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System capacity optimization design and optimal threshold $N^{*}$ for a $GEO/G/1$ discrete-time queue with single server vacation and under the control of Min($N, V$)-policy
1. | School of Mathematics and Software Science, Sichuan Normal University, Chengdu 610066, China, China |
2. | School of Science, Sichuan University of Science and Engineering, Zigong 643000, China |
References:
[1] |
A. S. Alfa and I. Frigui, Discrete NT-policy single server queue with Markovian arrival process and phase type service, European Journal of Operational Research, 88 (1996), 599-613.
doi: 10.1016/0377-2217(94)00206-1. |
[2] |
H. Bruneel and B. G. Kim, Discrete-Time Models for Communication Systems Including ATM, The Springer International Series In Engineering And Computer Science, 205, 1993.
doi: 10.1007/978-1-4615-3130-2. |
[3] |
B. Feyaerts, S. D. Vuyst, S. Wittevrongel and H. Bruneel, Analysis of a discrete-time queueing system with an NT-policy, in Analytical and Stochastic Modeling Techniques and Applications (eds. K. L. Begain, D. Fiems and W. J. Knottenbelt), Springer, 6148 (2010), 29-43.
doi: 10.1007/978-3-642-13568-2_3. |
[4] |
B. Feyaerts, S. D. Vuyst, H. Bruneel and S. Wittevrongel, The impact of the NT-policy on the behaviour of a discrete-time queue with general service times, Journal of Industrial and Management Optimization, 10 (2014), 131-149. |
[5] |
S. Gao and J. Wang, On a discrete-time $GI^X$/Geo/1/N-G queue with randomized working vacations and at most J vacations, Journal of Industrial and Management Optimization, 11 (2015), 779-806.
doi: 10.3934/jimo.2015.11.779. |
[6] |
J. J. Hunter, Mathematical Techniques of Applied Probability, Vol. 2, Discrete Time Models: Techniques and Applications, Academic Press, New York, 1983. |
[7] |
S. Hur, J. Kim and C. Kang, An analysis of the M/G/1 system with $N$ and $T$ policy, Applied Mathematical Modelling, 27 (2003), 665-675.
doi: 10.1016/S0307-904X(03)00074-X. |
[8] |
J. C. Ke, Bi-level control for batch arrival queues with an early startup and unreliable server, Applied Mathematical Modelling, 28 (2004), 469-485. |
[9] |
H. W. Lee, S. S. Lee, J. O. Park and K. C. Chae, Analysis of $M^X$ /G/1 queue with N-policy and multiple vacations, Journal of Applied Probability, 31 (1994), 476-496.
doi: 10.2307/3215040. |
[10] |
H. W. Lee and W. J. Seo, The performance of the M/G/1 queue under the dyadic Min(N, D)-policy and its cost optimization, Performance Evaluation, 65 (2008), 742-758. |
[11] |
C. Luo, Y. Tang, W. Li and K. Xiang, The recursive solution of queue length for Geo/G/1 queue with N-policy, Journal of Systems Science & Complexity, 25 (2012), 293-302.
doi: 10.1007/s11424-012-9313-3. |
[12] |
T. Meisling, Discrete time queue theory, Operations Research, 6 (1958), 96-105.
doi: 10.1287/opre.6.1.96. |
[13] |
S. K. Samanta, M. L. Chaudhry and U. C. Gupta, Discrete-time $Geo^X$/$G^{(a, b)}$/1/N queues with single and multiple vacations, Mathematical and Computer Modelling, 45 (2007), 93-108.
doi: 10.1016/j.mcm.2006.04.008. |
[14] |
H. Takagi, Queueing Analysis, A Foundation of Performance Evaluation, Vol. 3: Discrete-Time Systems, North-Holland, Amsterdam, The Netherlands, 1993. |
[15] |
Y. Tang, W. Wu, Y. Liu and X. Liu, The queue length distribution of M/G/1 queueing system with Min(N, V)-policy based on multiple server vacations, Systems Engineering-Theory & Practice, 34 (2014), 1525-1532. |
[16] |
N. Tian, X. Xu and Z. Ma, Dicrete-Time Queueing Theory, Science Press, Beijing, 2008. |
[17] |
N. Tian and Z. G. Zhang, Vacation Queueing Models: Theory and Applications, International Series in Operations Research & Management Science, 93, Springer, New York, 2006. |
[18] |
N. Tian and Z. G. Zhang, Discrete time Geo/G/1 queue with multiple adaptive vacations, Queueing System, 38 (2001), 419-429.
