-
Previous Article
Performance analysis of a P2P storage system with a lazy replica repair policy
- JIMO Home
- This Issue
-
Next Article
Performance analysis of large-scale parallel-distributed processing with backup tasks for cloud computing
The impact of the $NT$-policy on the behaviour of a discrete-time queue with general service times
1. | SMACS Research Group, TELIN Department, Ghent University, Sint-Pietersnieuwstraat 41, B-9000 Gent, Belgium, Belgium, Belgium |
2. | Supply Networks and Logistics Research Center, Department of Industrial Management, Ghent University, Technologiepark 903, B-9052 Zwijnaarde, Belgium |
We assume a Bernoulli arrival process of customers and independent and identically distributed service times. Using a probability generating functions approach, we obtain expressions for the steady-state distributions of the phase sojourn times, the cycle length, the system content and the customer delay. The influence of the threshold parameters $N$ and $T$ on the mean sojourn times and the expected delay is discussed by means of numerical examples.
References:
[1] |
A. S. Alfa and W. Li, Optimal ($N$,$T$)-policy for M/G/1 system with cost structures, Performance Evaluation, 42 (2000), 265-277.
doi: 10.1016/S0166-5316(00)00015-8. |
[2] |
W. Böhm and S. G. Mohanty, On discrete-time Markovian $N$-policy queues involving batches, Sankhya: The Indian Journal of Statistics, Series A, 56 (1994), 144-163. |
[3] |
O. J. Boxma and W. P. Groenendijk, Waiting times in discrete-time cyclic-service systems, IEEE Transactions on Communications, 36 (1988), 164-170.
doi: 10.1109/26.2746. |
[4] |
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. |
[5] |
B. Feyaerts, S. De Vuyst, S. Wittevrongel and H. Bruneel, Analysis of a discrete-time queueing system with an $NT$-policy, ASMTA '10: 17th International Conference on Analytical and Stochastic Modeling Techniques and Applications, Cardiff, United Kingdom, 2010, Lecture Notes in Computer Science, 6148 (2010), 29-43.
doi: 10.1007/978-3-642-13568-2_3. |
[6] |
P. Flajolet and R. Sedgewick, "Analytic Combinatorics," Cambridge University Press, Cambridge, 2009.
doi: 10.1017/CBO9780511801655. |
[7] |
A. G. Hernández-Díaz and P. Moreno, Analysis and optimal control of a discrete-time queueing system under the $(m,N)$-policy, Valuetools '06: Proceedings of the 1st International Conference on Performance Evaluation Methodologies and Tools, Pisa, Italy, 2006. |
[8] |
D. P. Heyman, The T-policy for the M/G/1 queue, Management Science, 23 (1977), 775-778.
doi: 10.1287/mnsc.23.7.775. |
[9] |
J.-C. Ke, Optimal $NT$ policies for M/G/1 system with a startup and unreliable server, Computers & Industrial Engineering, 50 (2006), 248-262.
doi: 10.1016/j.cie.2006.04.004. |
[10] |
J.-C. Ke, H.-I Huang and Y.-K. Chu, Batch arrival queue with $N$-policy and at most $J$ vacations, Applied Mathematical Modelling, 34 (2010), 451-466.
doi: 10.1016/j.apm.2009.06.003. |
[11] |
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. |
[12] |
S. S. Lee, H. W. Lee and K. C. Chae, Batch arrival queue with $N$-policy and single vacation, Computers & Operations Research, 22 (1995), 173-189. |
[13] |
P. Moreno, A discrete-time single-server queue with a modified $N$-policy, International Journal of Systems Science, 38 (2007), 483-492.
doi: 10.1080/00207720701353405. |
[14] |
H. Takagi, "Queueing Analysis, A Foundation of Performance Evaluation, Volume 3: Discrete-Time Systems," North-Holland, Amsterdam, The Netherlands, 1993. |
[15] |
K.-H. Wang, T.-Y. Wang and W. L. Pearn, Optimal control of the $N$-policy M/G/1 queueing system with server breakdowns and general startup times, Applied Mathematical Modelling, 31 (2007), 2199-2212.
doi: 10.1016/j.apm.2006.08.016. |
[16] |
T.-Y. Wang, K.-H. Wang and W. L. Pearn, Optimization of the $T$ policy M/G/1 queue with server breakdowns and general startup times, Journal of Computational and Applied Mathematics, 228 (2009), 270-278.
doi: 10.1016/j.cam.2008.09.021. |
[17] |
M. Yadin and P. Naor, Queueing systems with a removable service station, Operational Research Quarterly, 14 (1963), 393-405. |
show all references
References:
[1] |
A. S. Alfa and W. Li, Optimal ($N$,$T$)-policy for M/G/1 system with cost structures, Performance Evaluation, 42 (2000), 265-277.
