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On Optimization of a Coxian Queueing Model with Two Phases

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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  • 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

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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
  • Department of Statistics, Faculty of Science and Arts, Amasya University, Amasya, Turkey

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