# 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 & Management Optimization, 2011, 7 (3) : 641-653. doi: 10.3934/jimo.2011.7.641
##### References:
 [1] R. Cooper, "Introduction to Queueing Theory,", 2nd Edition, (1981). Google Scholar [2] D. G. Kendall, Some problems in the theory of queues,, Journal of the Royal Statistical Society, 13 (1951), 151. Google Scholar [3] Y. Levy and U. Yechiali, Utilization of idle time in an $M$/$G$/$1$ queueing system,, Management Science, 22 (1975), 202. doi: 10.1287/mnsc.22.2.202. Google Scholar [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. Google Scholar [5] T. Meisling, Discrete time queueing theory,, Operations Research, 6 (1958), 96. doi: 10.1287/opre.6.1.96. Google Scholar [6] M. Neuts, "Matrix-Geometric Solutions in Stochastic Models. An Algorithmic Approach,", John Hopkins Series in the Mathematical Sciences, 2 (1981). Google Scholar [7] L. Servi and S. Finn, $M$/$M$/$1$ queue with working vacations ($M$/$M$/$1$/$WV$),, Performance Evaluation, 50 (2002), 41. doi: 10.1016/S0166-5316(02)00057-3. Google Scholar [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. doi: 10.3934/jimo.2011.7.19. Google Scholar [9] H. Takagi, "Queueing Analysis: A Foundation of Performance Evaluation,", Vol. \textbf{1}: Vacation and Priority Systems, 1 (1991). Google Scholar [10] H. Takagi, "Queueing Analysis, Vol. 3: Discrete-Time Systems,", Elsevier Science Publishers, (1993). Google Scholar [11] Y. Tang and X. Tang, "Queueing Theory-Foundations And Analysis Technique,", Science Press, (2006). Google Scholar [12] N. Tian, X. Xu and Z. Ma, "Discrete Time Queueing Theory,", Science Press, (2008). Google Scholar [13] N. Tian, D. Zhang and C. Cao, The $GI$/$M$/$1$ queue with exponential vacations,, Queueing Systems Theory Applications, 5 (1989), 331. doi: 10.1007/BF01225323. Google Scholar [14] D. Wu and H. Takagi, $M$/$G$/$1$ queue with multiple working vacations,, Performance Evaluation, 63 (2006), 654. doi: 10.1016/j.peva.2005.05.005. Google Scholar [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. doi: 10.3934/jimo.2010.6.501. Google Scholar

