Science Journal of Applied Mathematics and Statistics

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A Study on a Tandem Stochastic Queueing Model with Parallel Phases and a Numerical Example

Received: 03 February 2015    Accepted: 19 February 2015    Published: 08 March 2015
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Abstract

In this study a two stage queueing model is analyzed. At first stage there is a single server having exponential service time with parameter μ_1 and no waiting is allowed in front of this server. There are two parallel phase-type servers at second stage and these parallel servers have exponential service time with parameter μ_2. Arrivals to this system is Poisson with parameter λ. An arriving customer to this system has service if the server at first stage is available or leaves the system if the server is busy where the first loss occurs. After having service in first stage the customer proceeds to the second stage, if both of the phase-type parallel servers in second stage are available the customer chooses one of these servers with probability 0.50 or leaves the system if any of these servers in second stage is busy so the second loss occurs. A customer who has service at both stages leaves the system. The number of customers in this model is represented by a 3-diamensional Markov chain and Kolmogorov differential equations are obtained. After that mean number of customers and mean waiting time in the system is obtained by limit probabilities. We have shown that the customer numbers at first and second stages are dependent to each other. The numerical analysis of obtained performance measures are shown by a numeric example. Finally the graphs of loss probabilities and measure of performances given for some values of arrival rate λ and the service parameters.

DOI 10.11648/j.sjams.20150302.12
Published in Science Journal of Applied Mathematics and Statistics (Volume 3, Issue 2, April 2015)
Page(s) 33-38
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), 2024. Published by Science Publishing Group

Keywords

3-diamensional Markov Chain, Tandem Queuing System, Poisson Current, Phase-Type Distributions, Loss Probabilities

References
[1] Jackson, R. R. P., “Queueing systems with Phase-Typeservice”, Operat. Res. Quart., 5, 109-120, 1954.
[2] Tembe, S. V., and Wolff, R. W., “The Optimal Order of Service in Tandem Queues”, Operations Research, 22, 824-832, 1974.
[3] R. K. Rana, “Queueing problems with arrivals in general stream and phase type service” Metrika, vol. 18, no. 1, pp. 69–80, 1972.
[4] V. Ramaswami and M. F. Neuts, “A duality theorem for phase type queues”, The Annals of Probability, Vol.8, No.5, 974-985, 1980.
[5] V. Ramaswami, “Algorithms for the Multi-Server Queue”, Commun. Statist.- Stochastic Models, 1(3), 393-417, 1985.
[6] D. D. Selvam and V. Sivasankaran, “A two-phase queueing system with server vacations”, Operations research letters, Vol.15, no.3, pp.163-169, 1994.
[7] J. R. Artalejo and G. Choudhury, “Steady State Analysis of an M/G/1 Queue with Repeated Attempts and Two-Phase Service”, Quality Technology & Quantative Management, Vol.1, No.2, pp. 189-199, 2004.
[8] B. V. Houdt and A. S. Alfa, “Response time in a tandem queue with blocking, Markovian arrivals and phase-type services”, Operations Researc Letters 33, pp.373-381, 2005.
[9] Gross, D., Harris, C. M., Thompson, M. J., Shortle, F. J., Fundementals of Queueing Theory, 4th ed., John Wiley & Sons, New York, 2008.
[10] Stewart, W.J., Probability, Markov Chains, Queues and Simulation, Princeton University Press, United Kingdom, 2009.
[11] M. Zobu, V. Sağlam, M. Sağır, E. Yücesoy, and T. Zaman, “The Simulation and Minimization of Loss Probability in the Tandem Queueing with Two Heterogeneous Channel” Mathematical Problems in Engineering, vol. 2013, Article ID 529010, 4 pages, 2013. doi:10.1155/2013/529010.
[12] M. Zobu and V.Sağlam, “Control of Traffic Intensity in Hyperexponential and Mixed Erlang Queueing Systems with a Method Based on SPRT” Mathematical Problems in Engineering, vol. 2013, Article ID 241241, 9 pages, 2013. doi:10.1155/2013/241241.
[13] V.Sağlam and M. Zobu, “A Two-Stage Model Queueing with No Waiting Line between Channels”, Mathematical Problems in Engineering, Volume 2013, Article ID 679369, 5 pages.
[14] V. Sağlam, M. Uğurlu, E.Yücesoy, M. Zobu, M. Sağır, “On Optimization of a Coxian Queueing Model with Two Phases”, Applied and Computational Mathematics. Vol. 3, No. 2, 2014, pp. 43-47. doi: 10.11648/j.acm.20140302.11.
Author Information
  • Department of Statistics, Faculty of Science and Arts, Amasya University, Amasya, Turkey

