scholarly journals Analysis of tri-cum biserial bulk queue model connected with a common server

2020 ◽  
Keyword(s):  
Bulk Queue ◽  
Queue Model

Stability and heavy traffic results for the general bulk queue

1978 ◽  
Vol 10 (1) ◽  
pp. 213-231 ◽  
Author(s):  
John Dagsvik
Keyword(s):  
Initial Conditions ◽  
Heavy Traffic ◽  
Mixing Processes ◽  
Absolute Value ◽  
Bulk Queue ◽  
Unstable Case ◽  
Queue Model ◽  
Functional Central Limit Theorems ◽  
Functional Central Limit ◽  
Gaussian Variable

In this paper we prove that the limiting distribution of the general bulk queue exists and is independent of the initial conditions if and only if the traffic intensity is less than one. We further generalize the following heavy traffic results of the GI/G/1 model to the general bulk queue model. When ρ > 1 or ρ = 1 the waiting time is distributed approximately as a Gaussian variable and the absolute value of a Gaussian variable, respectively. The exponential approximation is derived from the Wiener–Hopf matrix equations established in a previous paper while the unstable case ρ ≧ 1 is treated by means of functional central limit theorems for mixing processes.

Stability and heavy traffic results for the general bulk queue

1978 ◽  
Vol 10 (01) ◽  
pp. 213-231
Author(s):  
John Dagsvik
Keyword(s):  
Initial Conditions ◽  
Heavy Traffic ◽  
Mixing Processes ◽  
Absolute Value ◽  
Bulk Queue ◽  
Unstable Case ◽  
Queue Model ◽  
Functional Central Limit Theorems ◽  
Functional Central Limit ◽  
Gaussian Variable

In this paper we prove that the limiting distribution of the general bulk queue exists and is independent of the initial conditions if and only if the traffic intensity is less than one. We further generalize the following heavy traffic results of the GI/G/1 model to the general bulk queue model. When ρ > 1 or ρ = 1 the waiting time is distributed approximately as a Gaussian variable and the absolute value of a Gaussian variable, respectively. The exponential approximation is derived from the Wiener–Hopf matrix equations established in a previous paper while the unstable case ρ ≧ 1 is treated by means of functional central limit theorems for mixing processes.

Equilibrium probability calculations for a discrete-time bulk queue model

1996 ◽  
Vol 22 (1-2) ◽  
pp. 189-198 ◽  
Author(s):  
Y. Quennel Zhao ◽  
L. Lorne Campbell
Keyword(s):  
Discrete Time ◽  
Bulk Queue ◽  
Queue Model ◽  
Equilibrium Probability

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.

Sign in / Sign up

Export Citation Format

Share Document