zero with d = ω 1 1−λ at the steady-state based on Stein's method. Interested readers can find a comprehensive survey of recent results in [16].…”

“…From the best of our knowledge, there are only a few papers that deal with the steadystate analysis of many-server systems with distributed queues [1], [3], [10]. [1], [3] analyze the steady-state distribution of JSQ in the Halfin-Whitt regime and [10] studies the Pod with α < 1/6. This paper complements [1], [3], [10], as it applies to a class of load balancing algorithms and to any sub-Halfin-Whitt regime.…”

“…[1], [3] analyze the steady-state distribution of JSQ in the Halfin-Whitt regime and [10] studies the Pod with α < 1/6. This paper complements [1], [3], [10], as it applies to a class of load balancing algorithms and to any sub-Halfin-Whitt regime.…”