# 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 & 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.  doi: 10.1016/0377-2217(94)00206-1.  Google Scholar [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.  Google Scholar [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, 6148 (2010), 29.  doi: 10.1007/978-3-642-13568-2_3.  Google Scholar [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.   Google Scholar [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.  doi: 10.3934/jimo.2015.11.779.  Google Scholar [6] J. J. Hunter, Mathematical Techniques of Applied Probability, Vol. 2, Discrete Time Models: Techniques and Applications,, Academic Press, (1983).   Google Scholar [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.  doi: 10.1016/S0307-904X(03)00074-X.  Google Scholar [8] J. C. Ke, Bi-level control for batch arrival queues with an early startup and unreliable server,, Applied Mathematical Modelling, 28 (2004), 469.   Google Scholar [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.  doi: 10.2307/3215040.  Google Scholar [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.   Google Scholar [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.  doi: 10.1007/s11424-012-9313-3.  Google Scholar [12] T. Meisling, Discrete time queue theory,, Operations Research, 6 (1958), 96.  doi: 10.1287/opre.6.1.96.  Google Scholar [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.  doi: 10.1016/j.mcm.2006.04.008.  Google Scholar [14] H. Takagi, Queueing Analysis, A Foundation of Performance Evaluation, Vol. 3: Discrete-Time Systems,, North-Holland, (1993).   Google Scholar [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.   Google Scholar [16] N. Tian, X. Xu and Z. Ma, Dicrete-Time Queueing Theory,, Science Press, (2008).   Google Scholar [17] N. Tian and Z. G. Zhang, Vacation Queueing Models: Theory and Applications,, International Series in Operations Research & Management Science, (2006).   Google Scholar [18] N. Tian and Z. G. Zhang, Discrete time Geo/G/1 queue with multiple adaptive vacations,, Queueing System, 38 (2001), 419.  doi: 10.1023/A:1010947911863.  Google Scholar [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.  doi: 10.1016/j.apm.2010.11.021.  Google Scholar [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.  doi: 10.1016/j.mcm.2012.12.032.  Google Scholar [21] M. E. Woodward, Communication and Computer Networks: Modelling with Discrete-Time Queues,, IEEE Computer Society Press, (1994).   Google Scholar [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.  doi: 10.1016/j.cam.2010.10.013.  Google Scholar [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.  doi: 10.1007/s11424-013-1121-x.  Google Scholar [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.  doi: 10.3934/jimo.2016.12.653.  Google Scholar

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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.  doi: 10.1016/0377-2217(94)00206-1.  Google Scholar [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.  Google Scholar [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, 6148 (2010), 29.  doi: 10.1007/978-3-642-13568-2_3.  Google Scholar [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.   Google Scholar [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.  doi: 10.3934/jimo.2015.11.779.  Google Scholar [6] J. J. Hunter, Mathematical Techniques of Applied Probability, Vol. 2, Discrete Time Models: Techniques and Applications,, Academic Press, (1983).   Google Scholar [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.  doi: 10.1016/S0307-904X(03)00074-X.  Google Scholar [8] J. C. Ke, Bi-level control for batch arrival queues with an early startup and unreliable server,, Applied Mathematical Modelling, 28 (2004), 469.   Google Scholar [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.  doi: 10.2307/3215040.  Google Scholar [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.   Google Scholar [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.  doi: 10.1007/s11424-012-9313-3.  Google Scholar [12] T. Meisling, Discrete time queue theory,, Operations Research, 6 (1958), 96.  doi: 10.1287/opre.6.1.96.  Google Scholar [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.  doi: 10.1016/j.mcm.2006.04.008.  Google Scholar [14] H. Takagi, Queueing Analysis, A Foundation of Performance Evaluation, Vol. 3: Discrete-Time Systems,, North-Holland, (1993).   Google Scholar [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.   Google Scholar [16] N. Tian, X. Xu and Z. Ma, Dicrete-Time Queueing Theory,, Science Press, (2008).   Google Scholar [17] N. Tian and Z. G. Zhang, Vacation Queueing Models: Theory and Applications,, International Series in Operations Research & Management Science, (2006).   Google Scholar [18] N. Tian and Z. G. Zhang, Discrete time Geo/G/1 queue with multiple adaptive vacations,, Queueing System, 38 (2001), 419.  doi: 10.1023/A:1010947911863.  Google Scholar [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.  doi: 10.1016/j.apm.2010.11.021.  Google Scholar [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.  doi: 10.1016/j.mcm.2012.12.032.  Google Scholar [21] M. E. Woodward, Communication and Computer Networks: Modelling with Discrete-Time Queues,, IEEE Computer Society Press, (1994).   Google Scholar [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.  doi: 10.1016/j.cam.2010.10.013.  Google Scholar [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.  doi: 10.1007/s11424-013-1121-x.  Google Scholar [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.  doi: 10.3934/jimo.2016.12.653.  Google Scholar
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