On Optimization of a Coxian Queueing Model with Two Phases

Vedat Sağlam; Merve Uğurlu; Erdinç Yücesoy; Müjgan Zobu; Murat Sağır

doi:doi:10.11648/j.acm.20140302.11

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On Optimization of a Coxian Queueing Model with Two Phases

Vedat Sağlam, Merve Uğurlu, Erdinç Yücesoy, Müjgan Zobu, Murat Sağır

Published in Applied and Computational Mathematics (Volume 3, Issue 2)

Received: 30 January 2014 Published: 20 March 2014

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Abstract

In this study we have obtained stochastic equation systems of a Coxian queueing model with two phases where arrival stream of this model is according to the exponential distribution with λ parameter. The service time of any customer at server i (i=1,2) is exponential with parameter μ_i. In addition we have obtained state probabilities of this queueing model at any given t moment.Furthermore performance measures of this queueing system are calculated. Various queueing systems are found for some values of α probability and service parameters: if α=1and µ_1=µ_2taken then M/E_2/1/ 0 queueing model is obtained, for α=1it is shown that service time of a customer is according to hypoexponential, if α=0 is taken we have M/ M/1/ 0 queueing system. Lately,an application of this queueing model is done. The optimal value of the mean customer number in the system is found. Finally, optimal ordering according to the loss probability is obtained by changing the service parameters .A numerical example is given on the subject

Published in	Applied and Computational Mathematics (Volume 3, Issue 2)
DOI	10.11648/j.acm.20140302.11
Page(s)	43-47
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2014. Published by Science Publishing Group

Keywords

Coxian Model, Differential Equations, Loss Probability, Limiting Distribution, Optimal Ordering

References

[1]	D.R. Cox, "A use of complex probabilities in the theory of stochastic processes. Proc. Camb. Phil. Soc.,"313-19, 1955.
[2]	W. J. Stewart, "Probability, Markov Chains, Queues, Simulation, New Jersey,"2009.
[3]	F.Neuts, "Probability distributions of phase type, in Liber Amicorum Professor Emeritus H.Florin, Department of Mathematics", University of Louvain, Louvain, Belgium 173-206.
[4]	U. N. Bhat, "An Introduction to Queuing Theory," Boston, 2008.
[5]	R.Marie, "Calculating Equilibrim Probabilities for (((λ(n))⁄C_k )⁄1)⁄N Queues," ACM Sigmetrics Performance Evaluation Review, Vol. 9, No.2, 1980.
[6]	M. Zobu, V. Sağlam, M. Sağır, E. Yücesoy and T. Zaman"The Simulation and Minimization of Loss Probability in theTandem Queueing with Two Heterogeneous Channels," Mathematical Problems in Engineering, vol. 2013,Article ID 529010, 4, pages, 2013.
[7]	M.Zobu "Using Sequential Analysis for Hypothesis Tests in The Phase-Type Distribution Queueing Systems," Ph.D. Thesis, OndokuzMayis University, Samsun, Turkey, 2013.
[8]	D. Gross, C.M. Harris, "Fundamentals of Queuing Theory," Third Edition, John Wiley & Sons, New York, 1998.

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Vedat Sağlam, Merve Uğurlu, Erdinç Yücesoy, Müjgan Zobu, Murat Sağır. (2014). On Optimization of a Coxian Queueing Model with Two Phases. Applied and Computational Mathematics, 3(2), 43-47. https://doi.org/10.11648/j.acm.20140302.11

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ACS Style

Vedat Sağlam; Merve Uğurlu; Erdinç Yücesoy; Müjgan Zobu; Murat Sağır. On Optimization of a Coxian Queueing Model with Two Phases. Appl. Comput. Math. 2014, 3(2), 43-47. doi: 10.11648/j.acm.20140302.11

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AMA Style

Vedat Sağlam, Merve Uğurlu, Erdinç Yücesoy, Müjgan Zobu, Murat Sağır. On Optimization of a Coxian Queueing Model with Two Phases. Appl Comput Math. 2014;3(2):43-47. doi: 10.11648/j.acm.20140302.11

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@article{10.11648/j.acm.20140302.11,
  author = {Vedat Sağlam and Merve Uğurlu and Erdinç Yücesoy and Müjgan Zobu and Murat Sağır},
  title = {On Optimization of a Coxian Queueing Model with Two Phases},
  journal = {Applied and Computational Mathematics},
  volume = {3},
  number = {2},
  pages = {43-47},
  doi = {10.11648/j.acm.20140302.11},
  url = {https://doi.org/10.11648/j.acm.20140302.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20140302.11},
  abstract = {In this study we have obtained stochastic equation systems of a Coxian queueing model with two phases where arrival stream of this model is according to the exponential distribution with  λ parameter. The service time of any customer at server i (i=1,2) is exponential with parameter μ_i. In addition we have obtained state probabilities of  this queueing model at any given  t  moment.Furthermore  performance measures of this queueing system are calculated. Various queueing systems are found for some values of  α probability and service parameters: if α=1and µ_1=µ_2taken then M/E_2/1/ 0 queueing model is obtained, for α=1it is shown that service time of a customer is according to hypoexponential, if  α=0 is taken we have M/ M/1/ 0 queueing system. Lately,an application of this queueing model is done. The optimal value of the mean customer number   in the system is found. Finally, optimal ordering according to the  loss probability is  obtained  by changing the service parameters .A numerical example is given on the subject},
 year = {2014}
}