doi: 10.1023/A:1010947911863. |
[19] |
T. Y. Wang, J. C. Ke and F. M. Chang, On the discrete-time Geo/G/1 queue with randomized vacations and at most J vacations, Applied Mathematical Modelling, 35 (2011), 2297-2308.
doi: 10.1016/j.apm.2010.11.021. |
[20] |
Y. Wei, M. Yu, Y. Tang and J. Gu, Queue size distribution and capacity optimum design for N-policy $Geo^{(\lambda _1 , \lambda _2 , \lambda _3)}$/G/1 queue with setup time and variable input rate, Mathematical and Computer Modelling, 57 (2013), 1559-1571.
doi: 10.1016/j.mcm.2012.12.032. |
[21] |
M. E. Woodward, Communication and Computer Networks: Modelling with Discrete-Time Queues, IEEE Computer Society Press, Los Alamitos, California, 1994. |
[22] |
M. Yu, Y. Tang, Y. Fu and L. Pan, GI/Geom/1/N/MWV queue with changeover time and searching for the optimum service rate in working vacation period, Journal of Computational and Applied Mathematics, 235 (2011), 2170-2184.
doi: 10.1016/j.cam.2010.10.013. |
[23] |
D. Yue and F. Zhang, A discrete-time Geo/G/1 retrial queue with J-vacation policy and general retrial times, Journal of Systems Science & Complexity, 26 (2013), 556-571.
doi: 10.1007/s11424-013-1121-x. |
[24] |
D. Yue, W. Yue and G. Zhao, Analysis of an M/M/1 queue with vacations and impatience timers which depend on the server's states, Journal of Industrial and Management Optimization, 12 (2016), 653-666.
doi: 10.3934/jimo.2016.12.653. |
show all references
References:
[1] |
A. S. Alfa and I. Frigui, Discrete NT-policy single server queue with Markovian arrival process and phase type service, European Journal of Operational Research, 88 (1996), 599-613.
doi: 10.1016/0377-2217(94)00206-1. |
[2] |
H. Bruneel and B. G. Kim, Discrete-Time Models for Communication Systems Including ATM, The Springer International Series In Engineering And Computer Science, 205, 1993.
doi: 10.1007/978-1-4615-3130-2. |
[3] |
B. Feyaerts, S. D. Vuyst, S. Wittevrongel and H. Bruneel, Analysis of a discrete-time queueing system with an NT-policy, in Analytical and Stochastic Modeling Techniques and Applications (eds. K. L. Begain, D. Fiems and W. J. Knottenbelt), Springer, 6148 (2010), 29-43.
doi: 10.1007/978-3-642-13568-2_3. |
[4] |
B. Feyaerts, S. D. Vuyst, H. Bruneel and S. Wittevrongel, The impact of the NT-policy on the behaviour of a discrete-time queue with general service times, Journal of Industrial and Management Optimization, 10 (2014), 131-149. |
[5] |
S. Gao and J. Wang, On a discrete-time $GI^X$/Geo/1/N-G queue with randomized working vacations and at most J vacations, Journal of Industrial and Management Optimization, 11 (2015), 779-806.
doi: 10.3934/jimo.2015.11.779. |
[6] |
J. J. Hunter, Mathematical Techniques of Applied Probability, Vol. 2, Discrete Time Models: Techniques and Applications, Academic Press, New York, 1983. |
[7] |
S. Hur, J. Kim and C. Kang, An analysis of the M/G/1 system with $N$ and $T$ policy, Applied Mathematical Modelling, 27 (2003), 665-675.
