Brownian models of multiclass queueing networks: Current status and open problems
type: Item
instance of: scholarly article
title: Brownian models of multiclass queueing networks: Current status and open problems
zbMATH Open document ID: 0781.60090
DOI: 10.1007/BF01158927
author: J. Michael Harrison
author: Viên Nguyen
published in: Queueing Systems
publication date: 1994-02-17T00:00:00Z
review text: The problem of heavy traffic approximation by reflected Brownian motion (BM) for open queueing network with \(D\) nodes and \(C\) customer classes is surveyed. The network has Markovian routing, and customers are assumed to switch among classes in Markovian fashion also. It is assumed that external input processes can be correlated and obey some functional central limit theorem with limiting \(C\)-dimensional BM with zero drift. The same is right for cumulative service capacity process, too (with \(D\)- dimensional limiting BM). The relation between BM and heavy traffic theory is discussed. Continuous time workload, queue-size and some other basic processes are introduced for each customer type and the whole network. A natural scaling of time and space is considered. The main scaling factor is the vector of differences between long-time average service rates and traffic intensities for all nodes. Then the Brownian model is defined that approximates scaled system of the mentioned network processes. Heavy traffic convergence is considered. Two numerical examples are presented where complete sojourn time distributions are estimated.
Mathematics Subject Classification ID: 60K25
Mathematics Subject Classification ID: 90B22
Mathematics Subject Classification ID: 60J65
zbMATH DE Number: 218349
zbMATH Keywords: multiclass customers
zbMATH Keywords: heavy traffic approximation
zbMATH Keywords: functional central limit theorem
zbMATH Keywords: cumulative service capacity process
zbMATH Keywords: Brownian model
zbMATH Keywords: network processes
zbMATH Keywords: numerical examples
zbMATH Keywords: sojourn time distributions
reviewed by: Evsei Morozov
MaRDI profile type: Publication
cites work: Markov Processes Representing Traffic in Connecting Networks
cites work: Stochastic discrete flow networks: Diffusion approximations and bottlenecks
cites work: Diffusion approximations for open queueing networks with service interruptions
cites work: Reflected Brownian motion in an orthant: Numerical methods for steady- state analysis
cites work: Heavy-Traffic Limits for Queues with Many Exponential Servers
cites work: The diffusion approximation for tandem queues in heavy traffic
cites work: Q3808989
cites work: The QNET method for two-moment analysis of open queueing networks
cites work: Reflected Brownian motion on an orthant
cites work: Multidimensional reflected Brownian motions having exponential stationary distributions
cites work: Brownian models of open queueing networks with homogeneous customer populations∗
cites work: Brownian models of feedforward queueing networks: Quasireversibility and product form solutions
cites work: Multiple channel queues in heavy traffic. I
cites work: Diffusion approximations for queues with server vacations
cites work: Q5584435
cites work: A Heavy Traffic Limit Theorem for Networks of Queues with Multiple Customer Types
cites work: Open Queueing Networks in Heavy Traffic
cites work: A multiclass feedback queue in heavy traffic
cites work: A boundary property of semimartingale reflecting Brownian motions
cites work: Existence and uniqueness of semimartingale reflecting Brownian motions in an orthant
cites work: Heavy Traffic Limit Theorems for Queues: A Survey
cites work: Refining diffusion approximations for queues
cites work: Large Fluctuations in a Deterministic Multiclass Network of Queues
cites work: Reflected Brownian motion with skew symmetric data in a polyhedral domain
full work available at URL: https://doi.org/10.1007/bf01158927
OpenAlex ID: W1974927663
description: scientific article; zbMATH DE number 218349