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TY  - JOUR
T1  - On Optimization of a Coxian Queueing Model with Two Phases
AU  - Vedat Sağlam
AU  - Merve Uğurlu
AU  - Erdinç Yücesoy
AU  - Müjgan Zobu
AU  - Murat Sağır
Y1  - 2014/03/20
PY  - 2014
N1  - https://doi.org/10.11648/j.acm.20140302.11
DO  - 10.11648/j.acm.20140302.11
T2  - Applied and Computational Mathematics
JF  - Applied and Computational Mathematics
JO  - Applied and Computational Mathematics
SP  - 43
EP  - 47
PB  - Science Publishing Group
SN  - 2328-5613
UR  - https://doi.org/10.11648/j.acm.20140302.11
AB  - In this study we have obtained stochastic equation systems of a Coxian queueing model with two phases where arrival stream of this model is according to the exponential distribution with  λ parameter. The service time of any customer at server i (i=1,2) is exponential with parameter μ_i. In addition we have obtained state probabilities of  this queueing model at any given  t  moment.Furthermore  performance measures of this queueing system are calculated. Various queueing systems are found for some values of  α probability and service parameters: if α=1and µ_1=µ_2taken then M/E_2/1/ 0 queueing model is obtained, for α=1it is shown that service time of a customer is according to hypoexponential, if  α=0 is taken we have M/ M/1/ 0 queueing system. Lately,an application of this queueing model is done. The optimal value of the mean customer number   in the system is found. Finally, optimal ordering according to the  loss probability is  obtained  by changing the service parameters .A numerical example is given on the subject
VL  - 3
IS  - 2
ER  -

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Author Information

Vedat Sağlam
Merve Uğurlu
Erdinç Yücesoy
Müjgan Zobu

Department of Statistics, Faculty of Science and Arts, Amasya University, Amasya, Turkey
Murat Sağır

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Plain Text BibTeX RIS

APA Style

Vedat Sağlam, Merve Uğurlu, Erdinç Yücesoy, Müjgan Zobu, Murat Sağır. (2014). On Optimization of a Coxian Queueing Model with Two Phases. Applied and Computational Mathematics, 3(2), 43-47. https://doi.org/10.11648/j.acm.20140302.11

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ACS Style

Vedat Sağlam; Merve Uğurlu; Erdinç Yücesoy; Müjgan Zobu; Murat Sağır. On Optimization of a Coxian Queueing Model with Two Phases. Appl. Comput. Math. 2014, 3(2), 43-47. doi: 10.11648/j.acm.20140302.11

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AMA Style

Vedat Sağlam, Merve Uğurlu, Erdinç Yücesoy, Müjgan Zobu, Murat Sağır. On Optimization of a Coxian Queueing Model with Two Phases. Appl Comput Math. 2014;3(2):43-47. doi: 10.11648/j.acm.20140302.11

Copy | Download

@article{10.11648/j.acm.20140302.11,
  author = {Vedat Sağlam and Merve Uğurlu and Erdinç Yücesoy and Müjgan Zobu and Murat Sağır},
  title = {On Optimization of a Coxian Queueing Model with Two Phases},
  journal = {Applied and Computational Mathematics},
  volume = {3},
  number = {2},
  pages = {43-47},
  doi = {10.11648/j.acm.20140302.11},
  url = {https://doi.org/10.11648/j.acm.20140302.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20140302.11},
  abstract = {In this study we have obtained stochastic equation systems of a Coxian queueing model with two phases where arrival stream of this model is according to the exponential distribution with  λ parameter. The service time of any customer at server i (i=1,2) is exponential with parameter μ_i. In addition we have obtained state probabilities of  this queueing model at any given  t  moment.Furthermore  performance measures of this queueing system are calculated. Various queueing systems are found for some values of  α probability and service parameters: if α=1and µ_1=µ_2taken then M/E_2/1/ 0 queueing model is obtained, for α=1it is shown that service time of a customer is according to hypoexponential, if  α=0 is taken we have M/ M/1/ 0 queueing system. Lately,an application of this queueing model is done. The optimal value of the mean customer number   in the system is found. Finally, optimal ordering according to the  loss probability is  obtained  by changing the service parameters .A numerical example is given on the subject},
 year = {2014}
}

Copy | Download

TY  - JOUR
T1  - On Optimization of a Coxian Queueing Model with Two Phases
AU  - Vedat Sağlam
AU  - Merve Uğurlu
AU  - Erdinç Yücesoy
AU  - Müjgan Zobu
AU  - Murat Sağır
Y1  - 2014/03/20
PY  - 2014
N1  - https://doi.org/10.11648/j.acm.20140302.11
DO  - 10.11648/j.acm.20140302.11
T2  - Applied and Computational Mathematics
JF  - Applied and Computational Mathematics
JO  - Applied and Computational Mathematics
SP  - 43
EP  - 47
PB  - Science Publishing Group
SN  - 2328-5613
UR  - https://doi.org/10.11648/j.acm.20140302.11
AB  - In this study we have obtained stochastic equation systems of a Coxian queueing model with two phases where arrival stream of this model is according to the exponential distribution with  λ parameter. The service time of any customer at server i (i=1,2) is exponential with parameter μ_i. In addition we have obtained state probabilities of  this queueing model at any given  t  moment.Furthermore  performance measures of this queueing system are calculated. Various queueing systems are found for some values of  α probability and service parameters: if α=1and µ_1=µ_2taken then M/E_2/1/ 0 queueing model is obtained, for α=1it is shown that service time of a customer is according to hypoexponential, if  α=0 is taken we have M/ M/1/ 0 queueing system. Lately,an application of this queueing model is done. The optimal value of the mean customer number   in the system is found. Finally, optimal ordering according to the  loss probability is  obtained  by changing the service parameters .A numerical example is given on the subject
VL  - 3
IS  - 2
ER  -

Copy | Download