doi: 10.1016/S0307-904X(03)00074-X. |
[8] |
J. C. Ke, Bi-level control for batch arrival queues with an early startup and unreliable server, Applied Mathematical Modelling, 28 (2004), 469-485. |
[9] |
H. W. Lee, S. S. Lee, J. O. Park and K. C. Chae, Analysis of $M^X$ /G/1 queue with N-policy and multiple vacations, Journal of Applied Probability, 31 (1994), 476-496.
doi: 10.2307/3215040. |
[10] |
H. W. Lee and W. J. Seo, The performance of the M/G/1 queue under the dyadic Min(N, D)-policy and its cost optimization, Performance Evaluation, 65 (2008), 742-758. |
[11] |
C. Luo, Y. Tang, W. Li and K. Xiang, The recursive solution of queue length for Geo/G/1 queue with N-policy, Journal of Systems Science & Complexity, 25 (2012), 293-302.
doi: 10.1007/s11424-012-9313-3. |
[12] |
T. Meisling, Discrete time queue theory, Operations Research, 6 (1958), 96-105.
doi: 10.1287/opre.6.1.96. |
[13] |
S. K. Samanta, M. L. Chaudhry and U. C. Gupta, Discrete-time $Geo^X$/$G^{(a, b)}$/1/N queues with single and multiple vacations, Mathematical and Computer Modelling, 45 (2007), 93-108.
doi: 10.1016/j.mcm.2006.04.008. |
[14] |
H. Takagi, Queueing Analysis, A Foundation of Performance Evaluation, Vol. 3: Discrete-Time Systems, North-Holland, Amsterdam, The Netherlands, 1993. |
[15] |
Y. Tang, W. Wu, Y. Liu and X. Liu, The queue length distribution of M/G/1 queueing system with Min(N, V)-policy based on multiple server vacations, Systems Engineering-Theory & Practice, 34 (2014), 1525-1532. |
[16] |
N. Tian, X. Xu and Z. Ma, Dicrete-Time Queueing Theory, Science Press, Beijing, 2008. |
[17] |
N. Tian and Z. G. Zhang, Vacation Queueing Models: Theory and Applications, International Series in Operations Research & Management Science, 93, Springer, New York, 2006. |
[18] |
N. Tian and Z. G. Zhang, Discrete time Geo/G/1 queue with multiple adaptive vacations, Queueing System, 38 (2001), 419-429.
doi: 10.1023/A:1010947911863. |
[19] |
T. Y. Wang, J. C. Ke and F. M. Chang, On the discrete-time Geo/G/1 queue with randomized vacations and at most J vacations, Applied Mathematical Modelling, 35 (2011), 2297-2308.
doi: 10.1016/j.apm.2010.11.021. |
[20] |
Y. Wei, M. Yu, Y. Tang and J. Gu, Queue size distribution and capacity optimum design for N-policy $Geo^{(\lambda _1 , \lambda _2 , \lambda _3)}$/G/1 queue with setup time and variable input rate, Mathematical and Computer Modelling, 57 (2013), 1559-1571.
doi: 10.1016/j.mcm.2012.12.032. |
[21] |
M. E. Woodward, Communication and Computer Networks: Modelling with Discrete-Time Queues, IEEE Computer Society Press, Los Alamitos, California, 1994. |
[22] |
M. Yu, Y. Tang, Y. Fu and L. Pan, GI/Geom/1/N/MWV queue with changeover time and searching for the optimum service rate in working vacation period, Journal of Computational and Applied Mathematics, 235 (2011), 2170-2184.
doi: 10.1016/j.cam.2010.10.013. |
[23] |
D. Yue and F. Zhang, A discrete-time Geo/G/1 retrial queue with J-vacation policy and general retrial times, Journal of Systems Science & Complexity, 26 (2013), 556-571.
doi: 10.1007/s11424-013-1121-x. |
[24] |
D. Yue, W. Yue and G. Zhao, Analysis of an M/M/1 queue with vacations and impatience timers which depend on the server's states, Journal of Industrial and Management Optimization, 12 (2016), 653-666.
doi: 10.3934/jimo.2016.12.653. |
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