# American Institute of Mathematical Sciences

July  2011, 7(3): 641-653. doi: 10.3934/jimo.2011.7.641

## Performance analysis of a Geom/Geom/1 queueing system with variable input probability

 1 Department of Applied Mathematics, College of Science, Yanshan University, Qinhuangdao 066004, China 2 Department of Intelligence and Informatics, Konan University, Kobe 658-8501 3 Department of Applied Mathematics, College of Science Yanshan University, Qinhuangdao 066004, China

Received  September 2010 Revised  May 2011 Published  June 2011

In this paper, we present a Geom/Geom/1 queueing model with variable input probability. In this queueing model, an arriving customer, who sees many customers waiting for service in the queueing system, will consider whether to enter the system or not. We consider the possibility that the customer enters the system to receive service to be a probability called the "Input Probability". We derive the transition probability matrix of the birth and death chain of the queueing model. Using a birth and death process, we gain the probability distributions of the stationary queue length and the waiting time in the queueing model. Then we derive special cases of the considered model by applying different input probability distributions, which lead to several known specific queueing models. We also derive some performance measures of these specific queueing models. Therefore, this queueing model that we present in this paper has a certain extension, extending to the existing models. Finally, we compare the effect of the parameters on the stationary queue length and waiting time by using numerical results.
Citation: Zhanyou Ma, Wuyi Yue, Xiaoli Su. Performance analysis of a Geom/Geom/1 queueing system with variable input probability. Journal of Industrial and Management Optimization, 2011, 7 (3) : 641-653. doi: 10.3934/jimo.2011.7.641
##### References:
 [1] R. Cooper, "Introduction to Queueing Theory," 2nd Edition, North Holland Publishing Co., New York-Amsterdam, 1981. [2] D. G. Kendall, Some problems in the theory of queues, Journal of the Royal Statistical Society, Series B, 13 (1951), 151-173. [3] Y. Levy and U. Yechiali, Utilization of idle time in an $M$/$G$/$1$ queueing system, Management Science, 22 (1975), 202-211. doi: 10.1287/mnsc.22.2.202. [4] Z. Ma and N. Tian, Pure limited service $Geom$/$G$/$1$ queue with multiple adaptive vacations, Journal of Computational Information Systems, 1 (2005), 515-521. [5] T. Meisling, Discrete time queueing theory, Operations Research, 6 (1958), 96-105. doi: 10.1287/opre.6.1.96. [6] M. Neuts, "Matrix-Geometric Solutions in Stochastic Models. An Algorithmic Approach," John Hopkins Series in the Mathematical Sciences, 2, Johns Hopkins University Press, Baltimore, 1981. [7] L. Servi and S. Finn, $M$/$M$/$1$ queue with working vacations ($M$/$M$/$1$/$WV$), Performance Evaluation, 50 (2002), 41-52. doi: 10.1016/S0166-5316(02)00057-3. [8] L. Tadj, G. Zhang and C. Tadj, A queueing analysis of multi-purpose production facility's operations, Journal of Industrial and Management Optimization, 7 (2011), 19-30. doi: 10.3934/jimo.2011.7.19. [9] H. Takagi, "Queueing Analysis: A Foundation of Performance Evaluation," Vol. 1: Vacation and Priority Systems, Part 1, North-Holland Publishing Co., Amsterdam, 1991. [10] H. Takagi, "Queueing Analysis, Vol. 3: Discrete-Time Systems," Elsevier Science Publishers, Amsterdam, 1993. [11] Y. Tang and X. Tang, "Queueing Theory-Foundations And Analysis Technique," Science Press, Beijing, 2006 (in Chinese). [12] N. Tian, X. Xu and Z. Ma, "Discrete Time Queueing Theory," Science Press, Beijing, 2008 (in Chinese). [13] N. Tian, D. Zhang and C. Cao, The $GI$/$M$/$1$ queue with exponential vacations, Queueing Systems Theory Applications, 5 (1989), 331-344. doi: 10.1007/BF01225323. [14] D. Wu and H. Takagi, $M$/$G$/$1$ queue with multiple working vacations, Performance Evaluation, 63 (2006), 654-681. doi: 10.1016/j.peva.2005.05.005. [15] D. Yue and W. Yue, A heterogeneous two-server network system with balking and a Bernoulli vacation schedule, Journal of Industrial and Management Optimization, 6 (2010), 501-516. doi: 10.3934/jimo.2010.6.501.

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##### References:
 [1] R. Cooper, "Introduction to Queueing Theory," 2nd Edition, North Holland Publishing Co., New York-Amsterdam, 1981. [2] D. G. Kendall, Some problems in the theory of queues, Journal of the Royal Statistical Society, Series B, 13 (1951), 151-173. [3] Y. Levy and U. Yechiali, Utilization of idle time in an $M$/$G$/$1$ queueing system, Management Science, 22 (1975), 202-211. doi: 10.1287/mnsc.22.2.202. [4] Z. Ma and N. Tian, Pure limited service $Geom$/$G$/$1$ queue with multiple adaptive vacations, Journal of Computational Information Systems, 1 (2005), 515-521. [5] T. Meisling, Discrete time queueing theory, Operations Research, 6 (1958), 96-105. doi: 10.1287/opre.6.1.96. [6] M. Neuts, "Matrix-Geometric Solutions in Stochastic Models. An Algorithmic Approach," John Hopkins Series in the Mathematical Sciences, 2, Johns Hopkins University Press, Baltimore, 1981. [7] L. Servi and S. Finn, $M$/$M$/$1$ queue with working vacations ($M$/$M$/$1$/$WV$), Performance Evaluation, 50 (2002), 41-52. doi: 10.1016/S0166-5316(02)00057-3. [8] L. Tadj, G. Zhang and C. Tadj, A queueing analysis of multi-purpose production facility's operations, Journal of Industrial and Management Optimization, 7 (2011), 19-30. doi: 10.3934/jimo.2011.7.19. [9] H. Takagi, "Queueing Analysis: A Foundation of Performance Evaluation," Vol. 1: Vacation and Priority Systems, Part 1, North-Holland Publishing Co., Amsterdam, 1991. [10] H. Takagi, "Queueing Analysis, Vol. 3: Discrete-Time Systems," Elsevier Science Publishers, Amsterdam, 1993. [11] Y. Tang and X. Tang, "Queueing Theory-Foundations And Analysis Technique," Science Press, Beijing, 2006 (in Chinese). [12] N. Tian, X. Xu and Z. Ma, "Discrete Time Queueing Theory," Science Press, Beijing, 2008 (in Chinese). [13] N. Tian, D. Zhang and C. Cao, The $GI$/$M$/$1$ queue with exponential vacations, Queueing Systems Theory Applications, 5 (1989), 331-344. doi: 10.1007/BF01225323. [14] D. Wu and H. Takagi, $M$/$G$/$1$ queue with multiple working vacations, Performance Evaluation, 63 (2006), 654-681. doi: 10.1016/j.peva.2005.05.005. [15] D. Yue and W. Yue, A heterogeneous two-server network system with balking and a Bernoulli vacation schedule, Journal of Industrial and Management Optimization, 6 (2010), 501-516. doi: 10.3934/jimo.2010.6.501.
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