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

2019 ◽  
Vol 29 (2) ◽  
pp. 1262-1309 ◽  
Author(s):  
Sayan Banerjee ◽  
Debankur Mukherjee
Keyword(s):  
Diffusion Limit ◽  
Tail Asymptotics ◽  
Join The Shortest Queue ◽  
Shortest Queue

scholarly journals Steady-State Analysis of the Join-the-Shortest-Queue Model in the Halfin–Whitt Regime

2020 ◽  
Vol 45 (3) ◽  
pp. 1069-1103
Author(s):  
Anton Braverman
Keyword(s):  
Steady State ◽  
Diffusion Limit ◽  
Fluid Limit ◽  
Time Intervals ◽  
Dimensional Diffusion ◽  
Queue Model ◽  
Join The Shortest Queue ◽  
Shortest Queue ◽  
Process Level ◽  
General Tool

This paper studies the steady-state properties of the join-the-shortest-queue model in the Halfin–Whitt regime. We focus on the process tracking the number of idle servers and the number of servers with nonempty buffers. Recently, Eschenfeldt and Gamarnik proved that a scaled version of this process converges, over finite time intervals, to a two-dimensional diffusion limit as the number of servers goes to infinity. In this paper, we prove that the diffusion limit is exponentially ergodic and that the diffusion scaled sequence of the steady-state number of idle servers and nonempty buffers is tight. Combined with the process-level convergence proved by Eschenfeldt and Gamarnik, our results imply convergence of steady-state distributions. The methodology used is the generator expansion framework based on Stein’s method, also referred to as the drift-based fluid limit Lyapunov function approach in Stolyar. One technical contribution to the framework is to show how it can be used as a general tool to establish exponential ergodicity.

scholarly journals Asymptotic Optimality of Power-of-d Load Balancing in Large-Scale Systems

2020 ◽  
Vol 45 (4) ◽  
pp. 1535-1571 ◽  
Author(s):  
Debankur Mukherjee ◽  
Sem C. Borst ◽  
Johan S. H. van Leeuwaarden ◽  
Philip A. Whiting
Keyword(s):  
Large Scale ◽  
Asymptotic Optimality ◽  
Diffusion Limit ◽  
Fluid Limit ◽  
Large Scale Systems ◽  
Minimum Number ◽  
Join The Shortest Queue ◽  
The Difference ◽  
Shortest Queue ◽  
And Diffusion

We consider a system of N identical server pools and a single dispatcher in which tasks with unit-exponential service requirements arrive at rate [Formula: see text]. In order to optimize the experienced performance, the dispatcher aims to evenly distribute the tasks across the various server pools. Specifically, when a task arrives, the dispatcher assigns it to the server pool with the minimum number of tasks among d(N) randomly selected server pools. We construct a stochastic coupling to bound the difference in the system occupancy processes between the join-the-shortest-queue (JSQ) policy and a scheme with an arbitrary value of d(N). We use the coupling to derive the fluid limit in case [Formula: see text] and [Formula: see text] as [Formula: see text] along with the associated fixed point. The fluid limit turns out to be insensitive to the exact growth rate of d(N) and coincides with that for the JSQ policy. We further establish that the diffusion limit corresponds to that for the JSQ policy as well, as long as [Formula: see text], and characterize the common limiting diffusion process. These results indicate that the JSQ optimality can be preserved at the fluid and diffusion levels while reducing the overhead by nearly a factor O(N) and O([Formula: see text]), respectively.

scholarly journals Universality of load balancing schemes on the diffusion scale

2016 ◽  
Vol 53 (4) ◽  
pp. 1111-1124 ◽  
Author(s):  
Debankur Mukherjee ◽  
Sem C. Borst ◽  
Johan S. H. van Leeuwaarden ◽  
Philip A. Whiting
Keyword(s):  
Load Balancing ◽  
Poisson Process ◽  
Heavy Traffic ◽  
Diffusion Limit ◽  
Parallel Queues ◽  
Special Cases ◽  
Service Rates ◽  
Join The Shortest Queue ◽  
Shortest Queue ◽  
Diffusion Scale

Abstract We consider a system of N parallel queues with identical exponential service rates and a single dispatcher where tasks arrive as a Poisson process. When a task arrives, the dispatcher always assigns it to an idle server, if there is any, and to a server with the shortest queue among d randomly selected servers otherwise (1≤d≤N). This load balancing scheme subsumes the so-called join-the-idle queue policy (d=1) and the celebrated join-the-shortest queue policy (d=N) as two crucial special cases. We develop a stochastic coupling construction to obtain the diffusion limit of the queue process in the Halfin‒Whitt heavy-traffic regime, and establish that it does not depend on the value of d, implying that assigning tasks to idle servers is sufficient for diffusion level optimality.

scholarly journals Multiple-server system with flexible arrivals

2011 ◽  
Vol 43 (4) ◽  
pp. 985-1004 ◽  
Author(s):  
Osman T. Akgun ◽  
Rhonda Righter ◽  
Ronald Wolff
Keyword(s):  
Finite Buffers ◽  
Service Production ◽  
Weak Majorization ◽  
Service Rates ◽  
Join The Shortest Queue ◽  
Traffic Systems ◽  
Shortest Queue ◽  
Performance Gains ◽  
Number Of Customers ◽  
Server System

In many service, production, and traffic systems there are multiple types of customers requiring different types of ‘servers’, i.e. different services, products, or routes. Often, however, a proportion of the customers are flexible, i.e. they are willing to change their type in order to achieve faster service, and even if this proportion is small, it has the potential of achieving large performance gains. We generalize earlier results on the optimality of ‘join the shortest queue’ (JSQ) for flexible arrivals to the following: arbitrary arrivals where only a subset are flexible, multiple-server stations, and abandonments. Surprisingly, with abandonments, the optimality of JSQ for minimizing the number of customers in the system depends on the relative abandonment and service rates. We extend our model to finite buffers and resequencing. We assume exponential service. Our optimality results are very strong; we minimize the queue length process in the weak majorization sense.

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