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.
| Published in | Science Journal of Applied Mathematics and Statistics (Volume 3, Issue 2) |
| DOI | 10.11648/j.sjams.20150302.12 |
| 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), 2015. Published by Science Publishing Group |
3-diamensional Markov Chain, Tandem Queuing System, Poisson Current, Phase-Type Distributions, Loss Probabilities
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APA Style
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
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
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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Download@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://article.sciencepublishinggroup.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}
}
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 AU - Müjgan Zobu Y1 - 2015/03/08 PY - 2015 N1 - https://doi.org/10.11648/j.sjams.20150302.12 DO - 10.11648/j.sjams.20150302.12 T2 - Science Journal of Applied Mathematics and Statistics 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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