scholarly journals Delay-Optimal Scheduling for Two-Hop Relay Networks with Randomly Varying Connectivity: Join the Shortest Queue-Longest Connected Queue Policy

2017 ◽  
Vol 2017 ◽  
pp. 1-15
Author(s):  
Seung Jun Baek ◽  
Joon-Sang Park
Keyword(s):  
Queueing Network ◽  
Stochastic Ordering ◽  
Time Sharing ◽  
Dynamic Interactions ◽  
Join The Shortest Queue ◽  
State Changes ◽  
Shortest Queue ◽  
Multiple Relay ◽  
Symmetric Statistics ◽  
Symmetric Systems

We consider a scheduling problem for a two-hop queueing network where the queues have randomly varying connectivity. Customers arrive at the source queue and are later routed to multiple relay queues. A relay queue can be served only if it is in connected state, and the state changes randomly over time. The source queue and relay queues are served in a time-sharing manner; that is, only one customer can be served at any instant. We propose Join the Shortest Queue-Longest Connected Queue (JSQ-LCQ) policy as follows: (1) if there exist nonempty relay queues in connected state, serve the longest queue among them; (2) if there are no relay queues to serve, route a customer from the source queue to the shortest relay queue. For symmetric systems in which the connectivity has symmetric statistics across the relay queues, we show that JSQ-LCQ is strongly optimal, that is, minimizes the delay in the stochastic ordering sense. We use stochastic coupling and show that the systems under coupling exist in two distinct phases, due to dynamic interactions among source and relay queues. By careful construction of coupling in both phases, we establish the stochastic dominance in delay between JSQ-LCQ and any arbitrary policy.

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 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.

scholarly journals Positive correlations in a three-node Jackson queueing network

1989 ◽  
Vol 21 (01) ◽  
pp. 241-242 ◽  
Author(s):  
Robert D. Foley ◽  
Peter C. Kiessler
Keyword(s):  
Random Variables ◽  
Queueing Network ◽  
Stochastic Ordering ◽  
Jackson Network ◽  
Associated Random Variables ◽  
Sojourn Times ◽  
Positive Correlations

We show for the three-node Jackson network studied in [3] that a customer’s sojourn times in nodes 1 and 3 are positively correlated. We actually prove a stronger result, that the two sojourn times are associated random variables. Our proof uses a stochastic ordering argument similar to that in [4].

Subdiffusive Load Balancing in Time-Varying Queueing Systems

2019 ◽  
Vol 67 (6) ◽  
pp. 1678-1698
Author(s):  
Rami Atar ◽  
Isaac Keslassy ◽  
Gal Mendelson
Keyword(s):  
Load Balancing ◽  
Queue Length ◽  
Queueing Systems ◽  
Load Condition ◽  
Service Rates ◽  
Queue Lengths ◽  
Join The Shortest Queue ◽  
The Difference ◽  
Shortest Queue ◽  
Diffusion Scale

The degree to which delays or queue lengths equalize under load-balancing algorithms gives a good indication of their performance. Some of the most well-known results in this context are concerned with the asymptotic behavior of the delay or queue length at the diffusion scale under a critical load condition, where arrival and service rates do not vary with time. For example, under the join-the-shortest-queue policy, the queue length deviation process, defined as the difference between the greatest and smallest queue length as it varies over time, is at a smaller scale (subdiffusive) than that of queue lengths (diffusive).

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