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A Taylor series approach for coupled queueing systems with intermediate load

Ekaterina Evdokimova UGent, Sabine Wittevrongel UGent and Dieter Fiems UGent (2017) PROCEEDINGS OF THE INTERNATIONAL CONFERENCE ON NUMERICAL ANALYSIS AND APPLIED MATHEMATICS 2016 (ICNAAM-2016). In AIP Conference Proceedings 1863.
abstract
We focus on the numerical analysis of a coupled queueing system with Poisson arrivals and exponentially distributed service times. Such a system consists of multiple queues served by a single server. Service is synchronised meaning that there is a departure from every queue upon service completion and there is no service whenever one of the queues is empty. It was shown before that the terms in the Maclaurin series expansion of the steady-state distribution of this queueing system when the service rate is sent to 0 (overload) can be calculated efficiently. In the present paper we extend this approach to lower loads. We focus on a sequence of Taylor series expansions of the stationary distribution around increasing service rates. For each series expansion, we use Jacobi iteration to calculate the terms in the series expansion where the initial solution is the approximation found by the preceding series expansion. As the generator matrix of the queueing system at hand is sparse, the numerical complexity of a single Jacobi iteration is O(NMK), where N is the order of the series expansion, K is the number of queues and M is the size of the state space. Having a good initial solution reduces the number of Jacobi iterations considerably, meaning that we can find a sequence of good approximations of the steady state probabilities fast.
Please use this url to cite or link to this publication:
author
organization
year
type
conference (proceedingsPaper)
publication status
published
subject
in
PROCEEDINGS OF THE INTERNATIONAL CONFERENCE ON NUMERICAL ANALYSIS AND APPLIED MATHEMATICS 2016 (ICNAAM-2016)
series title
AIP Conference Proceedings
volume
1863
article number
UNSP 200003-1
pages
4 pages
publisher
Amer Inst Physics
place of publication
Melville
conference name
International Conference on Numerical Analysis and Applied Mathematics (ICNAAM)
conference location
Rhodes, GREECE
conference start
2016-09-19
conference end
2016-09-25
Web of Science type
Proceedings Paper
Web of Science id
000410159800214
ISSN
0094-243X
ISBN
978-0-7354-1538-6
DOI
10.1063/1.4992374
language
English
UGent publication?
yes
classification
P1
id
8538553
handle
http://hdl.handle.net/1854/LU-8538553
date created
2017-11-22 13:46:21
date last changed
2017-12-04 10:02:58
@inproceedings{8538553,
  abstract     = {We focus on the numerical analysis of a coupled queueing system with Poisson arrivals and exponentially distributed service times. Such a system consists of multiple queues served by a single server. Service is synchronised meaning that there is a departure from every queue upon service completion and there is no service whenever one of the queues is empty. It was shown before that the terms in the Maclaurin series expansion of the steady-state distribution of this queueing system when the service rate is sent to 0 (overload) can be calculated efficiently. In the present paper we extend this approach to lower loads. We focus on a sequence of Taylor series expansions of the stationary distribution around increasing service rates. For each series expansion, we use Jacobi iteration to calculate the terms in the series expansion where the initial solution is the approximation found by the preceding series expansion. As the generator matrix of the queueing system at hand is sparse, the numerical complexity of a single Jacobi iteration is O(NMK), where N is the order of the series expansion, K is the number of queues and M is the size of the state space. Having a good initial solution reduces the number of Jacobi iterations considerably, meaning that we can find a sequence of good approximations of the steady state probabilities fast.},
  articleno    = {UNSP 200003-1 },
  author       = {Evdokimova, Ekaterina and Wittevrongel, Sabine and Fiems, Dieter},
  booktitle    = {PROCEEDINGS OF THE INTERNATIONAL CONFERENCE ON NUMERICAL ANALYSIS AND APPLIED MATHEMATICS 2016 (ICNAAM-2016)},
  isbn         = {978-0-7354-1538-6},
  issn         = {0094-243X},
  language     = {eng},
  location     = {Rhodes, GREECE},
  pages        = {4},
  publisher    = {Amer Inst Physics},
  title        = {A Taylor series approach for coupled queueing systems with intermediate load},
  url          = {http://dx.doi.org/10.1063/1.4992374},
  volume       = {1863},
  year         = {2017},
}

Chicago
Evdokimova, Ekaterina, Sabine Wittevrongel, and Dieter Fiems. 2017. “A Taylor Series Approach for Coupled Queueing Systems with Intermediate Load.” In PROCEEDINGS OF THE INTERNATIONAL CONFERENCE ON NUMERICAL ANALYSIS AND APPLIED MATHEMATICS 2016 (ICNAAM-2016). Vol. 1863. Melville: Amer Inst Physics.
APA
Evdokimova, E., Wittevrongel, S., & Fiems, D. (2017). A Taylor series approach for coupled queueing systems with intermediate load. PROCEEDINGS OF THE INTERNATIONAL CONFERENCE ON NUMERICAL ANALYSIS AND APPLIED MATHEMATICS 2016 (ICNAAM-2016) (Vol. 1863). Presented at the International Conference on Numerical Analysis and Applied Mathematics (ICNAAM) , Melville: Amer Inst Physics.
Vancouver
1.
Evdokimova E, Wittevrongel S, Fiems D. A Taylor series approach for coupled queueing systems with intermediate load. PROCEEDINGS OF THE INTERNATIONAL CONFERENCE ON NUMERICAL ANALYSIS AND APPLIED MATHEMATICS 2016 (ICNAAM-2016). Melville: Amer Inst Physics; 2017.
MLA
Evdokimova, Ekaterina, Sabine Wittevrongel, and Dieter Fiems. “A Taylor Series Approach for Coupled Queueing Systems with Intermediate Load.” PROCEEDINGS OF THE INTERNATIONAL CONFERENCE ON NUMERICAL ANALYSIS AND APPLIED MATHEMATICS 2016 (ICNAAM-2016). Vol. 1863. Melville: Amer Inst Physics, 2017. Print.