doi: 10.1016/S0166-5316(00)00015-8. |
[2] |
W. Böhm and S. G. Mohanty, On discrete-time Markovian $N$-policy queues involving batches, Sankhya: The Indian Journal of Statistics, Series A, 56 (1994), 144-163. |
[3] |
O. J. Boxma and W. P. Groenendijk, Waiting times in discrete-time cyclic-service systems, IEEE Transactions on Communications, 36 (1988), 164-170.
doi: 10.1109/26.2746. |
[4] |
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. |
[5] |
B. Feyaerts, S. De Vuyst, S. Wittevrongel and H. Bruneel, Analysis of a discrete-time queueing system with an $NT$-policy, ASMTA '10: 17th International Conference on Analytical and Stochastic Modeling Techniques and Applications, Cardiff, United Kingdom, 2010, Lecture Notes in Computer Science, 6148 (2010), 29-43.
doi: 10.1007/978-3-642-13568-2_3. |
[6] |
P. Flajolet and R. Sedgewick, "Analytic Combinatorics," Cambridge University Press, Cambridge, 2009.
doi: 10.1017/CBO9780511801655. |
[7] |
A. G. Hernández-Díaz and P. Moreno, Analysis and optimal control of a discrete-time queueing system under the $(m,N)$-policy, Valuetools '06: Proceedings of the 1st International Conference on Performance Evaluation Methodologies and Tools, Pisa, Italy, 2006. |
[8] |
D. P. Heyman, The T-policy for the M/G/1 queue, Management Science, 23 (1977), 775-778.
doi: 10.1287/mnsc.23.7.775. |
[9] |
J.-C. Ke, Optimal $NT$ policies for M/G/1 system with a startup and unreliable server, Computers & Industrial Engineering, 50 (2006), 248-262.
doi: 10.1016/j.cie.2006.04.004. |
[10] |
J.-C. Ke, H.-I Huang and Y.-K. Chu, Batch arrival queue with $N$-policy and at most $J$ vacations, Applied Mathematical Modelling, 34 (2010), 451-466.
doi: 10.1016/j.apm.2009.06.003. |
[11] |
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. |
[12] |
S. S. Lee, H. W. Lee and K. C. Chae, Batch arrival queue with $N$-policy and single vacation, Computers & Operations Research, 22 (1995), 173-189. |
[13] |
P. Moreno, A discrete-time single-server queue with a modified $N$-policy, International Journal of Systems Science, 38 (2007), 483-492.
doi: 10.1080/00207720701353405. |
[14] |
H. Takagi, "Queueing Analysis, A Foundation of Performance Evaluation, Volume 3: Discrete-Time Systems," North-Holland, Amsterdam, The Netherlands, 1993. |
[15] |
K.-H. Wang, T.-Y. Wang and W. L. Pearn, Optimal control of the $N$-policy M/G/1 queueing system with server breakdowns and general startup times, Applied Mathematical Modelling, 31 (2007), 2199-2212.
doi: 10.1016/j.apm.2006.08.016. |
[16] |
T.-Y. Wang, K.-H. Wang and W. L. Pearn, Optimization of the $T$ policy M/G/1 queue with server breakdowns and general startup times, Journal of Computational and Applied Mathematics, 228 (2009), 270-278.
