# American Institute of Mathematical Sciences

October  2016, 12(4): 1435-1464. doi: 10.3934/jimo.2016.12.1435

## 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

Received  July 2014 Revised  May 2015 Published  January 2016

In this paper, we deal with a discrete-time $Geo/G/1$ queueing system under the control of Min($N, V$)-policy in which the server takes single vacation whenever the system becomes empty. The Min($N, V$)-policy means that the server commences its service once the number of waiting customers reaches threshold $N$ or when its vacation time ends with at least one but less than $N$ customers waiting for processing, whichever occurs first. Otherwise, if no customer is presenting at the end of the server vacation, the server remains idle until the first arrival occurs. Under these assumptions, the $z$-transform expressions for the transient queue size distribution at time epoch $n^+$ are obtained by employing the renewal process theory and the total probability decomposition technique. Based on the transient analysis, the explicit recursive formulas of the steady-state queue length distribution at time epochs $n^+$, $n$, $n^-$ and outside observer's time epoch are derived, respectively. Additionally, the stochastic decomposition structure is presented and some other performance measures are also discussed. Furthermore, some computational experiments are implemented to demonstrate the significant application value of the recursive formulas for the steady-state queue size in designing system capacity. Finally, the optimal threshold of $N$ for economizing the system cost is numerically determined.
Citation: Shaojun Lan, Yinghui Tang, Miaomiao Yu. 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. Journal of Industrial and Management Optimization, 2016, 12 (4) : 1435-1464. doi: 10.3934/jimo.2016.12.1435
##### 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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##### 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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