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    Vedat Sağlam, Erdinç Yücesoy, Murat Sağır, Müjgan Zobu. (2015). A Study on a Tandem Stochastic Queueing Model with Parallel Phases and a Numerical Example. Science Journal of Applied Mathematics and Statistics, 3(2), 33-38. https://doi.org/10.11648/j.sjams.20150302.12

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    ACS Style

    Vedat Sağlam; Erdinç Yücesoy; Murat Sağır; Müjgan Zobu. A Study on a Tandem Stochastic Queueing Model with Parallel Phases and a Numerical Example. Sci. J. Appl. Math. Stat. 2015, 3(2), 33-38. doi: 10.11648/j.sjams.20150302.12

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    AMA Style

    Vedat Sağlam, Erdinç Yücesoy, Murat Sağır, Müjgan Zobu. A Study on a Tandem Stochastic Queueing Model with Parallel Phases and a Numerical Example. Sci J Appl Math Stat. 2015;3(2):33-38. doi: 10.11648/j.sjams.20150302.12

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  • @article{10.11648/j.sjams.20150302.12,
      author = {Vedat Sağlam and Erdinç Yücesoy and Murat Sağır and Müjgan Zobu},
      title = {A Study on a Tandem Stochastic Queueing Model with Parallel Phases and a Numerical Example},
      journal = {Science Journal of Applied Mathematics and Statistics},
      volume = {3},
      number = {2},
      pages = {33-38},
      doi = {10.11648/j.sjams.20150302.12},
      url = {https://doi.org/10.11648/j.sjams.20150302.12},
      eprint = {https://download.sciencepg.com/pdf/10.11648.j.sjams.20150302.12},
      abstract = {In this study a two stage queueing model is analyzed. At first stage there is a single server having exponential service time with parameter μ_1 and no waiting is allowed in front of this server. There are two parallel phase-type servers at second stage and these parallel servers have exponential service time with parameter μ_2. Arrivals to this system is Poisson with parameter λ. An arriving customer to this system has service if the server at first stage is available  or leaves the system if the server is busy where the first loss occurs. After having service in first stage the customer proceeds to the second stage, if both of the phase-type  parallel servers in second stage are available the customer chooses one of these servers with probability 0.50 or leaves the system if any of these servers in second stage is busy so the second loss occurs. A customer who has service at both stages leaves the system. The number of customers in this model is represented by a 3-diamensional Markov chain and Kolmogorov differential equations are obtained. After that mean number of customers and mean waiting time in the system is obtained by limit probabilities. We have shown that the customer numbers at first and second stages are dependent to each other. The numerical analysis of obtained performance measures are shown by a numeric example. Finally the graphs of loss probabilities and measure of performances given for some values of arrival rate λ and the service parameters.},
     year = {2015}
    }
    

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  • TY  - JOUR
    T1  - A Study on a Tandem Stochastic Queueing Model with Parallel Phases and a Numerical Example
    AU  - Vedat Sağlam
    AU  - Erdinç Yücesoy
    AU  - Murat Sağır
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    DO  - 10.11648/j.sjams.20150302.12
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    JF  - Science Journal of Applied Mathematics and Statistics
    JO  - Science Journal of Applied Mathematics and Statistics
    SP  - 33
    EP  - 38
    PB  - Science Publishing Group
    SN  - 2376-9513
    UR  - https://doi.org/10.11648/j.sjams.20150302.12
    AB  - In this study a two stage queueing model is analyzed. At first stage there is a single server having exponential service time with parameter μ_1 and no waiting is allowed in front of this server. There are two parallel phase-type servers at second stage and these parallel servers have exponential service time with parameter μ_2. Arrivals to this system is Poisson with parameter λ. An arriving customer to this system has service if the server at first stage is available  or leaves the system if the server is busy where the first loss occurs. After having service in first stage the customer proceeds to the second stage, if both of the phase-type  parallel servers in second stage are available the customer chooses one of these servers with probability 0.50 or leaves the system if any of these servers in second stage is busy so the second loss occurs. A customer who has service at both stages leaves the system. The number of customers in this model is represented by a 3-diamensional Markov chain and Kolmogorov differential equations are obtained. After that mean number of customers and mean waiting time in the system is obtained by limit probabilities. We have shown that the customer numbers at first and second stages are dependent to each other. The numerical analysis of obtained performance measures are shown by a numeric example. Finally the graphs of loss probabilities and measure of performances given for some values of arrival rate λ and the service parameters.
    VL  - 3
    IS  - 2
    ER  - 

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