doi: 10.1016/j.cam.2008.09.021. |
[17] |
M. Yadin and P. Naor, Queueing systems with a removable service station, Operational Research Quarterly, 14 (1963), 393-405. |
[1] |
Zhanyou Ma, Pengcheng Wang, Wuyi Yue. Performance analysis and optimization of a pseudo-fault Geo/Geo/1 repairable queueing system with N-policy, setup time and multiple working vacations. Journal of Industrial and Management Optimization, 2017, 13 (3) : 1467-1481. doi: 10.3934/jimo.2017002 |
[2] |
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 |
[3] |
Sin-Man Choi, Ximin Huang, Wai-Ki Ching. Minimizing equilibrium expected sojourn time via performance-based mixed threshold demand allocation in a multiple-server queueing environment. Journal of Industrial and Management Optimization, 2012, 8 (2) : 299-323. doi: 10.3934/jimo.2012.8.299 |
[4] |
Dequan Yue, Wuyi Yue, Zsolt Saffer, Xiaohong Chen. Analysis of an M/M/1 queueing system with impatient customers and a variant of multiple vacation policy. Journal of Industrial and Management Optimization, 2014, 10 (1) : 89-112. doi: 10.3934/jimo.2014.10.89 |
[5] |
Shunfu Jin, Yuan Zhao, Wuyi Yue, Lingling Chen. Performance analysis of a P2P storage system with a lazy replica repair policy. Journal of Industrial and Management Optimization, 2014, 10 (1) : 151-166. doi: 10.3934/jimo.2014.10.151 |
[6] |
Pikkala Vijaya Laxmi, Seleshi Demie. Performance analysis of renewal input $(a,c,b)$ policy queue with multiple working vacations and change over times. Journal of Industrial and Management Optimization, 2014, 10 (3) : 839-857. doi: 10.3934/jimo.2014.10.839 |
[7] |
Rakesh Nandi, Sujit Kumar Samanta, Chesoong Kim. Analysis of $ D $-$ BMAP/G/1 $ queueing system under $ N $-policy and its cost optimization. Journal of Industrial and Management Optimization, 2021, 17 (6) : 3603-3631. doi: 10.3934/jimo.2020135 |
[8] |
Gang Chen, Zaiming Liu, Jinbiao Wu. Optimal threshold control of a retrial queueing system with finite buffer. Journal of Industrial and Management Optimization, 2017, 13 (3) : 1537-1552. doi: 10.3934/jimo.2017006 |
[9] |
Alexander O. Brown, Christopher S. Tang. The impact of alternative performance measures on single-period inventory policy. Journal of Industrial and Management Optimization, 2006, 2 (3) : 297-318. doi: 10.3934/jimo.2006.2.297 |
[10] |
Gang Chen, Zaiming Liu, Jingchuan Zhang. Analysis of strategic customer behavior in fuzzy queueing systems. Journal of Industrial and Management Optimization, 2020, 16 (1) : 371-386. doi: 10.3934/jimo.2018157 |
[11] |
Jianyu Cao, Weixin Xie. Optimization of a condition-based duration-varying preventive maintenance policy for the stockless production system based on queueing model. Journal of Industrial and Management Optimization, 2019, 15 (3) : 1049-1083. doi: 10.3934/jimo.2018085 |
[12] |
Ruiling Tian, Dequan Yue, Wuyi Yue. Optimal balking strategies in an M/G/1 queueing system with a removable server under N-policy. Journal of Industrial and Management Optimization, 2015, 11 (3) : 715-731. doi: 10.3934/jimo.2015.11.715 |
[13] |
Lotfi Tadj, Zhe George Zhang, Chakib Tadj. A queueing analysis of multi-purpose production facility's operations. Journal of Industrial and Management Optimization, 2011, 7 (1) : 19-30. doi: 10.3934/jimo.2011.7.19 |
[14] |
Yoshiaki Kawase, Shoji Kasahara. Priority queueing analysis of transaction-confirmation time for Bitcoin. Journal of Industrial and Management Optimization, 2020, 16 (3) : 1077-1098. doi: 10.3934/jimo.2018193 |
[15] |
Zhanqiang Huo, Wuyi Yue, Naishuo Tian, Shunfu Jin. Performance evaluation for the sleep mode in the IEEE 802.16e based on a queueing model with close-down time and multiple vacations. Journal of Industrial and Management Optimization, 2009, 5 (3) : 511-524. doi: 10.3934/jimo.2009.5.511 |
[16] |
Wai-Ki Ching, Sin-Man Choi, Min Huang. Optimal service capacity in a multiple-server queueing system: A game theory approach. Journal of Industrial and Management Optimization, 2010, 6 (1) : 73-102. doi: 10.3934/jimo.2010.6.73 |
[17] |
Kyosuke Hashimoto, Hiroyuki Masuyama, Shoji Kasahara, Yutaka Takahashi. Performance analysis of backup-task scheduling with deadline time in cloud computing. Journal of Industrial and Management Optimization, 2015, 11 (3) : 867-886. doi: 10.3934/jimo.2015.11.867 |
[18] |
Tuan Phung-Duc, Wouter Rogiest, Sabine Wittevrongel. Single server retrial queues with speed scaling: Analysis and performance evaluation. Journal of Industrial and Management Optimization, 2017, 13 (4) : 1927-1943. doi: 10.3934/jimo.2017025 |
[19] |
Bart Feyaerts, Stijn De Vuyst, Herwig Bruneel, Sabine Wittevrongel. Performance analysis of buffers with train arrivals and correlated output interruptions. Journal of Industrial and Management Optimization, 2015, 11 (3) : 829-848. doi: 10.3934/jimo.2015.11.829 |
[20] |
Arnaud Devos, Joris Walraevens, Tuan Phung-Duc, Herwig Bruneel. Analysis of the queue lengths in a priority retrial queue with constant retrial policy. Journal of Industrial and Management Optimization, 2020, 16 (6) : 2813-2842. doi: 10.3934/jimo.2019082 |
2020 Impact Factor: 1.801
Tools
Metrics
Other articles
by authors
[Back to Top]