show all references

##### References:
 [1] R. Cooper, "Introduction to Queueing Theory,", 2nd Edition, (1981). Google Scholar [2] D. G. Kendall, Some problems in the theory of queues,, Journal of the Royal Statistical Society, 13 (1951), 151. Google Scholar [3] Y. Levy and U. Yechiali, Utilization of idle time in an $M$/$G$/$1$ queueing system,, Management Science, 22 (1975), 202. doi: 10.1287/mnsc.22.2.202. Google Scholar [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. Google Scholar [5] T. Meisling, Discrete time queueing theory,, Operations Research, 6 (1958), 96. doi: 10.1287/opre.6.1.96. Google Scholar [6] M. Neuts, "Matrix-Geometric Solutions in Stochastic Models. An Algorithmic Approach,", John Hopkins Series in the Mathematical Sciences, 2 (1981). Google Scholar [7] L. Servi and S. Finn, $M$/$M$/$1$ queue with working vacations ($M$/$M$/$1$/$WV$),, Performance Evaluation, 50 (2002), 41. doi: 10.1016/S0166-5316(02)00057-3. Google Scholar [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. doi: 10.3934/jimo.2011.7.19. Google Scholar [9] H. Takagi, "Queueing Analysis: A Foundation of Performance Evaluation,", Vol. \textbf{1}: Vacation and Priority Systems, 1 (1991). Google Scholar [10] H. Takagi, "Queueing Analysis, Vol. 3: Discrete-Time Systems,", Elsevier Science Publishers, (1993). Google Scholar [11] Y. Tang and X. Tang, "Queueing Theory-Foundations And Analysis Technique,", Science Press, (2006). Google Scholar [12] N. Tian, X. Xu and Z. Ma, "Discrete Time Queueing Theory,", Science Press, (2008). Google Scholar [13] N. Tian, D. Zhang and C. Cao, The $GI$/$M$/$1$ queue with exponential vacations,, Queueing Systems Theory Applications, 5 (1989), 331. doi: 10.1007/BF01225323. Google Scholar [14] D. Wu and H. Takagi, $M$/$G$/$1$ queue with multiple working vacations,, Performance Evaluation, 63 (2006), 654. doi: 10.1016/j.peva.2005.05.005. Google Scholar [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. doi: 10.3934/jimo.2010.6.501. Google Scholar
 [1] Sung-Seok Ko. A nonhomogeneous quasi-birth-death process approach for an $(s, S)$ policy for a perishable inventory system with retrial demands. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-19. doi: 10.3934/jimo.2019009 [2] Jacek Banasiak, Marcin Moszyński. Dynamics of birth-and-death processes with proliferation - stability and chaos. Discrete & Continuous Dynamical Systems - A, 2011, 29 (1) : 67-79. doi: 10.3934/dcds.2011.29.67 [3] Genni Fragnelli, A. Idrissi, L. Maniar. The asymptotic behavior of a population equation with diffusion and delayed birth process. Discrete & Continuous Dynamical Systems - B, 2007, 7 (4) : 735-754. doi: 10.3934/dcdsb.2007.7.735 [4] Jacques Henry. For which objective is birth process an optimal feedback in age structured population dynamics?. Discrete & Continuous Dynamical Systems - B, 2007, 8 (1) : 107-114. doi: 10.3934/dcdsb.2007.8.107 [5] Xianlong Fu, Dongmei Zhu. Stability results for a size-structured population model with delayed birth process. Discrete & Continuous Dynamical Systems - B, 2013, 18 (1) : 109-131. doi: 10.3934/dcdsb.2013.18.109 [6] Hilla Behar, Alexandra Agranovich, Yoram Louzoun. Diffusion rate determines balance between extinction and proliferation in birth-death processes. Mathematical Biosciences & Engineering, 2013, 10 (3) : 523-550. doi: 10.3934/mbe.2013.10.523 [7] Jacek Banasiak, Mustapha Mokhtar-Kharroubi. Universality of dishonesty of substochastic semigroups: Shattering fragmentation and explosive birth-and-death processes. Discrete & Continuous Dynamical Systems - B, 2005, 5 (3) : 529-542. doi: 10.3934/dcdsb.2005.5.529 [8] Valery Y. Glizer, Vladimir Turetsky, Emil Bashkansky. Statistical process control optimization with variable sampling interval and nonlinear expected loss. Journal of Industrial & Management Optimization, 2015, 11 (1) : 105-133. doi: 10.3934/jimo.2015.11.105 [9] Doria Affane, Meriem Aissous, Mustapha Fateh Yarou. Almost mixed semi-continuous perturbation of Moreau's sweeping process. Evolution Equations & Control Theory, 2020, 9 (1) : 27-38. doi: 10.3934/eect.2020015 [10] Dongxue Yan, Xianlong Fu. Asymptotic analysis of a spatially and size-structured population model with delayed birth process. Communications on Pure & Applied Analysis, 2016, 15 (2) : 637-655. doi: 10.3934/cpaa.2016.15.637 [11] Dongxue Yan, Yu Cao, Xianlong Fu. Asymptotic analysis of a size-structured cannibalism population model with delayed birth process. Discrete & Continuous Dynamical Systems - B, 2016, 21 (6) : 1975-1998. doi: 10.3934/dcdsb.2016032 [12] Oscar Patterson-Lomba, Muntaser Safan, Sherry Towers, Jay Taylor. Modeling the role of healthcare access inequalities in epidemic outcomes. Mathematical Biosciences & Engineering, 2016, 13 (5) : 1011-1041. doi: 10.3934/mbe.2016028 [13] Motahhareh Gharahi, Massoud Hadian Dehkordi. Average complexities of access structures on five participants. Advances in Mathematics of Communications, 2013, 7 (3) : 311-317. doi: 10.3934/amc.2013.7.311 [14] Pikkala Vijaya Laxmi, Obsie Mussa Yesuf. Analysis of a finite buffer general input queue with Markovian service process and accessible and non-accessible batch service. Journal of Industrial & Management Optimization, 2010, 6 (4) : 929-944. doi: 10.3934/jimo.2010.6.929 [15] Rongfei Liu, Dingcheng Wang, Jiangyan Peng. Infinite-time ruin probability of a renewal risk model with exponential Levy process investment and dependent claims and inter-arrival times. Journal of Industrial & Management Optimization, 2017, 13 (2) : 995-1007. doi: 10.3934/jimo.2016058 [16] Shunfu Jin, Wuyi Yue, Shiying Ge. Equilibrium analysis of an opportunistic spectrum access mechanism with imperfect sensing results. Journal of Industrial & Management Optimization, 2017, 13 (3) : 1255-1271. doi: 10.3934/jimo.2016071 [17] Motahhareh Gharahi, Shahram Khazaei. Reduced access structures with four minimal qualified subsets on six participants. Advances in Mathematics of Communications, 2018, 12 (1) : 199-214. doi: 10.3934/amc.2018014 [18] Jinsong Xu. Reversible hidden data access algorithm in cloud computing environment. Discrete & Continuous Dynamical Systems - S, 2019, 12 (4&5) : 1219-1232. doi: 10.3934/dcdss.2019084 [19] Ummugul Bulut, Edward J. Allen. Derivation of SDES for a macroevolutionary process. Discrete & Continuous Dynamical Systems - B, 2013, 18 (7) : 1777-1792. doi: 10.3934/dcdsb.2013.18.1777 [20] Dariush Mohamadi Zanjirani, Majid Esmaelian. An integrated approach based on Fuzzy Inference System for scheduling and process planning through multiple objectives. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-25. doi: 10.3934/jimo.2018202

2018 Impact Factor: 1.025